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PART 01LLM basics
Is an LLM the same thing as an AI agent?
How do I call an LLM from JavaScript?
Install your provider's official SDK, for example the openai package, and put your API key in an environment variable. Read the full answer in the LLM lesson
Does an LLM plan its whole answer before writing it?
Not the way a person does. Read the full answer in the Next-token prediction lesson
Why does next-token prediction matter when building agents?
Because the model only reacts to the text you give it. Read the full answer in the Next-token prediction lesson
What is the difference between an LLM integration, a workflow and an agent?
An integration calls the model for one job, like summarising a ticket. Read the full answer in the Who decides the next step lesson
How do I explain where my project sits on the AI scale in an interview?
Name the level and say who decides the next step. Read the full answer in the Who decides the next step lesson
How many tokens is a word?
In English, one token is about four characters, or roughly three quarters of a word. Read the full answer in the Token lesson
Why do tokens matter for cost?
Providers bill per token, for both the text you send and the text the model writes. Read the full answer in the Token lesson
What happens when a conversation is bigger than the context window?
The call fails, or the oldest text gets cut off, depending on the provider and your settings. Read the full answer in the Context window lesson
What counts toward the context window?
Everything in one call: the system prompt, every earlier user and assistant message, every tool call and tool result, and the answer the model writes back. Read the full answer in the Context window lesson
Does ChatGPT remember my conversation between API calls?
The API does not. Read the full answer in the Stateless lesson
How do agents have memory if the model is stateless?
Your code keeps the memory. Read the full answer in the Stateless lesson
What are the system, user, assistant and tool roles?
System holds the rules for the whole conversation. Read the full answer in the Message roles lesson
Why does an agent need the tool role?
The tool role is how results get back to the model. Read the full answer in the Message roles lesson
What should go in a system prompt for an agent?
Its job, its limits and its hints. Read the full answer in the System prompt lesson
Is the system prompt a security boundary?
What temperature should I use for an agent?
Keep it low, between 0 and 0.3. Read the full answer in the Temperature lesson
Does temperature 0 make an LLM fully deterministic?
Almost, but not always. Read the full answer in the Temperature lesson
Why do LLMs hallucinate?
The model predicts text that sounds right, not text it has checked. Read the full answer in the Hallucination lesson
How do I reduce hallucinations in an agent?
Give the model real data to read. Read the full answer in the Hallucination lesson
PART 02OpenAI API
How do I set up the OpenAI SDK in Node?
Run npm i openai, put your key in a .env file as OPENAI_API_KEY, and create a client with new OpenAI(). Read the full answer in the The OpenAI SDK lesson
Should I learn the raw OpenAI SDK before LangChain or the Vercel AI SDK?
What does a chat completion request send and return?
You send a model name and an ordered list of messages, plus optional settings like temperature or tools. Read the full answer in the Chat completion lesson
How is an agent related to a chat completion?
An agent is the same chat completion call, made again and again inside a loop. Read the full answer in the Chat completion lesson
Why do I have to send the whole messages array every time?
The model is stateless and sees nothing except the request in front of it. Read the full answer in the Messages array lesson
What does the messages array look like inside an agent?
It starts with the system prompt and the user's goal. Read the full answer in the Messages array lesson
How do I calculate the cost of an OpenAI API call?
Read the usage object on the response. Read the full answer in the Usage and cost lesson
Why are agents more expensive than single calls?
One task makes many model calls, and each call resends the growing history, including every earlier tool result. Read the full answer in the Usage and cost lesson
What is structured output in the OpenAI API?
It forces the model's reply to match a JSON schema you provide. Read the full answer in the Structured output lesson
When should I use structured output instead of plain text?
Whenever code reads the answer rather than a person. Read the full answer in the Structured output lesson
How do I decide between free text and JSON output?
Ask who reads it. Read the full answer in the Who reads the answer? lesson
Why do agents mostly use structured output?
Inside an agent, most outputs are not shown to anyone. Read the full answer in the Who reads the answer? lesson
How do I stream an OpenAI response in Node?
Pass stream: true to chat.completions.create, then loop over the result with for await. Read the full answer in the Streaming lesson
Why should agents stream their answers?
Agents can take many seconds, because they call tools and the model several times. Read the full answer in the Streaming lesson
Which OpenAI feature should I use for my task?
Match it to the job. Read the full answer in the Picking the right feature lesson
Do I need embeddings or tool calling to look up data?
They solve different problems. Read the full answer in the Picking the right feature lesson
What is the OpenAI Responses API?
It is a newer OpenAI API for building assistants and agents, alongside an Agents SDK for JavaScript. Read the full answer in the Responses API and Agents SDK lesson
Should I learn chat completions or the Responses API first?
Start with chat completions. Read the full answer in the Responses API and Agents SDK lesson
PART 03Tool calling
Does the model run my tools itself?
Why are tools what turn a chatbot into an agent?
Without tools a model can only talk about what it already knows. Read the full answer in the What a tool is lesson
How do I describe a function for OpenAI tool calling?
Give each tool a type of function, a name, a description, and a parameters object written as JSON schema: the property names, their types and which ones are required. Read the full answer in the Function schema lesson
Can I write tool schemas with Zod instead of raw JSON schema?
How do I write a good tool description?
Say what the tool does, when to use it and what comes back. Read the full answer in the Tool description lesson
Why does my agent pick the wrong tool?
Most often the descriptions are vague or overlap, so the model cannot tell the tools apart. Read the full answer in the Tool description lesson
How do I know the model wants to call a tool?
Check the reply message for a tool_calls array. Read the full answer in the The tool_calls reply lesson
Why is the message content null when the model calls a tool?
Because the reply is a request, not an answer. Read the full answer in the The tool_calls reply lesson
Why is function.arguments a string and not an object?
The model writes text, so the arguments arrive as a JSON string. Read the full answer in the Arguments are a string lesson
How should I validate tool arguments from the model?
Parse the JSON string, then check it with a Zod schema using safeParse. Read the full answer in the Arguments are a string lesson
How do I send a tool result back to the model?
First append the assistant message that contained the tool_calls. Read the full answer in the The tool message lesson
What should a tool result contain?
Only what the model needs for its next step, as short JSON. Read the full answer in the The tool message lesson
Why does tool calling need two model calls?
In the first call the model decides it needs a tool and asks for it. Read the full answer in the Calling the model again lesson
How do I turn the tool calling round trip into an agent?
Put it in a loop. Read the full answer in the Calling the model again lesson
What does tool_choice do?
It controls tool use for one call. auto lets the model decide, none forbids tools, required makes it call at least one, and naming a specific function forces that function. Read the full answer in the tool_choice lesson
When should I force a specific tool?
When a step must always happen, for example classifying every request before routing it, or always running a search before answering. Read the full answer in the tool_choice lesson
Can the model call several tools at once?
What happens if I skip one of the parallel tool results?
The API expects a tool message for every tool call id in the assistant's message. Read the full answer in the Parallel tool calls lesson
What kinds of tools does an agent need?
Two kinds. Read the full answer in the Tools give data and actions lesson
Should write tools need human approval?
Usually, yes, for anything that is hard to undo or reaches customers: deleting data, refunds, payments and outgoing emails. Read the full answer in the Tools give data and actions lesson
PART 04Agents
What is an AI agent in simple terms?
An AI agent is a language model running in a loop with tools. Read the full answer in the Agent lesson
Do I need a framework to build an AI agent in JavaScript?
How does the agent loop work?
Call the model with the messages and the tools. Read the full answer in the The agent loop lesson
What does the agent loop look like in code?
A for loop that counts steps up to a maximum. Read the full answer in the The agent loop lesson
How does an AI agent know when to stop?
Two ways. Read the full answer in the Stop conditions lesson
What step limit should I set for an agent?
Start around 10 steps for most tasks and adjust it from real traces. Read the full answer in the Stop conditions lesson
When should I use a workflow instead of an agent?
Use a workflow when you can write the steps down in advance, like extract an invoice, check it, then save it. Read the full answer in the Agent vs workflow lesson
Are agents better than workflows?
Not better, different. Read the full answer in the Agent vs workflow lesson
How do I know if my task even needs an agent?
Try to draw it as a flowchart. Read the full answer in the Use the simplest thing that works lesson
Why do interviewers like the simplest thing that works answer?
It shows you have shipped real systems. Read the full answer in the Use the simplest thing that works lesson
What does it mean that the model plans the steps?
You give the agent a goal and tools, not a sequence. Read the full answer in the The model plans the steps lesson
How do I control an agent if the model chooses the steps?
You control the boundaries, not the sequence. Read the full answer in the The model plans the steps lesson
What is the ReAct pattern for AI agents?
ReAct stands for reason and act. Read the full answer in the ReAct lesson
Is ReAct the same as the agent loop I write by hand?
Yes, in practice. Read the full answer in the ReAct lesson
What is a router in an AI agent system?
A router is a first step that classifies the request and sends it to the right specialist: a docs agent, a tickets agent, or a simple prompt. Read the full answer in the Router lesson
Why use a router instead of one agent with every tool?
Models choose tools better from a short list. Read the full answer in the Router lesson
What is the plan-and-execute pattern?
The model first writes a plan, a list of steps, and then works through them one by one, possibly updating the plan as it learns more. Read the full answer in the Plan-and-execute lesson
When is plan-and-execute better than a simple agent loop?
For long tasks with many steps, where seeing the plan helps. Read the full answer in the Plan-and-execute lesson
When should I split one agent into multiple agents?
Only when one agent clearly struggles: its prompt carries too many tools or responsibilities and tool choice accuracy drops. Read the full answer in the Multi-agent lesson
What is a supervisor in a multi-agent system?
A supervisor is an agent or router that receives the request, decides which specialist agent should handle it, passes the work on, and combines the results. Read the full answer in the Multi-agent lesson
What is human-in-the-loop for AI agents?
The agent pauses before a risky action and waits for a person to approve it. Read the full answer in the Human-in-the-loop lesson
How do I add an approval step to an agent?
Mark write tools as needing approval. Read the full answer in the Human-in-the-loop lesson
PART 05Embeddings + RAG
What is an embedding in AI?
An embedding is a list of numbers, a vector often 384 or 1536 long, that captures the meaning of a piece of text. Read the full answer in the Embedding lesson
How do I create embeddings in JavaScript?
Call an embeddings endpoint, for example client.embeddings.create with model text-embedding-3-small and your text as input. Read the full answer in the Embedding lesson
What does cosine similarity measure?
How close two embeddings are in meaning, by comparing the angle between the vectors. Read the full answer in the Cosine similarity lesson
Do I need to calculate cosine similarity myself?
Usually not. Read the full answer in the Cosine similarity lesson
What is the difference between semantic search and keyword search?
Keyword search matches exact words, like a SQL LIKE query. Read the full answer in the Semantic search lesson
Why do agents need semantic search?
Users rarely phrase questions the way the docs are written. Read the full answer in the Semantic search lesson
Which vector database should a JavaScript developer use?
If you already run Postgres, start with the pgvector extension: one extension, one new column type, and your vectors live next to your normal data. Read the full answer in the Vector database lesson
What does a vector database actually do?
It stores embeddings with their text and finds the nearest ones to a query vector quickly, even across millions of rows. Read the full answer in the Vector database lesson
What is RAG in simple terms?
RAG, retrieval-augmented generation, means searching your own data first, pasting the best matches into the prompt, and then letting the model answer from them. Read the full answer in the RAG lesson
What problems does RAG solve?
Two at once. Read the full answer in the RAG lesson
How do I index documents for RAG?
Split each document into chunks, create an embedding for each chunk, and store the chunk text with its vector and some metadata in a vector database. Read the full answer in the Indexing (run once) lesson
How do I update a RAG system when my documents change?
Re-index the changed documents: delete their old chunks, split the new version, embed the chunks and store them. Read the full answer in the Indexing (run once) lesson
What happens when a user asks a RAG system a question?
The question is turned into an embedding, the vector database returns the closest chunks, and those chunks are added to the prompt with an instruction to answer only from them. Read the full answer in the Querying (every question) lesson
How many steps does a RAG query take?
Three. Read the full answer in the Querying (every question) lesson
How do I stop a RAG chatbot from making things up?
Tell it to answer only from the provided context, and to say it does not know when the answer is not there. Read the full answer in the Answer only from context lesson
Why should a RAG model be allowed to say I don't know?
Because a confident wrong answer is worse than no answer. Read the full answer in the Answer only from context lesson
What chunk size should I use for RAG?
Start with 300 to 800 tokens per chunk and about 10 percent overlap, splitting on headings and paragraphs, or on functions for code. Read the full answer in the Chunk size lesson
Why do bad chunks cause bad RAG answers?
Retrieval returns whole chunks. Read the full answer in the Chunk size lesson
How many chunks should RAG retrieve?
Start with 3 to 5 chunks per question. Read the full answer in the Top K lesson
Is retrieving more chunks always better?
What metadata should I store with each RAG chunk?
At least the source document, the section heading and a date. Read the full answer in the Metadata lesson
How does metadata help an agent cite sources?
Each retrieved chunk carries its file name and section, so the model can name them in the answer and your UI can link back to the original page. Read the full answer in the Metadata lesson
What is hybrid search in RAG?
Hybrid search combines keyword search with vector search and merges the results. Read the full answer in the Hybrid search lesson
When do I need hybrid search?
When users search for exact things: error codes like ERR_502, function names, ticket IDs, product SKUs or people's names. Read the full answer in the Hybrid search lesson
What is re-ranking in RAG?
Re-ranking is a second pass over search results. Read the full answer in the Re-ranking lesson
Is re-ranking worth the extra cost?
Often, yes, when answers suffer from irrelevant chunks. Read the full answer in the Re-ranking lesson
What is agentic RAG?
In agentic RAG, retrieval becomes a tool, such as searchDocs with a query argument. Read the full answer in the Agentic RAG lesson
When is agentic RAG better than classic RAG?
When questions need several lookups or a rephrased search, like comparing two policies or following up on what the first search found. Read the full answer in the Agentic RAG lesson
PART 06LangChain / LangGraph
What is LangChain.js used for?
LangChain.js is a TypeScript toolbox of ready-made parts for LLM apps: model wrappers, document loaders, text splitters, vector stores, retrievers and tools. Read the full answer in the LangChain.js lesson
Is LangChain worth using, or should I use the raw SDK?
For a single call or a simple chat, the raw SDK is enough and easier to read. Read the full answer in the LangChain.js lesson
How do I switch between OpenAI, Claude and Gemini in LangChain?
Use LangChain's chat model classes, such as ChatOpenAI or ChatAnthropic. Read the full answer in the Chat models lesson
Can an agent use different models for different steps?
Yes, and it often should. Read the full answer in the Chat models lesson
What are document loaders in LangChain?
Document loaders are ready-made readers that turn sources such as PDFs, web pages, GitHub repos or Notion pages into text documents with metadata. Read the full answer in the Document loaders lesson
Where do document loaders fit in a RAG pipeline?
At the very start. Read the full answer in the Document loaders lesson
How do LangChain text splitters work?
A splitter such as RecursiveCharacterTextSplitter cuts text into chunks of a chosen size with a chosen overlap. Read the full answer in the Text splitters lesson
What chunk size and overlap should I give a text splitter?
A common starting point is a chunk size around 1,000 characters with about 100 characters of overlap, which is roughly in line with 300 to 800 tokens. Read the full answer in the Text splitters lesson
What is a retriever in LangChain?
A retriever is an object with one job: given a query, return the most relevant chunks. Read the full answer in the Retrievers lesson
How do I let an agent search my documents with a retriever?
Wrap the retriever in a tool, for example searchDocs with a query string argument, and give that tool to the agent. Read the full answer in the Retrievers lesson
How do I define a tool with Zod in LangChain?
Use the tool function from @langchain/core/tools. Read the full answer in the tool() with Zod lesson
Why define agent tools with Zod schemas?
One schema does two jobs. Read the full answer in the tool() with Zod lesson
What is LangGraph.js?
LangGraph.js is a framework for building agents as a graph of steps that share saved state. Read the full answer in the LangGraph.js lesson
What is the difference between LangChain and LangGraph?
LangChain gives you building blocks: model wrappers, loaders, splitters, retrievers and tools. Read the full answer in the LangGraph.js lesson
What is state in LangGraph?
State is the data every node in the graph can read, most often the messages array plus a few fields of your own, such as the current route or a draft. Read the full answer in the Graph state lesson
Why does graph state matter for pausing and resuming agents?
Because the state is what gets saved. Read the full answer in the Graph state lesson
What are nodes and edges in LangGraph?
Nodes are functions that each do one step, such as calling the model or running tools. Read the full answer in the Nodes and edges lesson
Why model an agent as a graph?
A graph makes the control flow visible. Read the full answer in the Nodes and edges lesson
What is a conditional edge in LangGraph?
A conditional edge runs a function on the current state and returns the name of the next node. Read the full answer in the Conditional edges lesson
How do I build a router with conditional edges?
Make a router node that classifies the request, ideally with structured output, and writes the label into the state. Read the full answer in the Conditional edges lesson
What does createReactAgent do in LangGraph?
It builds a ready-made ReAct agent from a model and a list of tools. Read the full answer in the createReactAgent lesson
When should I move from createReactAgent to a custom graph?
When you need more than one loop: a router in front, approvals before write tools, extra fields in state, or several specialist agents. createReactAgent is like create-next-app, a fast start. Read the full answer in the createReactAgent lesson
What is checkpointing in LangGraph?
Checkpointing saves the graph's state after every step, in memory for development or in a database such as Postgres for production. Read the full answer in the Checkpointing lesson
How do LangGraph checkpoints give an agent memory?
Pass the same thread_id in the config on every call. Read the full answer in the Checkpointing lesson
What does interrupt do in LangGraph?
interrupt pauses the graph at a chosen point, usually just before a risky node such as sending an email or issuing a refund. Read the full answer in the interrupt() lesson
How do I add human approval to a LangGraph agent?
Call interrupt in the node before the risky action, or interrupt before that node when compiling the graph, and use a checkpointer. Read the full answer in the interrupt() lesson
When should I use the raw SDK, LangChain or LangGraph?
Use the raw SDK for one call or a simple chat. Read the full answer in the When to use which lesson
How do I answer the interview criticism that LangChain hides too much?
Agree that abstractions hide details, then say you built the raw loop first, so you know exactly what the framework does underneath. Read the full answer in the When to use which lesson
PART 07Hugging Face
What is the Hugging Face Hub?
The Hugging Face Hub is a public site with hundreds of thousands of open models and datasets, plus libraries to run them. Read the full answer in the The Hugging Face Hub lesson
When should I use a Hugging Face model instead of OpenAI?
When a small, focused model can do the job: embeddings, classification or sentiment at high volume, or work on private data that must not leave your servers. Read the full answer in the The Hugging Face Hub lesson
What is Transformers.js?
Transformers.js is a library, published as @huggingface/transformers, that runs Hugging Face models directly in Node or in the browser, with no Python and no separate model server. Read the full answer in the Transformers.js lesson
Can I run AI models in Node without Python?
What does the pipeline function do in Transformers.js?
pipeline takes a task name, such as feature-extraction or zero-shot-classification, and a model id, then loads the model and returns a ready-to-call function. Read the full answer in the pipeline() lesson
How do I use a Transformers.js pipeline as an agent tool?
Load the pipeline once when your server starts, not on every request, because loading downloads and prepares the model. Read the full answer in the pipeline() lesson
How do I create embeddings locally without an API key?
Use Transformers.js with pipeline feature-extraction and a small model such as Xenova/all-MiniLM-L6-v2. Read the full answer in the Local embeddings lesson
Are local embeddings good enough for RAG?
For many projects, yes. Read the full answer in the Local embeddings lesson
What is zero-shot classification?
Zero-shot classification labels text with categories you choose at the moment you call it, without training anything. Read the full answer in the Zero-shot classification lesson
Can a small classifier replace an LLM for routing?
Often, yes. Read the full answer in the Zero-shot classification lesson
What is the Hugging Face Inference API?
It lets you call models hosted by Hugging Face over HTTP with an access token, using the @huggingface/inference package. Read the full answer in the Inference API lesson
When should I use the Inference API instead of running a model locally?
When the model is too big for your server, especially larger language models that need a GPU, or when you are prototyping and do not want to manage hardware. Read the full answer in the Inference API lesson
How do I self-host an open-source LLM?
Use a serving tool such as Ollama for simple setups or Text Generation Inference for production, and run an open model like Llama or Mistral on your own machine or server. Read the full answer in the Self-hosted models lesson
Why would a company self-host its models?
Mainly data control: prompts and documents never leave the company's own infrastructure, which matters for client code, health or financial data. Read the full answer in the Self-hosted models lesson
What is a Hugging Face model card?
A model card is the README of a model. Read the full answer in the Model card lesson
What should I check on a model card before using it in production?
The license, to confirm commercial use is allowed. Read the full answer in the Model card lesson
What are open-weight models?
Open-weight models, such as Llama, Mistral and Qwen, let you download the trained model files and run them yourself. Read the full answer in the Open weights lesson
Why do open weights matter for agents?
They make private, self-hosted agents possible. Read the full answer in the Open weights lesson
What is fine-tuning an LLM?
Fine-tuning means training an existing model further on your own examples, so it changes its behaviour, tone or output format. Read the full answer in the Fine-tuning lesson
Should I use RAG or fine-tuning for company knowledge?
RAG, almost always. Read the full answer in the Fine-tuning lesson
What is model quantization?
Quantisation shrinks a model by storing its numbers with fewer bits, for example 4-bit instead of 16-bit. Read the full answer in the Quantization lesson
When should I use a quantized model?
When you want to run an open model on a laptop, a cheap server or limited GPU memory, especially for narrow tasks such as classification or simple extraction. Read the full answer in the Quantization lesson
Should I use OpenAI or Hugging Face models for my agent?
Usually both. Read the full answer in the OpenAI vs Hugging Face lesson
How do I mix OpenAI and open-source models in one agent?
Give each model the job it is best at. Read the full answer in the OpenAI vs Hugging Face lesson
PART 08n8n
What is n8n?
n8n is an open-source, visual workflow tool. Read the full answer in the n8n lesson
How does n8n fit next to an AI agent?
n8n handles the plumbing around the agent: what starts it and which apps it touches. Read the full answer in the n8n lesson
What is a trigger node in n8n?
A trigger node starts a workflow. Read the full answer in the Trigger node lesson
How can an agent run without anyone opening a chat?
Use a trigger. Read the full answer in the Trigger node lesson
What is an action node in n8n?
An action node does one thing in an app: send a Slack message, create a Jira issue, add a row to a database or update a spreadsheet. Read the full answer in the Action node lesson
How do n8n action nodes become an agent's tools?
Put an action node behind a Webhook trigger in its own small workflow. Read the full answer in the Action node lesson
What is $json in n8n?
$json is the data of the current item as it moves from one node to the next, similar to req.body passed down a chain of middleware. Read the full answer in the $json lesson
How do I use an agent's response in the next n8n node?
When an HTTP Request node calls your agent, the response body arrives as $json on the next node. Read the full answer in the $json lesson
What is the n8n Code node for?
The Code node runs your own JavaScript inside a workflow. Read the full answer in the Code node lesson
Should agent logic go in an n8n Code node?
Keep it small there. Read the full answer in the Code node lesson
How do I branch a workflow in n8n?
Use the IF node for a yes or no condition and the Switch node for several cases. Read the full answer in the IF and Switch lesson
How do I route an agent's result in n8n?
Have the agent return structured output, such as a risk field with low, medium or high. Read the full answer in the IF and Switch lesson
How do I call my own AI agent API from n8n?
Add an HTTP Request node, set the method and your agent's URL, for example POST /api/agent/review on your Next.js app, and send the data from earlier nodes as the JSON body. Read the full answer in the HTTP Request node lesson
Can n8n call any API?
How does n8n store API keys and logins?
n8n keeps API keys and OAuth logins as credentials, stored separately from workflows and encrypted. Read the full answer in the Credentials lesson
Why keep app authentication in n8n instead of the agent?
It keeps the agent's code simple and safer. Read the full answer in the Credentials lesson
What is the n8n AI Agent node?
It is a built-in node, built on LangChain, that runs an agent loop on the canvas: you choose a model, attach tools and memory, and it decides which tools to call. Read the full answer in the AI Agent node lesson
Should I build my agent in the n8n AI Agent node or in code?
Use the node for simple agents and prototypes. Read the full answer in the AI Agent node lesson
How do I debug a workflow in n8n?
Open the Executions list. Read the full answer in the Executions lesson
How do I tell whether n8n or the agent caused a wrong result?
Open the execution and check the data at the step that called the agent. Read the full answer in the Executions lesson
How do I trigger an AI agent from n8n?
Make a workflow with a trigger, such as a GitHub webhook or a schedule, followed by an HTTP Request node that posts the event data to your agent's API. Read the full answer in the n8n as the trigger lesson
Why use n8n to start agent runs?
n8n already speaks to hundreds of apps and handles webhooks, schedules and retries. Read the full answer in the n8n as the trigger lesson
How do I give my agent n8n workflows as tools?
Build each action, such as createJiraTicket or sendSlackMessage, as a small n8n workflow that starts with a Webhook node. Read the full answer in the n8n as the hands lesson
Is it safe to let an agent call n8n webhooks?
Yes, with the usual guardrails. Read the full answer in the n8n as the hands lesson
What should live in code and what in n8n?
Keep the thinking in code: the agent's reasoning, tools, prompts and tests, behind an API. Read the full answer in the Code for thinking, n8n for plumbing lesson
Why split agent reasoning from integrations?
Each part then uses the right tool. Read the full answer in the Code for thinking, n8n for plumbing lesson
PART 09Production
What are guardrails for AI agents?
Guardrails are rules and checks that stop an agent from doing damage: a maximum number of steps, validation of every tool argument, limited permissions for each tool, and human approval for risky actions. Read the full answer in the Guardrails lesson
How do I explain agent safety in an interview?
Name concrete guardrails. Read the full answer in the Guardrails lesson
How do I validate tool arguments in an agent?
Parse the arguments string and check it with a Zod schema using safeParse before running anything. Read the full answer in the Validate tool arguments lesson
Why treat the model like an untrusted client?
Because its arguments can be wrong, malformed, or shaped by text an attacker planted in a document or web page. Read the full answer in the Validate tool arguments lesson
What is least privilege for AI agents?
Give each tool and each agent only the access it needs, and never more than the user it acts for. Read the full answer in the Least privilege lesson
Should an agent use the same permissions as the user?
At most the same, and usually fewer. Read the full answer in the Least privilege lesson
What is prompt injection?
Prompt injection is when instructions hidden in content the agent reads try to take over its behaviour. Read the full answer in the Prompt injection lesson
How do I defend an agent against prompt injection?
Treat every tool result and document as data, never as instructions. Read the full answer in the Prompt injection lesson
Do agent tools need timeouts and retries?
Yes. Read the full answer in the Timeouts and retries lesson
What should a tool return when it fails?
A short, clear error as the tool result, for example the order service timed out, try again or tell the user it is unavailable. Read the full answer in the Timeouts and retries lesson
What are the types of memory in an AI agent?
Three. Read the full answer in the Short-term, long-term and working memory lesson
How do I answer how do you handle memory in an interview?
Name all three kinds and how you store each. Read the full answer in the Short-term, long-term and working memory lesson
What is tracing for AI agents?
Tracing records every step of an agent run: the prompt, each tool call with its arguments and result, tokens used, time taken and cost. Read the full answer in the Tracing lesson
How does tracing help when the agent gives a wrong answer?
The trace shows exactly where it went wrong. Read the full answer in the Tracing lesson
How do I test an AI agent?
Build an eval set: 20 to 50 real questions with expected results, such as which tool should be called or what the answer must contain. Read the full answer in the Evals lesson
What is the difference between evals and unit tests?
Unit tests check exact behaviour and should all pass. Read the full answer in the Evals lesson
What is LLM-as-judge?
LLM-as-judge means using a second model to grade an answer, for example scoring it from 1 to 5 against a reference answer or a rubric. Read the full answer in the LLM-as-judge lesson
Can I trust an LLM judge's scores?
How do I reduce the cost of an AI agent?
Route by difficulty. Read the full answer in the Small models for small jobs lesson
Will a smaller model make my agent worse?
Only if it does a job that needs more reasoning. Read the full answer in the Small models for small jobs lesson
How do I make an AI agent faster?
Run independent tool calls in parallel, keep tool results short so each model call is smaller, cache repeated answers and embeddings, use small models for simple steps, and stream the final answer so users see progress immediately. Read the full answer in the Cost and latency tricks lesson
Why should tool results be short?
Every token in a tool result stays in the messages and is paid for again on every later loop step. Read the full answer in the Cost and latency tricks lesson
What is MCP, the Model Context Protocol?
MCP is an open standard for connecting AI apps to tools and data. Read the full answer in the MCP lesson
How do I expose my agent's tools over MCP in TypeScript?
Use the official @modelcontextprotocol/sdk package to create an MCP server and register each tool with a name, description and input schema, plus the function that runs it. Read the full answer in the MCP lesson
What does a production AI agent architecture look like?
In the DevPilot example, a pull request opens and n8n calls the DevPilot API. Read the full answer in the DevPilot, end to end lesson
How do I walk an interviewer through an agent I built?
Use four parts: the problem, the architecture, one hard problem you solved, and the result in numbers. Read the full answer in the DevPilot, end to end lesson
Biology (school)
PART 01Cells
Who discovered cells?
Robert Hooke first described cells in 1665 when he looked at thin slices of cork through an early microscope. Read the full answer in the Cell lesson
Are there living things with only one cell?
Do all cells have a nucleus?
What is inside the nucleus?
The nucleus contains the cell's DNA, which is packed into structures called chromosomes. Read the full answer in the Nucleus lesson
What is the cytoplasm made of?
Cytoplasm is mostly water with many substances dissolved in it, such as sugars, salts and proteins. Read the full answer in the Cytoplasm lesson
What is the difference between cytoplasm and cytosol?
Cytosol is the liquid part only. Read the full answer in the Cytoplasm lesson
Why is the cell membrane called partially permeable?
Partially permeable means it lets some substances through but blocks others. Read the full answer in the Cell membrane lesson
What is the difference between a cell membrane and a cell wall?
The membrane is a thin, flexible skin found in every cell. Read the full answer in the Cell membrane lesson
Why are mitochondria called the powerhouse of the cell?
They are where aerobic respiration takes place. Read the full answer in the Mitochondria lesson
Do plant cells have mitochondria?
Where are ribosomes found in a cell?
Ribosomes float in the cytoplasm or sit on a network of membranes inside the cell. Read the full answer in the Ribosome lesson
How do ribosomes make proteins?
The nucleus sends out a copy of a gene's instructions. Read the full answer in the Ribosome lesson
What is the cell wall made of?
In plant cells the wall is made mainly of cellulose, a tough material built from long chains of glucose. Read the full answer in the Cell wall lesson
Can water pass through a plant cell wall?
Why are chloroplasts green?
They contain a green pigment called chlorophyll. Read the full answer in the Chloroplast lesson
Why do root cells not have chloroplasts?
Roots grow underground where there is no light, so photosynthesis cannot happen there. Read the full answer in the Chloroplast lesson
What does the vacuole do in a plant cell?
The vacuole is filled with cell sap, a watery mix of sugars and salts. Read the full answer in the Vacuole lesson
Do animal cells have vacuoles?
Animal cells can have very small, temporary vacuoles for storing or moving things. Read the full answer in the Vacuole lesson
What are some examples of specialised cells?
A red blood cell is shaped like a disc with no nucleus to carry more oxygen. Read the full answer in the Specialised cell lesson
How does a cell become specialised?
This process is called differentiation. Read the full answer in the Specialised cell lesson
How do you calculate magnification?
Use the formula magnification equals image size divided by real size. Read the full answer in the Magnification lesson
What is the difference between magnification and resolution?
Magnification is how much bigger the image looks. Read the full answer in the Magnification lesson
Where do stem cells come from?
Embryonic stem cells come from very early embryos and can become any type of cell. Read the full answer in the Stem cell lesson
Why are stem cells useful in medicine?
Because stem cells can turn into many cell types, doctors hope to use them to replace damaged cells, for example in diseases of the nerves or blood. Read the full answer in the Stem cell lesson
PART 02Moving materials, enzymes and energy
What affects the rate of diffusion?
Three things matter most. Read the full answer in the Diffusion lesson
What is a concentration gradient?
A concentration gradient is the difference in how many particles there are in two places. Read the full answer in the Diffusion lesson
What is the difference between diffusion and osmosis?
Diffusion is the movement of any particles from high to low concentration. Read the full answer in the Osmosis lesson
Why do plant cells not burst in pure water?
When a plant cell takes in water, it swells and presses against its stiff cell wall. Read the full answer in the Osmosis lesson
Why does active transport need energy?
Particles are moved against the concentration gradient, which is like pushing something uphill. Read the full answer in the Active transport lesson
Where does active transport happen in the body?
In the small intestine, active transport takes glucose from the gut into the blood when there is more glucose in the blood than in the gut. Read the full answer in the Active transport lesson
What is the active site of an enzyme?
The active site is a small dent in the enzyme's surface with a precise shape. Read the full answer in the Enzyme lesson
Why are enzymes called biological catalysts?
A catalyst speeds up a reaction without being used up. Read the full answer in the Enzyme lesson
Can a denatured enzyme work again?
Usually not. Read the full answer in the Denaturing lesson
Why do different enzymes work at different pH values?
Each enzyme's shape is held together by chemical bonds that depend on pH. Read the full answer in the Denaturing lesson
Where does photosynthesis take place?
It takes place in chloroplasts, which are most plentiful in the palisade cells near the top of a leaf. Read the full answer in the Photosynthesis lesson
What does a plant do with the glucose it makes?
The plant uses glucose for respiration to release energy. Read the full answer in the Photosynthesis lesson
How do you read a limiting factor graph?
On the rising part of the graph, the factor on the x-axis is limiting, because increasing it raises the rate. Read the full answer in the Limiting factor lesson
Why do farmers use greenhouses?
Greenhouses let farmers control the limiting factors of photosynthesis. Read the full answer in the Limiting factor lesson
Is respiration the same as breathing?
What is the energy from respiration used for?
Cells use it to build large molecules from smaller ones, such as proteins. Read the full answer in the Aerobic respiration lesson
Why does anaerobic respiration release less energy?
The glucose is only partly broken down. Read the full answer in the Anaerobic respiration lesson
What is anaerobic respiration in yeast?
Yeast breaks down glucose without oxygen into ethanol and carbon dioxide. Read the full answer in the Anaerobic respiration lesson
How do guard cells open and close stomata?
When guard cells take in water by osmosis, they swell and bend apart, opening the stoma. Read the full answer in the Stomata lesson
Why are stomata mostly on the underside of leaves?
The underside is shaded and cooler, so less water evaporates through the openings. Read the full answer in the Stomata lesson
What is the transpiration stream?
It is the steady flow of water from roots up through xylem vessels to the leaves, where it evaporates through the stomata. Read the full answer in the Transpiration lesson
What conditions increase the rate of transpiration?
Transpiration is faster in higher temperatures, because water evaporates quicker. Read the full answer in the Transpiration lesson
PART 03The human body
What is the difference between a tissue and an organ?
A tissue is a group of similar cells that do the same job, like muscle tissue. Read the full answer in the Organ system lesson
How many organ systems does the human body have?
Textbooks usually list around ten to twelve, including the digestive, circulatory, respiratory, nervous, endocrine, skeletal, muscular, excretory, immune and reproductive systems. Read the full answer in the Organ system lesson
What does bile do in digestion?
Bile is made in the liver and stored in the gall bladder. Read the full answer in the Digestive system lesson
Which enzymes digest which food?
Carbohydrases such as amylase break starch into sugars. Read the full answer in the Digestive system lesson
Why are villi walls only one cell thick?
A thin wall means the digested food has a very short distance to travel to reach the blood. Read the full answer in the Villi lesson
What happens to food that is not absorbed?
Undigested material, mostly fibre, passes into the large intestine. Read the full answer in the Villi lesson
Why does exercise make you breathe faster?
Working muscles respire faster, so they use more oxygen and make more carbon dioxide. Read the full answer in the Alveoli lesson
How does smoking damage the alveoli?
Smoke contains tar and chemicals that irritate the lungs. Read the full answer in the Alveoli lesson
Why does the left side of the heart have a thicker wall?
The left ventricle must pump blood all the way around the body at high pressure, while the right ventricle only pumps blood a short distance to the lungs. Read the full answer in the Double circulation lesson
What are arteries, veins and capillaries?
Arteries carry blood away from the heart and have thick, elastic walls to cope with high pressure. Read the full answer in the Double circulation lesson
What do white blood cells do?
White blood cells defend the body against disease. Read the full answer in the Blood lesson
Why do red blood cells have no nucleus?
Without a nucleus, there is more room inside the cell for haemoglobin, the red pigment that carries oxygen. Read the full answer in the Blood lesson
What causes coronary heart disease?
A build-up of fatty material in the coronary arteries narrows them. Read the full answer in the Coronary heart disease lesson
How is coronary heart disease treated?
Stents are small metal tubes that hold arteries open. Read the full answer in the Coronary heart disease lesson
Why are reflexes faster than normal actions?
In a reflex, the signal travels from the sensory neurone to the spinal cord and straight out via a motor neurone, without going through the brain's decision-making. Read the full answer in the Reflex arc lesson
What is a synapse?
A synapse is the tiny gap between two nerve cells. Read the full answer in the Reflex arc lesson
How are hormones different from nerve signals?
Nerve signals are electrical impulses that act very fast and for a short time on a precise place. Read the full answer in the Hormone lesson
What is the difference between type 1 and type 2 diabetes?
In type 1 diabetes, the pancreas stops making enough insulin, so patients inject insulin. Read the full answer in the Hormone lesson
Why is it important to control body temperature?
Enzymes in the body work best at about 37 degrees Celsius. Read the full answer in the Homeostasis lesson
What other things does the body keep steady?
Besides temperature, the body controls blood glucose, water content and ion levels. Read the full answer in the Homeostasis lesson
Why is it called negative feedback?
The response goes in the opposite direction to the change. Read the full answer in the Negative feedback lesson
What is an example of negative feedback in temperature control?
If blood temperature rises above normal, the brain triggers sweating and wider skin vessels, which cool the body. Read the full answer in the Negative feedback lesson
PART 04Health and disease
What is the difference between bacteria and viruses?
Bacteria are living single cells that can reproduce by themselves and are killed by antibiotics. Read the full answer in the Pathogen lesson
Are all bacteria harmful?
How do communicable diseases spread?
There are several routes. Read the full answer in the Communicable disease lesson
How can you stop a communicable disease spreading?
Good hygiene such as handwashing, covering coughs and cleaning surfaces helps. Read the full answer in the Communicable disease lesson
Why is malaria hard to control?
Malaria is spread by mosquitoes, which breed in standing water and are hard to remove completely. Read the full answer in the Vector lesson
Is a vector the same as a pathogen?
What are the three ways white blood cells attack pathogens?
Phagocytes engulf pathogens and digest them. Read the full answer in the Immune system lesson
Why do you get a fever when you are ill?
A higher body temperature makes it harder for many pathogens to multiply and helps white blood cells work faster. Read the full answer in the Immune system lesson
Why does each antibody only work on one pathogen?
An antibody has a specific shape that fits only one antigen, in the same way a key fits one lock. Read the full answer in the Antibody lesson
What are memory cells?
After an infection, some white blood cells stay in the body as memory cells. Read the full answer in the Antibody lesson
What is in a vaccine?
A vaccine contains a safe form of the pathogen, such as a dead or weakened version, or just a piece of it. Read the full answer in the Vaccination lesson
Why do some vaccines need boosters?
For some diseases, one dose does not give long-lasting immunity, so extra doses remind the immune system. Read the full answer in the Vaccination lesson
What percentage of people need to be vaccinated for herd immunity?
It depends on how easily the disease spreads. Read the full answer in the Herd immunity lesson
Who is protected by herd immunity?
Mostly people who cannot be vaccinated, such as newborn babies and people with weakened immune systems. Read the full answer in the Herd immunity lesson
How do antibiotics work?
Different antibiotics attack bacteria in different ways. Read the full answer in the Antibiotic lesson
Why must you finish a course of antibiotics?
Stopping early may leave some bacteria alive, especially the toughest ones. Read the full answer in the Antibiotic lesson
How does antibiotic resistance develop?
Bacteria reproduce quickly, and random mutations can make one resistant to a drug. Read the full answer in the Antibiotic resistance lesson
What can we do about antibiotic resistance?
Doctors should prescribe antibiotics only when necessary and not for viral infections. Read the full answer in the Antibiotic resistance lesson
What lifestyle factors raise the risk of non-communicable disease?
Smoking raises the risk of lung disease, heart disease and cancer. Read the full answer in the Non-communicable disease lesson
What is cancer?
Cancer is uncontrolled cell division. Read the full answer in the Non-communicable disease lesson
PART 05Genes and inheritance
What does DNA stand for?
DNA stands for deoxyribonucleic acid. Read the full answer in the DNA lesson
Where is DNA found in a cell?
In animal and plant cells, most DNA is inside the nucleus, packed into chromosomes. Read the full answer in the DNA lesson
How does a gene make a protein?
A copy of the gene is made and travels out of the nucleus to a ribosome. Read the full answer in the Gene lesson
How many genes do humans have?
Humans have roughly 20,000 genes that code for proteins. Read the full answer in the Gene lesson
Why do chromosomes come in pairs?
You inherit one set of 23 chromosomes from your mother's egg and one set of 23 from your father's sperm. Read the full answer in the Chromosome lesson
What is Down syndrome?
Down syndrome happens when a person has three copies of chromosome 21 instead of two. Read the full answer in the Chromosome lesson
Where do alleles come from?
New alleles arise by mutation, a change in the DNA sequence of a gene. Read the full answer in the Allele lesson
What is the difference between a gene and an allele?
A gene is the section of DNA that controls a feature, such as hair colour. Read the full answer in the Allele lesson
Does dominant mean more common?
No. Read the full answer in the Dominant and recessive alleles lesson
How do you write dominant and recessive alleles?
Use the same letter for both, with a capital letter for the dominant allele and a lower-case letter for the recessive one, such as T for tall and t for short. Read the full answer in the Dominant and recessive alleles lesson
What do homozygous and heterozygous mean?
Homozygous means having two identical alleles for a gene, such as TT or tt. Read the full answer in the Genotype and phenotype lesson
Does the environment affect phenotype?
Yes. Read the full answer in the Genotype and phenotype lesson
How do you fill in a Punnett square?
Draw a two by two grid. Read the full answer in the Punnett square lesson
Does a Punnett square predict exactly what will happen?
What are the stages of mitosis?
First the cell copies its DNA, so each chromosome has two identical strands. Read the full answer in the Mitosis lesson
Why is cell division in cancer a problem?
Normal cells divide only when the body needs them to. Read the full answer in the Mitosis lesson
Why do sex cells need half the chromosomes?
When an egg and sperm join at fertilisation, their chromosomes combine. Read the full answer in the Meiosis lesson
How does meiosis create variation?
During meiosis the chromosomes are shuffled, so each egg or sperm gets a random mix of the mother's and father's chromosomes. Read the full answer in the Meiosis lesson
Which parent decides the sex of a baby?
The father's sperm decides. Read the full answer in the Sex chromosomes lesson
Why are some conditions more common in males?
Some genes, like those for red-green colour blindness, are on the X chromosome. Read the full answer in the Sex chromosomes lesson
Can inherited disorders be screened for?
What is cystic fibrosis?
Cystic fibrosis is an inherited disorder caused by a recessive allele. Read the full answer in the Inherited disorder lesson
PART 06Variation and evolution
What is the difference between inherited and environmental variation?
Inherited variation comes from the genes you get from your parents, like eye colour or blood group. Read the full answer in the Variation lesson
Why is variation important for a species?
If the environment changes, such as a new disease or a hotter climate, some individuals may have features that help them survive. Read the full answer in the Variation lesson
Are all mutations bad?
What causes mutations?
Mutations happen by chance when DNA is copied, but the rate increases with ionising radiation such as X-rays, ultraviolet light and some chemicals. Read the full answer in the Mutation lesson
Who came up with the theory of natural selection?
Charles Darwin published it in 1859 in his book On the Origin of Species. Read the full answer in the Natural selection lesson
Does natural selection mean animals choose to change?
What are some examples of adaptation?
Polar bears have thick fur, a layer of fat and white colouring to stay warm and hidden. Read the full answer in the Adaptation lesson
What is an extremophile?
An extremophile is an organism that lives in extreme conditions, such as very hot springs, very salty water or deep ocean vents. Read the full answer in the Adaptation lesson
Is evolution just a theory?
In science, a theory is a well-tested explanation backed by a large amount of evidence, not just a guess. Read the full answer in the Evolution lesson
Did humans evolve from monkeys?
What is a species?
A species is a group of organisms that can breed together to produce fertile offspring. Read the full answer in the Speciation lesson
How does isolation lead to new species?
If a population is split by a mountain, sea or river, the two groups live in different conditions and face different selection pressures. Read the full answer in the Speciation lesson
How are fossils formed?
Most fossils form when hard parts such as bones or shells are buried in sediment and slowly replaced by minerals. Read the full answer in the Fossil lesson
Why is the fossil record incomplete?
Most dead organisms are eaten or rot away before they can fossilise. Read the full answer in the Fossil lesson
What causes species to go extinct?
Causes include changes to the environment such as climate change, a new predator or disease, competition from other species, habitat destruction and over-hunting. Read the full answer in the Extinction lesson
How were the dinosaurs wiped out?
Most scientists think a large asteroid hit Earth about 66 million years ago. Read the full answer in the Extinction lesson
What are examples of selective breeding?
Dog breeds were created from wolves by choosing for size, coat and behaviour. Read the full answer in the Selective breeding lesson
What is the difference between selective breeding and natural selection?
In natural selection the environment decides which individuals survive and breed. Read the full answer in the Selective breeding lesson
How does genetic engineering work?
Scientists cut out the useful gene using enzymes. Read the full answer in the Genetic engineering lesson
What are the pros and cons of GM crops?
Benefits include higher yields, resistance to pests or drought and added vitamins. Read the full answer in the Genetic engineering lesson
What are the levels of classification?
From largest to smallest the levels are kingdom, phylum, class, order, family, genus and species. Read the full answer in the Classification lesson
What is the binomial system?
Binomial means two names. Read the full answer in the Classification lesson
PART 07Ecosystems and food chains
What is the difference between habitat and ecosystem?
A habitat is the place where one type of organism lives, such as a rock pool for a crab. Read the full answer in the Ecosystem lesson
What is a community?
A community is all the populations of different species living in the same area. Read the full answer in the Ecosystem lesson
What is a microhabitat?
A microhabitat is a small area within a larger habitat with its own conditions. Read the full answer in the Habitat lesson
How do scientists estimate population size?
They sample the area with a quadrat, a square frame, in several random places. Read the full answer in the Habitat lesson
What are examples of biotic factors?
Biotic factors include food availability, new predators, new diseases and competition between species. Read the full answer in the Biotic and abiotic factors lesson
What abiotic factors affect plants?
Light intensity, temperature, moisture, soil pH, soil minerals, wind speed and carbon dioxide level all affect plants. Read the full answer in the Biotic and abiotic factors lesson
What is interdependence?
Interdependence means that organisms in a community depend on each other for food, shelter, pollination or seed dispersal. Read the full answer in the Competition lesson
What is a predator-prey cycle?
When prey numbers rise, predators have more food, and their numbers rise too. Read the full answer in the Competition lesson
Why are producers called the base of the food chain?
Only producers turn light energy into chemical energy in food. Read the full answer in the Producer lesson
Are there producers that don't use sunlight?
What are herbivores, carnivores and omnivores?
Herbivores eat only plants, like cows. Read the full answer in the Consumer lesson
What does an arrow in a food chain mean?
The arrow shows the direction in which energy and biomass flow, so it points from the food to the animal that eats it. Read the full answer in the Consumer lesson
What happens to a food web if one species is removed?
Species that fed on it lose a food source and may decline. Read the full answer in the Food web lesson
What is bioaccumulation?
Some poisons, such as certain pesticides or mercury, are not broken down or excreted. Read the full answer in the Food web lesson
Why is only about 10 percent of biomass passed on?
Animals use much of their food for respiration to release energy for moving and staying warm, and lose some as heat. Read the full answer in the Biomass lesson
Why are there fewer top predators than herbivores?
Energy decreases at each step in the food chain. Read the full answer in the Biomass lesson
Why are decomposers important?
They break down dead organisms and waste, releasing minerals such as nitrates back into the soil, where plants can take them up again. Read the full answer in the Decomposer lesson
What conditions help decay?
Decay is fastest when it is warm, moist and has plenty of oxygen, because decomposers respire and multiply well in these conditions. Read the full answer in the Decomposer lesson
How does carbon dioxide get into the air?
Carbon dioxide enters the air through respiration by plants, animals and microorganisms. Read the full answer in the Carbon cycle lesson
What are fossil fuels?
Coal, oil and natural gas are fossil fuels. Read the full answer in the Carbon cycle lesson
What are the main steps of the water cycle?
The Sun heats water in seas and lakes, which evaporates into vapour. Read the full answer in the Water cycle lesson
Why is the water cycle important for living things?
Fresh water is needed by all living things for chemical reactions and for transporting substances. Read the full answer in the Water cycle lesson
PART 08Humans and the environment
Why does biodiversity matter?
Many species depend on each other, so losing one can affect many others. Read the full answer in the Biodiversity lesson
How can biodiversity be measured?
Scientists count the number of different species in an area using methods such as quadrats for plants and traps or nets for animals. Read the full answer in the Biodiversity lesson
Is the greenhouse effect bad?
The natural greenhouse effect is vital. Read the full answer in the Greenhouse effect lesson
How do greenhouse gases trap heat?
The Sun's light passes through the air and warms the ground. Read the full answer in the Greenhouse effect lesson
What are the effects of global warming?
Likely effects include melting ice caps and glaciers, rising sea levels and more extreme weather such as heatwaves and floods. Read the full answer in the Global warming lesson
How can we reduce global warming?
We can use less fossil fuel by switching to renewable energy such as wind and solar, improving energy efficiency and using public transport. Read the full answer in the Global warming lesson
Why do people cut down forests?
Forests are cleared to create farmland for crops and cattle, including palm oil and soya. Read the full answer in the Deforestation lesson
What can be done to reduce deforestation?
Governments can protect forests by law and punish illegal logging. Read the full answer in the Deforestation lesson
Why should peat bogs be protected?
Peat takes thousands of years to form, so it is effectively non-renewable. Read the full answer in the Peat bog lesson
How does peat form?
In a waterlogged bog there is little oxygen, so decomposers cannot break down dead plants fully. Read the full answer in the Peat bog lesson
How can eutrophication be prevented?
Farmers can use less fertiliser, apply it at the right time and not before heavy rain, and leave a strip of plants along rivers to soak up extra nutrients. Read the full answer in the Eutrophication lesson
Why does eutrophication kill fish?
Extra nutrients make algae spread across the surface, blocking light from plants below, which die. Read the full answer in the Eutrophication lesson
What is acid rain and what does it do?
Burning fossil fuels releases sulfur dioxide and nitrogen oxides. Read the full answer in the Air pollution lesson
How does air pollution affect human health?
Tiny particles from smoke and vehicle exhaust can enter the lungs and cause breathing problems, such as asthma attacks, and raise the risk of heart disease. Read the full answer in the Air pollution lesson
What is captive breeding?
Captive breeding means breeding endangered animals in zoos or special centres where they are protected. Read the full answer in the Conservation lesson
What is a seed bank?
A seed bank stores seeds from many plant species in cold, dry conditions so they stay alive for years. Read the full answer in the Conservation lesson
What is a fishing quota?
A quota limits how many fish of a species can be caught in a year. Read the full answer in the Sustainable development lesson
Why is recycling more sustainable?
Recycling reuses materials such as paper, glass and metal, so fewer new raw materials need to be mined or cut down. Read the full answer in the Sustainable development lesson
What factors threaten food security?
A growing population needs more food. Read the full answer in the Food security lesson
How can food production be made more sustainable?
Farmers can use biological control, such as natural predators, to reduce pesticide use. Read the full answer in the Food security lesson
Biology (university)
PART 01Cell structure and membranes
Do prokaryotes have any organelles at all?
Prokaryotes have no membrane-bound organelles, but they are not empty. Read the full answer in the Prokaryotic cell lesson
Are bacteria and archaea the same thing?
Why are eukaryotic cells usually bigger than prokaryotic cells?
Internal compartments let eukaryotes organise chemistry inside a large volume. Read the full answer in the Eukaryotic cell lesson
Do all eukaryotic cells have the same organelles?
Why does the nucleus have a double membrane?
The double membrane gives the genome a strong barrier, separating DNA from the reactions in the cytoplasm. Read the full answer in the Nucleus lesson
What is the nucleolus?
The nucleolus is a dense region inside the nucleus, not a separate organelle with its own membrane. Read the full answer in the Nucleus lesson
What is the difference between rough and smooth ER?
Rough ER is studded with ribosomes and makes proteins that will be secreted or placed in membranes. Read the full answer in the Endomembrane system lesson
What does the Golgi apparatus actually do?
The Golgi receives proteins from the ER, modifies them (for example by adding sugar chains), sorts them and sends them out in vesicles. Read the full answer in the Endomembrane system lesson
Which organelles came from endosymbiosis?
Mitochondria came from a bacterium that could use oxygen to make energy. Read the full answer in the Endosymbiosis lesson
Why do mitochondria have their own DNA?
Because they were once independent bacteria. Read the full answer in the Endosymbiosis lesson
What do microtubules do besides give shape?
Microtubules act as tracks. Read the full answer in the Cytoskeleton lesson
Do prokaryotes have a cytoskeleton?
Yes, in a simpler form. Read the full answer in the Cytoskeleton lesson
Why can membrane proteins move around?
Phospholipids are held together by weak forces rather than strong bonds, so they slide past each other like people in a crowd. Read the full answer in the Fluid mosaic model lesson
What are the heads and tails of a phospholipid?
The head contains a phosphate group and likes water, so it faces outward toward the watery inside and outside of the cell. Read the full answer in the Fluid mosaic model lesson
Is osmosis just a type of diffusion?
Diffusion is the spread of any particles from high to low concentration. Read the full answer in the Osmosis lesson
What is turgor pressure in plant cells?
Plant cells have a rigid cell wall. Read the full answer in the Osmosis lesson
What is the difference between primary and secondary active transport?
Primary active transport uses ATP directly, as the sodium-potassium pump does. Read the full answer in the Active transport lesson
Why do cells spend so much energy on pumps?
Gradients store energy like a charged battery. Read the full answer in the Active transport lesson
PART 02Enzymes and energy
What is a metabolic pathway?
A metabolic pathway is a series of linked reactions, where the product of one step becomes the substrate of the next. Read the full answer in the Metabolism lesson
Does a faster metabolism mean burning more energy?
In everyday talk, yes. Read the full answer in the Metabolism lesson
Why is ATP called the energy currency of the cell?
Because nearly every energy-needing job in a cell is paid for with ATP, whatever the source of the energy was. Read the full answer in the ATP lesson
Why does breaking ATP release energy?
The three phosphate groups are all negatively charged and crowd close together, so the molecule is under strain. Read the full answer in the ATP lesson
What is the difference between exergonic and endergonic?
An exergonic reaction releases free energy and has a negative delta G, such as burning sugar. Read the full answer in the Free energy lesson
If a reaction is spontaneous, why might it not happen quickly?
Spontaneous only says the reaction is energetically downhill. Read the full answer in the Free energy lesson
How does ATP couple to another reaction in practice?
Often the end phosphate of ATP is transferred to the other molecule, making it unstable and more reactive. Read the full answer in the Energy coupling lesson
Is any energy lost in coupled reactions?
Why are enzymes so specific?
The active site has a particular shape and pattern of charges, which fits only certain substrates, like a lock fits certain keys. Read the full answer in the Enzyme lesson
What happens to an enzyme at high temperature?
As temperature rises, reactions speed up until the enzyme begins to denature. Read the full answer in the Enzyme lesson
What is the difference between lock-and-key and induced fit?
Lock-and-key pictured a rigid enzyme with a perfectly matching shape. Read the full answer in the Active site and induced fit lesson
What are cofactors and coenzymes?
Cofactors are non-protein helpers that many enzymes need, such as zinc or magnesium ions. Read the full answer in the Active site and induced fit lesson
What does the Michaelis-Menten equation describe?
It describes how reaction rate depends on substrate concentration for a simple enzyme. Read the full answer in the Enzyme kinetics lesson
Besides substrate, what else changes enzyme rate?
Enzyme concentration, temperature and pH all matter. Read the full answer in the Enzyme kinetics lesson
Are inhibitors always harmful?
How is irreversible inhibition different?
A reversible inhibitor binds weakly and can come off again. Read the full answer in the Enzyme inhibition lesson
What is allosteric regulation?
Allosteric regulation happens when a molecule binds an enzyme at a site away from the active site and changes the enzyme's shape. Read the full answer in the Feedback inhibition lesson
Why is feedback inhibition more efficient than switching genes off?
It acts within seconds, because it works on enzymes that are already there. Read the full answer in the Feedback inhibition lesson
PART 03Respiration and photosynthesis
What does OIL RIG mean?
It is a memory aid: Oxidation Is Loss, Reduction Is Gain, of electrons. Read the full answer in the Redox reaction lesson
What is NAD+ and why is it important?
NAD+ is an electron carrier made from a vitamin. Read the full answer in the Redox reaction lesson
Why does glycolysis use 2 ATP before making 4?
The first steps add phosphate groups to glucose using 2 ATP. Read the full answer in the Glycolysis lesson
Is glycolysis found in all organisms?
Almost all. Read the full answer in the Glycolysis lesson
Why is it called a cycle?
Because the last molecule, oxaloacetate, is the same one that picks up carbon at the start. Read the full answer in the Citric acid cycle lesson
Does the cycle use oxygen directly?
Why do we need to breathe oxygen?
Oxygen is the final electron acceptor. Read the full answer in the Electron transport chain lesson
What does cyanide do to the body?
Cyanide binds to an enzyme complex at the end of the electron transport chain, called cytochrome c oxidase. Read the full answer in the Electron transport chain lesson
Where does the ATP synthase get its energy?
From the flow of protons down their gradient. Read the full answer in the Chemiosmosis lesson
Can uncoupling the gradient be useful?
Why do muscles produce lactate?
During hard exercise, muscles need ATP faster than oxygen can arrive. Read the full answer in the Fermentation lesson
How much ATP does fermentation produce?
Only the 2 ATP made in glycolysis, per glucose. Read the full answer in the Fermentation lesson
What are photosystems?
Photosystems are clusters of pigment molecules, including chlorophyll, and proteins in the thylakoid membrane. Read the full answer in the Light reactions lesson
Why are plants green?
Chlorophyll absorbs mostly red and blue light and reflects green. Read the full answer in the Light reactions lesson
What is carbon fixation?
Carbon fixation is turning carbon from inorganic carbon dioxide into an organic molecule. Read the full answer in the Calvin cycle lesson
Why is it sometimes called the dark reactions?
Because the Calvin cycle does not use light directly. Read the full answer in the Calvin cycle lesson
PART 04DNA, transcription and translation
Why is DNA double-stranded?
Two strands give a built-in backup. Read the full answer in the DNA structure lesson
What does 5' to 3' mean?
It describes the direction of a DNA strand. Read the full answer in the DNA structure lesson
Who proved DNA replication is semi-conservative?
Meselson and Stahl showed it in 1958. Read the full answer in the Semi-conservative replication lesson
How accurate is DNA replication?
Very accurate. Read the full answer in the Semi-conservative replication lesson
Why does the lagging strand need Okazaki fragments?
DNA polymerase can only add nucleotides in one direction, 5' to 3'. Read the full answer in the Lagging strand lesson
What is a primer and why is it needed?
DNA polymerase cannot start a new strand from nothing. Read the full answer in the Lagging strand lesson
Is the central dogma always true?
It describes the usual flow, but there are exceptions. Read the full answer in the Central dogma lesson
What is the difference between genotype and phenotype?
Genotype is the set of alleles an organism carries, the DNA instructions. Read the full answer in the Central dogma lesson
How does RNA polymerase know where to start?
It recognises a promoter, a DNA sequence just before the gene. Read the full answer in the Transcription lesson
What is the difference between DNA and RNA?
RNA is usually single-stranded, uses ribose sugar instead of deoxyribose, and has uracil in place of thymine. Read the full answer in the Transcription lesson
Why do eukaryotic genes have introns?
Introns can allow one gene to produce several proteins by alternative splicing, which greatly increases the variety of proteins. Read the full answer in the RNA splicing lesson
What do the cap and poly-A tail do?
The 5' cap and the poly-A tail protect mRNA from being broken down, help it leave the nucleus, and help the ribosome recognise and start reading it. Read the full answer in the RNA splicing lesson
Why is the genetic code called redundant?
With 64 codons but only 20 amino acids, most amino acids have more than one codon. Read the full answer in the Genetic code lesson
Is the genetic code the same in all organisms?
What is the role of the ribosome?
The ribosome is the machine that builds proteins. Read the full answer in the Translation lesson
What happens at the stop codon?
No tRNA matches a stop codon. Read the full answer in the Translation lesson
Are all mutations harmful?
What is a frameshift mutation?
A frameshift occurs when bases are added or deleted in a number that is not a multiple of three. Read the full answer in the Point mutation lesson
PART 05Gene regulation and biotechnology
Why is the lac operon called inducible?
Because the genes are normally off and are switched on by the presence of a molecule, the inducer. Read the full answer in the Lac operon lesson
Why does E. coli prefer glucose to lactose?
Glucose is quicker to use. Read the full answer in the Lac operon lesson
Why do different cells have different genes active?
Every cell holds the same DNA, but each cell type makes a different set of transcription factors. Read the full answer in the Transcription factor lesson
What is an enhancer?
An enhancer is a stretch of DNA that increases the transcription of a gene. Read the full answer in the Transcription factor lesson
Can epigenetic changes be inherited?
Within the body, yes: when a cell divides, the tags are copied, so daughter cells keep the same gene activity. Read the full answer in the Epigenetic modification lesson
What do histones do?
Histones are proteins that DNA winds around, like thread on a spool. Read the full answer in the Epigenetic modification lesson
Why don't bacteria cut their own DNA?
Bacteria add methyl tags to their own DNA at the recognition sites, which stops their enzymes cutting there. Read the full answer in the Restriction enzyme lesson
What are sticky ends?
Some restriction enzymes cut the two DNA strands in staggered places, leaving short single-stranded overhangs. Read the full answer in the Restriction enzyme lesson
What is a plasmid?
A plasmid is a small circular piece of DNA found in bacteria, separate from the main chromosome. Read the full answer in the Recombinant DNA lesson
How do scientists tell which bacteria took up the plasmid?
Plasmids used in cloning carry a marker, often an antibiotic resistance gene. Read the full answer in the Recombinant DNA lesson
Who invented PCR?
Kary Mullis developed PCR in the 1980s and shared the 1993 Nobel Prize in Chemistry for it. Read the full answer in the Polymerase chain reaction lesson
What are primers in PCR?
Primers are short pieces of single-stranded DNA, usually 18 to 25 bases long, that match the two ends of the region to be copied. Read the full answer in the Polymerase chain reaction lesson
Why does DNA move through the gel?
The phosphate groups in DNA carry negative charge, so DNA is pulled toward the positive electrode when voltage is applied. Read the full answer in the Gel electrophoresis lesson
How are DNA bands made visible?
The gel is stained with a dye that sticks to DNA and glows under ultraviolet or blue light, or the DNA is labelled in advance with a fluorescent tag. Read the full answer in the Gel electrophoresis lesson
How does CRISPR work as a bacterial defence?
Bacteria store short pieces of DNA from viruses that attacked them in the past, in regions called CRISPR. Read the full answer in the CRISPR-Cas9 lesson
What are the ethical concerns about gene editing?
Editing body cells to treat disease is widely accepted, with careful oversight. Read the full answer in the CRISPR-Cas9 lesson
PART 06Cell cycle, mitosis and meiosis
How long does the cell cycle take?
It depends on the cell. Read the full answer in the Cell cycle lesson
What is the G0 phase?
G0 is a resting state outside the active cycle. Read the full answer in the Cell cycle lesson
What are cyclins?
Cyclins are proteins whose levels rise and fall through the cell cycle. Read the full answer in the Cell cycle checkpoint lesson
What is apoptosis?
Apoptosis is programmed cell death, a tidy process in which the cell shrinks, breaks its DNA and is cleared away. Read the full answer in the Cell cycle checkpoint lesson
Why do chromosomes condense before division?
Chromosomes are very long threads of DNA. Read the full answer in the Mitosis lesson
What does the spindle do?
The spindle is a structure made of microtubules. Read the full answer in the Mitosis lesson
Can a cell divide its nucleus but not its cytoplasm?
Why do plant cells form a cell plate?
Plant cells have a stiff cell wall, so they cannot pinch in two. Read the full answer in the Cytokinesis lesson
What is the difference between benign and malignant tumours?
A benign tumour stays in one place and is usually enclosed, so it can often be removed. Read the full answer in the Cancer lesson
Why does cancer risk increase with age?
Cancer usually needs several mutations in different cell-cycle genes. Read the full answer in the Cancer lesson
Why does meiosis need to halve the chromosome number?
Fertilisation joins two gametes. Read the full answer in the Meiosis lesson
What are homologous chromosomes?
A homologous pair consists of two chromosomes of the same size and gene order, one from each parent. Read the full answer in the Meiosis lesson
What is a chiasma?
A chiasma is the place where two homologous chromosomes are joined while crossing over. Read the full answer in the Crossing over and independent assortment lesson
Why is genetic variation useful?
Variation means individuals in a population differ, so when conditions change, such as a new disease, some are likely to survive. Read the full answer in the Crossing over and independent assortment lesson
Does the risk of Down syndrome increase with maternal age?
Are all chromosome number changes fatal?
No, but most are. Read the full answer in the Nondisjunction lesson
PART 07Genetics and inheritance
Does dominant mean most common?
What does homozygous and heterozygous mean?
Homozygous means the two alleles of a gene are the same, such as AA or aa. Read the full answer in the Allele lesson
What is a test cross?
A test cross breeds an individual showing the dominant trait with a homozygous recessive one, to find out its genotype. Read the full answer in the Law of segregation lesson
Who was Gregor Mendel?
Mendel was a monk who bred pea plants in the 1850s and 1860s. Read the full answer in the Law of segregation lesson
Why does independent assortment not always hold?
It holds only for genes on different chromosomes, or far apart on the same one. Read the full answer in the Law of independent assortment lesson
How do I do a dihybrid Punnett square?
List the four types of gamete from each parent, such as AB, Ab, aB, ab. Read the full answer in the Law of independent assortment lesson
How is codominance different from incomplete dominance?
In incomplete dominance the heterozygote shows a new, in-between version, like pink flowers from red and white parents. Read the full answer in the Incomplete dominance and codominance lesson
How does ABO blood type inheritance work?
Three alleles exist: I^A, I^B and i. Read the full answer in the Incomplete dominance and codominance lesson
Why can a father not pass an X-linked trait to his son?
A son receives the Y chromosome from his father and his X from his mother. Read the full answer in the Sex-linked inheritance lesson
Can females have X-linked recessive disorders?
Yes, but it is rarer. Read the full answer in the Sex-linked inheritance lesson
What is a map unit or centimorgan?
A map unit is a measure of distance between genes on a chromosome. Read the full answer in the Linked genes lesson
Why can recombination frequency never be more than 50 percent?
If genes are far apart, many crossovers occur between them. Read the full answer in the Linked genes lesson
How can you tell if a trait is dominant from a pedigree?
Dominant traits usually appear in every generation, and an affected child has at least one affected parent. Read the full answer in the Pedigree lesson
What does it mean to be a carrier?
A carrier has one copy of a recessive disease allele and one normal allele, so they usually have no symptoms. Read the full answer in the Pedigree lesson
Why are traits like height not simple 3:1 ratios?
Height depends on many genes, each adding a small effect, and on the environment, such as nutrition. Read the full answer in the Polygenic trait lesson
Does the environment affect gene expression?
PART 08Evolution and diversity of life
What does theory mean in the theory of evolution?
In science, a theory is a well-tested explanation backed by many lines of evidence, not a guess. Read the full answer in the Natural selection lesson
Does natural selection make organisms better or perfect?
Is biological fitness the same as being strong?
In biology, fitness means reproductive success, how many offspring survive to breed, and not strength or health. Read the full answer in the Fitness lesson
What is sexual selection?
Sexual selection is a form of natural selection in which traits help an individual win mates. Read the full answer in the Fitness lesson
What is the founder effect?
The founder effect occurs when a small group leaves a large population to start a new one. Read the full answer in the Genetic drift lesson
Does drift help or harm a population?
Drift is random, so it is neither. Read the full answer in the Genetic drift lesson
How does gene flow affect speciation?
Gene flow keeps populations genetically similar, so it works against the splitting of one species into two. Read the full answer in the Gene flow lesson
Can gene flow happen between species?
Occasionally. Read the full answer in the Gene flow lesson
Why is Hardy-Weinberg useful if populations never meet it?
It gives a baseline. Read the full answer in the Hardy-Weinberg equilibrium lesson
How do I use the Hardy-Weinberg equation?
Start from what you can count, usually the frequency of the recessive phenotype, which equals q squared. Read the full answer in the Hardy-Weinberg equilibrium lesson
Why do biologists use more than one definition of a species?
A common definition is a group of organisms that can interbreed and produce fertile offspring. Read the full answer in the Speciation lesson
What are reproductive barriers?
Reproductive barriers stop different species interbreeding. Read the full answer in the Speciation lesson
What are vestigial structures?
Vestigial structures are body parts that have lost most or all of their original function, such as the pelvis bones in whales or the human tailbone. Read the full answer in the Evidence for evolution lesson
What is convergent evolution?
Convergent evolution is when unrelated species evolve similar features because they face similar challenges. Read the full answer in the Evidence for evolution lesson
How are phylogenetic trees built?
Scientists compare traits, most often DNA sequences, across species. Read the full answer in the Phylogenetic tree lesson
What are the three domains of life?
Bacteria, Archaea and Eukarya. Read the full answer in the Phylogenetic tree lesson
PART 09Animal physiology
What is the difference between positive and negative feedback?
Negative feedback reverses a change, keeping conditions steady, like a thermostat. Read the full answer in the Homeostasis lesson
How does the body control temperature?
The brain's hypothalamus acts as a thermostat. Read the full answer in the Homeostasis lesson
Why is the sodium-potassium pump important for nerves?
The pump restores the resting balance of ions after each signal by moving sodium out and potassium in. Read the full answer in the Action potential lesson
What is myelin and why does it speed up signals?
Myelin is a fatty sheath wrapped around many axons. Read the full answer in the Action potential lesson
How are neurotransmitters removed from the synapse?
If the neurotransmitter stayed, the next cell would keep firing. Read the full answer in the Synapse lesson
What is the difference between excitatory and inhibitory synapses?
Excitatory synapses make the next neuron more likely to fire, by letting positive ions in. Read the full answer in the Synapse lesson
Why do hormones only affect certain cells?
Hormones travel through the blood to every cell, but only target cells have the matching receptor. Read the full answer in the Hormone lesson
What is diabetes?
Diabetes is a condition in which blood glucose stays too high. Read the full answer in the Hormone lesson
Why do mammals need a double circulation?
Blood passing through the lungs loses pressure. Read the full answer in the Double circulation lesson
What does a heartbeat consist of?
A heartbeat has a filling and a pumping part. Read the full answer in the Double circulation lesson
What is carbon monoxide poisoning?
Carbon monoxide binds to haemoglobin about 200 times more strongly than oxygen does. Read the full answer in the Oxygen transport lesson
Why is foetal haemoglobin different?
A foetus gets oxygen from its mother's blood across the placenta. Read the full answer in the Oxygen transport lesson
Why is there glucose in the filtrate but not in urine?
The glomerulus filters small molecules, including glucose, out of the blood. Read the full answer in the Nephron lesson
What is urea and where does it come from?
Urea is a waste made in the liver when excess amino acids are broken down. Read the full answer in the Nephron lesson
What are antibodies?
Antibodies are Y-shaped proteins made by B cells. Read the full answer in the Adaptive immunity lesson
How does HIV damage the immune system?
HIV infects and destroys helper T cells, which coordinate the immune response. Read the full answer in the Adaptive immunity lesson
What do phagocytes do?
Phagocytes such as macrophages and neutrophils are white blood cells that engulf germs and digest them. Read the full answer in the Innate immunity lesson
Why does inflammation cause swelling, heat and redness?
Damaged tissue releases chemicals that widen nearby blood vessels and make them leaky. Read the full answer in the Innate immunity lesson
PART 10Ecology and the environment
What are density-dependent factors?
These are limits whose effect becomes stronger as a population becomes more crowded, such as competition for food, disease and predation. Read the full answer in the Population growth lesson
What is the difference between r-selected and K-selected species?
r-selected species, such as insects and weeds, have many young, grow fast and give little care. Read the full answer in the Population growth lesson
What is the difference between habitat and niche?
A habitat is the place where an organism lives, its address. Read the full answer in the Ecological niche lesson
What is an invasive species?
An invasive species is a non-native species that spreads and harms the ecosystem it has entered. Read the full answer in the Ecological niche lesson
What is an example of mutualism between plants and fungi?
Mycorrhizae are partnerships between fungi and plant roots. Read the full answer in the Symbiosis lesson
What is the difference between a parasite and a predator?
A predator kills and eats its prey quickly. Read the full answer in the Symbiosis lesson
What is the difference between a food chain and a food web?
A food chain is a single line of who eats whom. Read the full answer in the Trophic level lesson
What do decomposers do in an ecosystem?
Decomposers, mainly fungi and bacteria, break down dead plants, animals and waste. Read the full answer in the Trophic level lesson
Why is eating lower on the food chain more energy efficient?
Each step in the chain loses about 90 percent of the energy. Read the full answer in the Energy transfer lesson
What is biomagnification?
Biomagnification is the build-up of a toxic substance as it moves up a food chain. Read the full answer in the Energy transfer lesson
Why does carbon dioxide cause warming?
Carbon dioxide is a greenhouse gas. Read the full answer in the Carbon cycle lesson
What is a carbon sink?
A carbon sink is something that takes in more carbon than it releases. Read the full answer in the Carbon cycle lesson
Why is nitrogen so important to living things?
Nitrogen is a key part of amino acids, so it is needed to make proteins, and of nucleotides, so it is needed to make DNA and RNA. Read the full answer in the Nitrogen cycle lesson
What is eutrophication?
Eutrophication occurs when fertiliser runoff adds too many nutrients to water. Read the full answer in the Nitrogen cycle lesson
What is a climax community?
A climax community is the stable, mature stage that succession tends toward, for example a forest. Read the full answer in the Ecological succession lesson
What are pioneer species?
Pioneer species are the first to colonise bare or newly disturbed ground, such as lichens, mosses and fast-growing weeds. Read the full answer in the Ecological succession lesson
Which ecosystem services depend on biodiversity?
Diverse ecosystems tend to be more stable and productive. Read the full answer in the Biodiversity lesson
What is a keystone species?
A keystone species has an effect on its ecosystem far larger than its numbers suggest. Read the full answer in the Biodiversity lesson
Chemistry (school)
PART 01Atoms and matter
How small is an atom?
An atom is about 0.0000000001 metres across, which is far too small to see with a normal microscope. Read the full answer in the Atom lesson
Can atoms be split?
Atoms can be split into protons, neutrons and electrons, but then they are no longer the same element. Read the full answer in the Atom lesson
How many elements are there?
There are 118 elements on the periodic table. Read the full answer in the Element lesson
Why do some element symbols not match their English names?
Some symbols come from old Latin names. Read the full answer in the Element lesson
Why does a compound look different from the elements in it?
When elements bond, their atoms rearrange and share or swap electrons. Read the full answer in the Compound lesson
How do you write the formula for a compound?
A formula uses symbols and small numbers to show how many atoms of each element are in one unit. Read the full answer in the Compound lesson
What are common ways to separate a mixture?
Filtration separates a solid from a liquid. Read the full answer in the Mixture lesson
Is air a mixture or a compound?
Air is a mixture of gases, mainly nitrogen at about 78 percent and oxygen at about 21 percent, plus small amounts of argon and carbon dioxide. Read the full answer in the Mixture lesson
Why does the number of protons decide the element?
The number of protons fixes how many electrons a neutral atom has, and electrons control how the atom reacts. Read the full answer in the Proton lesson
Where are protons found in an atom?
Protons sit in the nucleus, the tiny dense centre of the atom, together with the neutrons. Read the full answer in the Proton lesson
What do neutrons do in an atom?
Neutrons add mass and help hold the nucleus together. Read the full answer in the Neutron lesson
Do all atoms have neutrons?
Almost all do. Read the full answer in the Neutron lesson
Why are electrons important in chemistry?
Electrons on the outside of an atom take part in reactions. Read the full answer in the Electron lesson
Do electrons really orbit like planets?
The planet picture is a useful simple model for school. Read the full answer in the Electron lesson
Where do you find the atomic number on the periodic table?
On most periodic tables the atomic number is the whole number next to the symbol, and it is usually the smaller of the two numbers. Read the full answer in the Atomic number lesson
Can two different elements have the same atomic number?
Why is the mass number always a whole number?
It counts particles, and you cannot have half a proton or neutron. Read the full answer in the Mass number lesson
Do electrons count towards mass number?
Why are isotopes of an element chemically the same?
Chemical reactions depend on electrons, and isotopes of an element have the same number of electrons when neutral. Read the full answer in the Isotope lesson
Why is relative atomic mass often not a whole number?
Most elements are a mixture of isotopes with different masses. Read the full answer in the Isotope lesson
PART 02The periodic table
Why does the second shell hold only 8 electrons?
Each shell has a fixed number of spaces, set by how electrons are arranged in energy levels. Read the full answer in the Electron shell lesson
How do you write the electron arrangement of an atom?
Find the atomic number, which is the number of electrons in a neutral atom. Read the full answer in the Electron shell lesson
Why does the periodic table have rows of different lengths?
Each period begins when a new shell starts to fill. Read the full answer in the Period lesson
What changes as you move across a period?
The number of protons and outer electrons rises by one each step, so atoms go from metals on the left to non-metals on the right. Read the full answer in the Period lesson
How can you tell the group of an element from its electrons?
For main groups 1 to 7, the group number equals the number of outer electrons. Read the full answer in the Group lesson
Why did Mendeleev leave gaps in his periodic table?
Dmitri Mendeleev arranged elements by atomic mass and by chemical behaviour. Read the full answer in the Group lesson
Why do metals conduct electricity?
A metal holds its atoms in a structure where outer electrons move freely. Read the full answer in the Metal lesson
Are all metals hard and solid?
Why are most non-metals poor conductors?
Non-metal atoms hold on to their electrons or share them in bonds, so there are no free electrons to carry charge. Read the full answer in the Non-metal lesson
Where is the line between metals and non-metals on the table?
A stepped line runs down the right-hand side of the periodic table. Read the full answer in the Non-metal lesson
Why are alkali metals stored under oil?
They react quickly with oxygen and water vapour in air, forming a dull crust on the surface. Read the full answer in the Alkali metals lesson
Why do alkali metals have low melting points compared with other metals?
They have only one outer electron, so the bonds holding the metal together are weak. Read the full answer in the Alkali metals lesson
Why are halogens found as pairs of atoms?
Each halogen atom has 7 outer electrons and needs one more for a full shell. Read the full answer in the Halogens lesson
What is a halogen displacement reaction?
A more reactive halogen pushes a less reactive one out of its compound. Read the full answer in the Halogens lesson
Why are noble gases called inert?
Inert means they do not take part in chemical reactions. Read the full answer in the Noble gases lesson
What happens to the boiling point of noble gases down the group?
What are typical uses of transition metals?
Iron is used to make steel for buildings. Read the full answer in the Transition metals lesson
How are transition metals different from group 1 metals?
Transition metals are harder, stronger and denser, and have higher melting points. Read the full answer in the Transition metals lesson
PART 03Bonding and structure
Why do atoms form ions?
Atoms are more stable with a full outer shell of electrons. Read the full answer in the Ion lesson
How do you know the charge of an ion from its group?
Group 1 ions have a charge of 1+, group 2 ions 2+, group 6 ions 2- and group 7 ions 1-. Read the full answer in the Ion lesson
Why is an ionic compound neutral overall?
The positive and negative charges balance out. Read the full answer in the Ionic bond lesson
Do ionic compounds dissolve in water?
What is a double bond?
A double bond is two shared pairs of electrons between the same two atoms. Read the full answer in the Covalent bond lesson
Why do covalent bonds not conduct electricity?
In a covalent bond the electrons are fixed between two atoms. Read the full answer in the Covalent bond lesson
Why do metals conduct electricity and heat so well?
The sea of free electrons moves through the whole structure. Read the full answer in the Metallic bonding lesson
Why can metals be bent and hammered into shape?
The ions in a metal sit in regular layers. Read the full answer in the Metallic bonding lesson
Why do ionic solids not conduct electricity?
In a solid lattice the ions are locked in place and cannot move. Read the full answer in the Ionic lattice lesson
Why are ionic crystals brittle?
If a force pushes one layer sideways, ions of the same charge end up next to each other. Read the full answer in the Ionic lattice lesson
Why do simple molecules have low boiling points?
Boiling only needs the weak forces between molecules to be overcome, not the strong covalent bonds inside them. Read the full answer in the Simple molecule lesson
Do larger molecules have higher boiling points?
Why does silicon dioxide have a high melting point?
Silicon dioxide, the main substance in sand, is a giant network of silicon and oxygen atoms. Read the full answer in the Giant covalent structure lesson
What is diamond used for?
Diamond is the hardest natural substance, so it is used to cut and grind. Read the full answer in the Giant covalent structure lesson
What is graphene?
Graphene is a single layer of graphite, one atom thick, with carbon atoms in a hexagonal pattern. Read the full answer in the Allotrope lesson
What are fullerenes used for?
Fullerenes are hollow shapes of carbon atoms, such as balls and tubes. Read the full answer in the Allotrope lesson
Why is an alloy harder than a pure metal?
In a pure metal the atoms are all the same size, so layers slide over each other easily. Read the full answer in the Alloy lesson
What are some common alloys and their uses?
Steel, iron with a little carbon, builds bridges and cars. Read the full answer in the Alloy lesson
PART 04Chemical reactions
How can you tell that a chemical reaction has happened?
Signs include a temperature change, fizzing as gas forms, a colour change, a solid forming in a liquid, or a new smell. Read the full answer in the Chemical reaction lesson
What is the difference between a physical change and a chemical change?
In a physical change, such as melting or dissolving, no new substance forms and you can usually reverse it easily. Read the full answer in the Chemical reaction lesson
Why does a reaction in an open flask seem to lose mass?
If a gas is made, it escapes into the air and takes its mass with it, so the flask reads lighter. Read the full answer in the Conservation of mass lesson
Who discovered the conservation of mass?
Antoine Lavoisier, a French chemist in the 1700s, is usually credited. Read the full answer in the Conservation of mass lesson
Why can't you change the small numbers when balancing?
The small numbers are part of the formula and show what the substance is. Read the full answer in the Balanced equation lesson
What are the steps for balancing an equation?
Write the correct formulae. Read the full answer in the Balanced equation lesson
When do you need to include state symbols?
Include them when the question asks, or when the states matter, as in ionic equations and reactions that give off gases or make a solid. Read the full answer in the State symbols lesson
What does aqueous mean?
Aqueous means dissolved in water. Read the full answer in the State symbols lesson
What are everyday examples of exothermic reactions?
Burning fuels such as gas or petrol, the reaction of acids with alkalis, rusting, and the respiration that keeps your body warm. Read the full answer in the Exothermic reaction lesson
Why does an exothermic reaction release energy?
Breaking bonds in the reactants takes in energy, while making bonds in the products gives out energy. Read the full answer in the Exothermic reaction lesson
What are examples of endothermic reactions?
Photosynthesis takes in light energy. Read the full answer in the Endothermic reaction lesson
How would you show a reaction is endothermic in an experiment?
Put the reactants in an insulated cup and measure the temperature with a thermometer before and after mixing. Read the full answer in the Endothermic reaction lesson
What is collision theory?
For a reaction to happen, particles must collide with enough energy. Read the full answer in the Rate of reaction lesson
How do you measure the rate of a reaction that makes a gas?
Collect the gas in a gas syringe and record the volume at regular times. Read the full answer in the Rate of reaction lesson
Why isn't a catalyst used up?
A catalyst takes part in the reaction but is regenerated at the end. Read the full answer in the Catalyst lesson
What are enzymes?
Enzymes are biological catalysts, made of protein, found in living things. Read the full answer in the Catalyst lesson
What is equilibrium?
Equilibrium is reached in a closed container when the forward and backward reactions run at the same rate. Read the full answer in the Reversible reaction lesson
If a reaction is exothermic one way, what is it the other way?
Endothermic. Read the full answer in the Reversible reaction lesson
PART 05Moles and calculations
Why does chlorine have a relative atomic mass of 35.5?
Chlorine is a mixture of two isotopes, chlorine-35 and chlorine-37, in a ratio of about 3 to 1. Read the full answer in the Relative atomic mass lesson
Which number on the periodic table is the relative atomic mass?
It is the larger of the two numbers next to the symbol, and it is often a decimal. Read the full answer in the Relative atomic mass lesson
How do you calculate Mr for a compound with brackets?
Work out the mass of the group in the brackets, then multiply by the number outside. Read the full answer in the Relative formula mass lesson
Does relative formula mass have units?
No. Read the full answer in the Relative formula mass lesson
Why do chemists use moles?
Atoms are far too small to count one by one. Read the full answer in the Mole lesson
How many atoms are in one mole of carbon?
One mole of carbon contains 6.02 x 10^23 atoms, and weighs 12 g. Read the full answer in the Mole lesson
Who was Avogadro?
Amedeo Avogadro was an Italian scientist in the early 1800s. Read the full answer in the Avogadro constant lesson
How do you use the Avogadro constant in a calculation?
Multiply the number of moles by 6.02 x 10^23 to get the number of particles. Read the full answer in the Avogadro constant lesson
How do you rearrange the moles equation?
Moles = mass / Mr. Read the full answer in the Moles from mass lesson
How do you use moles to work out the mass of product?
Convert the known mass to moles. Read the full answer in the Moles from mass lesson
How do you find the limiting reactant?
Convert each reactant to moles. Read the full answer in the Limiting reactant lesson
What does excess mean?
A reactant in excess is present in a larger amount than needed. Read the full answer in the Limiting reactant lesson
Can percentage yield be over 100 percent?
In theory no. Read the full answer in the Percentage yield lesson
Why do industries care about percentage yield?
A higher yield means more product from the same raw materials, which saves money and reduces waste. Read the full answer in the Percentage yield lesson
How do you convert cm3 to dm3?
Divide by 1,000. Read the full answer in the Concentration lesson
What does a concentration of 2 mol/dm3 mean?
It means there are 2 moles of dissolved substance in every 1 dm3 of solution. Read the full answer in the Concentration lesson
Does the 24 dm3 value work for every gas?
Yes, at room temperature and pressure, which is about 20 degrees Celsius and 1 atmosphere. Read the full answer in the Molar gas volume lesson
How do you find the volume of gas made in a reaction?
Work out the moles of the gas using the balanced equation. Read the full answer in the Molar gas volume lesson
PART 06Acids, bases and salts
Are all acids dangerous?
What is the difference between an acid and a base?
An acid releases hydrogen ions in water. Read the full answer in the Acid lesson
Why do alkalis feel soapy?
Alkalis react with the oils in your skin and turn them into a kind of soap. Read the full answer in the Alkali lesson
Which common household items are alkaline?
Bleach, soap, oven cleaner, baking soda and toothpaste are alkaline. Read the full answer in the Alkali lesson
What does each step on the pH scale mean?
Each step is a tenfold change in acidity. Read the full answer in the pH scale lesson
How do you measure pH accurately?
A pH meter with a probe gives a precise number, while universal indicator paper or solution gives a rough value by colour. Read the full answer in the pH scale lesson
What is the difference between litmus and universal indicator?
Litmus has only two colours: red for acid and blue for alkali. Read the full answer in the Indicator lesson
Which indicator is best for a titration?
One with a sharp colour change at the end point, such as methyl orange or phenolphthalein. Read the full answer in the Indicator lesson
What happens to the pH during neutralisation?
The pH moves towards 7. Read the full answer in the Neutralisation lesson
What is the ionic equation for neutralisation?
H+ + OH- -> H2O. Read the full answer in the Neutralisation lesson
What are the other ways to make a salt?
Acids react with metals, metal oxides, hydroxides and carbonates to give a salt. Read the full answer in the Salt lesson
Is table salt the only salt?
Is a strong acid always more dangerous than a weak one?
Not always. Read the full answer in the Strong and weak acids lesson
How can you tell a strong from a weak acid in an experiment?
Compare solutions of the same concentration. Read the full answer in the Strong and weak acids lesson
Why is a pipette used for the alkali and a burette for the acid?
A pipette delivers a fixed volume very accurately, such as 25.0 cm3. Read the full answer in the Titration lesson
Why repeat a titration?
A first run, the rough titration, shows roughly where the end point is. Read the full answer in the Titration lesson
How do you test for hydrogen gas?
Hold a lit splint at the mouth of a test tube of the gas. Read the full answer in the Metal and acid reaction lesson
Why does copper not react with dilute acids?
Copper is below hydrogen in the reactivity series, so it cannot push hydrogen out of an acid. Read the full answer in the Metal and acid reaction lesson
PART 07Metals and electrolysis
How do you work out the reactivity of a metal?
Compare how vigorously metals react with water, acid and oxygen. Read the full answer in the Reactivity series lesson
Why is carbon in the reactivity series?
Carbon is not a metal, but it can displace metals from their oxides. Read the full answer in the Reactivity series lesson
How can you tell a displacement reaction has happened?
Look for a colour change in the solution and a new coating on the metal. Read the full answer in the Displacement reaction lesson
What is the thermite reaction?
Aluminium powder is mixed with iron oxide and ignited. Read the full answer in the Displacement reaction lesson
Is oxidation always about oxygen?
What is a redox reaction?
Redox is short for reduction and oxidation happening together. Read the full answer in the Oxidation lesson
How do you remember oxidation and reduction?
Use OIL RIG: Oxidation Is Loss, Reduction Is Gain, of electrons. Read the full answer in the Reduction lesson
Why is metal extraction called reduction?
A metal ore is usually an oxide. Read the full answer in the Reduction lesson
Why are most metals found as compounds?
Most metals react with oxygen and other substances in nature, so they sit in rocks as compounds. Read the full answer in the Ore lesson
What does it mean for an ore to be economic?
It contains enough metal that it is profitable to dig up and extract it. Read the full answer in the Ore lesson
What does the blast furnace do?
It is a tall furnace where iron ore, coke (carbon) and limestone are fed in at the top and hot air is blown in at the bottom. Read the full answer in the Extraction by carbon lesson
Why is limestone added to the blast furnace?
Iron ore contains impurities such as sand, silicon dioxide. Read the full answer in the Extraction by carbon lesson
Why is electrolysis expensive?
It uses large amounts of electricity. Read the full answer in the Electrolysis lesson
What happens when you electrolyse a salt solution?
Water also contributes ions, so the products can differ from a molten salt. Read the full answer in the Electrolysis lesson
Why can solid ionic compounds not be electrolytes?
In a solid, the ions are locked in a lattice and cannot move to the electrodes. Read the full answer in the Electrolyte lesson
What is an electrode?
An electrode is a rod or plate, often made from graphite or metal, that carries electricity into and out of the electrolyte. Read the full answer in the Electrolyte lesson
How can rusting be prevented?
Stop water and oxygen from reaching the iron. Read the full answer in the Rusting lesson
What is sacrificial protection?
A more reactive metal such as zinc or magnesium is attached to iron. Read the full answer in the Rusting lesson
PART 08Carbon chemistry and the Earth
Why is crude oil called a non-renewable resource?
It forms over millions of years from the remains of dead sea creatures buried under mud. Read the full answer in the Crude oil lesson
What is crude oil used for apart from fuel?
Many products start from crude oil, including plastics, solvents, detergents, lubricants and bitumen for roads. Read the full answer in the Crude oil lesson
Why do hydrocarbons burn so well?
They release a lot of energy when carbon and hydrogen combine with oxygen to form carbon dioxide and water. Read the full answer in the Hydrocarbon lesson
Why do bigger hydrocarbons burn less easily?
Long-chain hydrocarbons are thicker, less runny and have higher boiling points, so they are harder to vaporise and mix with oxygen. Read the full answer in the Hydrocarbon lesson
Why do different fractions have different boiling points?
The longer a hydrocarbon chain, the stronger the forces between its molecules. Read the full answer in the Fractional distillation lesson
What is each fraction used for?
Gases are used for heating and cooking. Read the full answer in the Fractional distillation lesson
What does saturated mean?
A saturated hydrocarbon has only single bonds, so each carbon is bonded to as many hydrogens as possible. Read the full answer in the Alkane lesson
How do you name the alkanes?
The name starts with a part showing how many carbons: meth for 1, eth for 2, prop for 3 and but for 4. Read the full answer in the Alkane lesson
Why are alkenes more reactive than alkanes?
The carbon-carbon double bond can open up, which lets other atoms add on. Read the full answer in the Alkene lesson
What does unsaturated mean?
An unsaturated compound has at least one carbon-carbon double bond, so it holds fewer hydrogen atoms than the maximum. Read the full answer in the Alkene lesson
Why is cracking needed?
Crude oil gives more large molecules than are needed, while demand for small ones like petrol is high. Read the full answer in the Cracking lesson
What conditions are used in cracking?
Long hydrocarbon vapours are heated to a high temperature, around 600 to 700 degrees Celsius, and passed over a hot catalyst such as aluminium oxide. Read the full answer in the Cracking lesson
Why are plastics a problem for the environment?
Many addition polymers do not break down naturally, so they stay in landfill and oceans for hundreds of years. Read the full answer in the Polymer lesson
What is a monomer?
A monomer is a small molecule that can join with many others to form a polymer. Read the full answer in the Polymer lesson
What is the difference between complete and incomplete combustion?
With plenty of oxygen, complete combustion gives carbon dioxide and water. Read the full answer in the Combustion lesson
Why is carbon monoxide dangerous?
Carbon monoxide binds to haemoglobin in red blood cells, stopping them carrying oxygen around the body. Read the full answer in the Combustion lesson
Why is the extra greenhouse effect from humans a problem?
The natural greenhouse effect is essential. Read the full answer in the Greenhouse effect lesson
How can we reduce carbon dioxide emissions?
Use less energy and switch to renewable sources like wind, solar and hydroelectric power. Read the full answer in the Greenhouse effect lesson
How did oxygen get into the atmosphere?
Early Earth had mostly carbon dioxide from volcanoes. Read the full answer in the Earth's atmosphere lesson
What causes acid rain?
Burning fossil fuels containing sulfur makes sulfur dioxide, and hot engines make nitrogen oxides. Read the full answer in the Earth's atmosphere lesson
Chemistry (university)
PART 01Atomic structure and periodic trends
Why do atoms emit only certain colours of light?
An electron can only sit at certain fixed energy levels. Read the full answer in the Atomic emission spectrum lesson
How does an emission spectrum differ from an absorption spectrum?
An emission spectrum shows bright lines on a dark background, made when excited atoms release light. Read the full answer in the Atomic emission spectrum lesson
Does brighter light mean more energy per photon?
Why do we use joules per photon and kilojoules per mole?
One photon's energy is tiny, around 10⁻¹⁹ J, so it is awkward to use. Read the full answer in the Photon energy lesson
Why don't we see a thrown ball behave like a wave?
A ball has a huge mass compared with an electron. Read the full answer in the Wave-particle duality lesson
What evidence shows electrons act as waves?
Electrons fired at a thin crystal make a pattern of bright and dark rings, called diffraction. Read the full answer in the Wave-particle duality lesson
What values can l take for n = 3?
For n = 3, l can be 0, 1 or 2, which is always 0 up to n − 1. Read the full answer in the Quantum numbers lesson
Why are there only two spin values?
Experiments show an electron behaves as though it has a tiny built-in magnet that can point in only two directions, called spin up and spin down. Read the full answer in the Quantum numbers lesson
Is an orbital the same as an orbit?
What is a node in an orbital?
A node is a place where the chance of finding the electron is zero. Read the full answer in the Atomic orbital lesson
Why does 4s fill before 3d?
In a neutral atom, the 4s orbital is slightly lower in energy than 3d as filling begins, so it gets electrons first. Read the full answer in the Electron configuration lesson
Why are chromium and copper exceptions?
Chromium is [Ar] 4s¹ 3d⁵ and copper is [Ar] 4s¹ 3d¹⁰. Read the full answer in the Electron configuration lesson
Why do electrons avoid pairing in the same orbital?
Two electrons in one orbital sit close together and repel each other strongly. Read the full answer in the Pauli exclusion principle and Hund's rule lesson
What does unpaired electron mean for magnetism?
Each electron acts like a tiny magnet. Read the full answer in the Pauli exclusion principle and Hund's rule lesson
Why do electrons in the same shell shield poorly?
Electrons in the same shell are at about the same distance from the nucleus. Read the full answer in the Effective nuclear charge lesson
How does Zeff explain trends across a period?
Moving across a period, each step adds one proton but the extra electron goes into the same shell. Read the full answer in the Effective nuclear charge lesson
Why is a cation smaller than its neutral atom?
Losing electrons often removes the whole outer shell, and the remaining electrons are pulled in by the same nuclear charge. Read the full answer in the Atomic and ionic radius lesson
Why is an anion larger than its neutral atom?
An anion has extra electrons but the same number of protons. Read the full answer in the Atomic and ionic radius lesson
Why does ionization energy dip from Be to B?
Beryllium's outer electrons are in the 2s subshell. Read the full answer in the Ionization energy lesson
Why does the second ionization energy exceed the first?
After the first electron leaves, the ion is positive. Read the full answer in the Ionization energy lesson
PART 02Chemical bonding and molecular shape
How is electronegativity different from electron affinity?
Electron affinity is the energy change when a lone gas atom gains an electron. Read the full answer in the Electronegativity lesson
What electronegativity difference makes a bond ionic?
There is no sharp line, but a difference above about 1.7 to 2.0 is usually treated as ionic. Read the full answer in the Electronegativity lesson
Why do ionic solids have high melting points?
Each ion is held by strong attractions to many neighbours in all directions. Read the full answer in the Ionic bonding and lattice energy lesson
Why do ionic compounds conduct electricity only when molten or dissolved?
In the solid, ions are locked in the lattice and cannot move. Read the full answer in the Ionic bonding and lattice energy lesson
What is the octet rule and when does it fail?
The octet rule says main-group atoms tend to end up with eight valence electrons. Read the full answer in the Lewis structure lesson
How do I choose the central atom?
The central atom is usually the one that makes the most bonds and is the least electronegative, not counting hydrogen. Read the full answer in the Lewis structure lesson
Is formal charge the real charge on an atom?
How do I choose between structures using formal charge?
Prefer the structure where formal charges are smallest, and where any negative charge sits on the more electronegative atom. Read the full answer in the Formal charge lesson
Does a molecule flip between resonance forms?
How does resonance affect stability?
Spreading electrons over more atoms lowers the energy, so the molecule is more stable than any single drawing suggests. Read the full answer in the Resonance lesson
Can a molecule with polar bonds be nonpolar?
Yes. Read the full answer in the Bond polarity and dipole moment lesson
How is dipole moment measured?
Dipole moment is charge times distance, measured in debye (D). Read the full answer in the Bond polarity and dipole moment lesson
Why is the H–O–H angle in water 104.5° and not 109.5°?
Water has four electron groups, which would give 109.5°. Read the full answer in the VSEPR theory lesson
What is the difference between electron geometry and molecular geometry?
Electron geometry counts every electron group, including lone pairs. Read the full answer in the VSEPR theory lesson
How do I find the hybridization of an atom?
Count its electron groups: bonded atoms plus lone pairs. Read the full answer in the Hybridization lesson
What is the difference between a sigma and a pi bond?
A sigma bond forms by head-on overlap of orbitals along the line between the nuclei. Read the full answer in the Hybridization lesson
Why does He₂ not exist as a stable molecule?
Two helium atoms bring four electrons. Read the full answer in the Molecular orbital theory lesson
Why is O₂ paramagnetic?
In the MO diagram for O₂, the last two electrons go into two separate, equal-energy antibonding π orbitals, one each, with the same spin (Hund's rule). Read the full answer in the Molecular orbital theory lesson
PART 03Intermolecular forces, gases and solutions
What are London dispersion forces?
London forces are weak attractions that exist between all molecules. Read the full answer in the Intermolecular forces lesson
Why do larger molecules usually boil at higher temperatures?
Bigger molecules have more electrons, and their electron clouds are easier to distort, which is called polarisability. Read the full answer in the Intermolecular forces lesson
Why does ice float on water?
In ice, hydrogen bonds hold each water molecule in an open six-sided lattice with gaps. Read the full answer in the Hydrogen bonding lesson
Do hydrogen bonds exist inside DNA?
Why does a liquid with strong intermolecular forces have low vapour pressure?
Strong attractions hold molecules in the liquid, so fewer have enough energy to escape into the gas. Read the full answer in the Vapour pressure lesson
What is the normal boiling point?
The normal boiling point is the temperature at which a liquid's vapour pressure equals 1 atm (101.3 kPa). Read the full answer in the Vapour pressure lesson
Why must temperature be in kelvin?
Gas volume and pressure are proportional to absolute temperature. Read the full answer in the Ideal gas law lesson
What is STP?
STP stands for standard temperature and pressure. Read the full answer in the Ideal gas law lesson
What is mole fraction?
Mole fraction is the moles of one gas divided by the total moles of all gases. Read the full answer in the Dalton's law of partial pressures lesson
Why do we correct for water vapour when collecting a gas over water?
A gas bubbled through water picks up water vapour, so the total pressure includes the vapour's pressure. Read the full answer in the Dalton's law of partial pressures lesson
Do all gas molecules at one temperature move at the same speed?
No. Read the full answer in the Kinetic molecular theory lesson
What is Graham's law of effusion?
Graham's law says the rate at which a gas leaks through a tiny hole is inversely proportional to the square root of its molar mass. Read the full answer in the Kinetic molecular theory lesson
When does a real gas behave most like an ideal gas?
A real gas is most ideal at low pressure and high temperature. Read the full answer in the Real gases and van der Waals equation lesson
Why is the pressure of a real gas lower than ideal at moderate pressures?
At moderate pressure, attractions between molecules slow their approach to the wall and soften their collisions. Read the full answer in the Real gases and van der Waals equation lesson
Why does temperature affect solid and gas solubility differently?
Most solids dissolve better when heated, because dissolving usually absorbs heat. Read the full answer in the Solubility and 'like dissolves like' lesson
What does ion-dipole attraction mean?
When salt dissolves in water, the partly negative oxygen ends of water molecules surround the positive ions, and the partly positive hydrogen ends surround the negative ions. Read the full answer in the Solubility and 'like dissolves like' lesson
What is the van 't Hoff factor, i?
The van 't Hoff factor is the number of particles one formula unit gives in solution. Read the full answer in the Colligative properties lesson
What is osmosis?
Osmosis is the movement of solvent through a membrane from a dilute solution to a more concentrated one. Read the full answer in the Colligative properties lesson
PART 04Stoichiometry and reactions in solution
Why is the mole defined using 6.022×10²³?
The number was chosen so that one mole of atoms has a mass in grams equal to the atomic mass in atomic mass units. Read the full answer in the Mole and molar mass lesson
How do I convert grams to number of molecules?
First divide the mass by the molar mass to get moles. Read the full answer in the Mole and molar mass lesson
How do I find the molecular formula from the empirical formula?
Divide the compound's molar mass by the empirical formula mass to get a whole number n. Read the full answer in the Empirical and molecular formula lesson
What if my ratios are not whole numbers?
Multiply all the ratios by the same small number to make them whole. Read the full answer in the Empirical and molecular formula lesson
Why must an equation be balanced?
The law of conservation of mass says atoms are not created or destroyed in a chemical reaction. Read the full answer in the Mole ratio from a balanced equation lesson
Can I use grams directly in the mole ratio?
No. Read the full answer in the Mole ratio from a balanced equation lesson
What is the quickest way to find the limiting reagent?
Divide the moles of each reactant by its coefficient in the balanced equation. Read the full answer in the Limiting reagent lesson
What happens to the excess reagent?
The excess reagent is left over after the reaction. Read the full answer in the Limiting reagent lesson
Why is percent yield usually below 100?
Some product stays on glassware, some reactant forms side products, and some reactions stop before all reactants are used because they reach equilibrium. Read the full answer in the Percent yield lesson
Can percent yield be more than 100?
If a calculation shows over 100 percent, something is wrong. Read the full answer in the Percent yield lesson
What is the difference between molarity and molality?
Molarity is moles of solute per litre of solution and changes slightly with temperature because volume changes. Read the full answer in the Molarity and dilution lesson
Why add acid to water, not water to acid?
Mixing strong acid with water releases a lot of heat. Read the full answer in the Molarity and dilution lesson
How do I know if a precipitate will form?
Use the solubility rules. Read the full answer in the Precipitation reaction and net ionic equation lesson
What is a spectator ion?
A spectator ion is present in the solution but does not take part in the reaction. Read the full answer in the Precipitation reaction and net ionic equation lesson
What do OIL RIG and LEO GER stand for?
These are memory aids. Read the full answer in the Oxidation state and redox lesson
What are the oxidising and reducing agents?
The oxidising agent takes electrons and so is itself reduced. Read the full answer in the Oxidation state and redox lesson
How does balancing change in basic solution?
First balance as if the solution were acidic. Read the full answer in the Balancing redox equations lesson
Why must the electrons cancel?
Electrons are not free in solution. Read the full answer in the Balancing redox equations lesson
PART 05Thermochemistry and thermodynamics
What is a system and what are the surroundings?
The system is the part of the universe you are studying, such as the reaction in a flask. Read the full answer in the First law of thermodynamics lesson
What is a state function?
A state function depends only on the present condition of the system, not on how it got there. Read the full answer in the First law of thermodynamics lesson
What is the difference between ΔU and ΔH?
ΔU is the heat flow at constant volume, and ΔH is the heat flow at constant pressure. Read the full answer in the Enthalpy lesson
What does a thermochemical equation tell you?
It gives the balanced equation together with its ΔH for the amounts shown. Read the full answer in the Enthalpy lesson
What is specific heat capacity?
Specific heat capacity is the heat needed to raise the temperature of 1 g of a substance by 1 °C. Read the full answer in the Calorimetry lesson
Why does a reaction in a coffee-cup calorimeter give ΔH?
A coffee cup is open to the air, so the reaction runs at constant pressure. Read the full answer in the Calorimetry lesson
Why does Hess's law work?
Enthalpy is a state function, so ΔH depends only on the starting and ending states, not on the path. Read the full answer in the Hess's law lesson
What do I do with ΔH when I reverse or multiply an equation?
Reversing an equation changes the sign of ΔH. Read the full answer in the Hess's law lesson
Why is ΔHf° of O₂ zero?
The enthalpy of formation is the energy change to make a compound from its elements. Read the full answer in the Standard enthalpy of formation lesson
Does a more negative ΔHf° mean a more stable compound?
A more negative ΔHf° means the compound is lower in enthalpy than its elements, so energy is released on forming it. Read the full answer in the Standard enthalpy of formation lesson
Why are bond enthalpies only approximate?
A bond's strength changes slightly depending on the rest of the molecule. Read the full answer in the Bond enthalpy lesson
Does breaking a bond release or absorb energy?
Breaking a bond always absorbs energy, because the atoms are held together by attraction and you must pull them apart. Read the full answer in the Bond enthalpy lesson
Does entropy always mean disorder?
Disorder is a handy picture, but a more exact idea is the number of ways energy can be spread over particles. Read the full answer in the Entropy lesson
What is the third law of thermodynamics?
The third law says a perfect crystal at absolute zero (0 K) has zero entropy, because there is only one possible arrangement. Read the full answer in the Entropy lesson
What does spontaneous mean in chemistry?
A spontaneous process happens by itself under the given conditions, without ongoing outside help. Read the full answer in the Second law of thermodynamics lesson
Why is heat given out good for the universe's entropy?
Heat released into cooler surroundings makes particles there move more, which raises the surroundings' entropy. Read the full answer in the Second law of thermodynamics lesson
What happens when ΔH and ΔS are both positive?
The reaction absorbs heat but gains entropy. Read the full answer in the Gibbs free energy lesson
Does negative ΔG mean a reaction is fast?
No. ΔG only tells you whether a reaction is thermodynamically favourable, not how fast it is. Read the full answer in the Gibbs free energy lesson
PART 06Chemical kinetics
Why does the rate change during a reaction?
Reactants collide less often as they are used up, so the rate drops over time. Read the full answer in the Reaction rate lesson
How do coefficients affect rate expressions?
For aA → bB, the rate of reactant loss and product gain differ by the coefficients. Read the full answer in the Reaction rate lesson
Can I find the order from the balanced equation?
No. Read the full answer in the Rate law and reaction order lesson
What are the units of k?
The units of k depend on the overall order. Read the full answer in the Rate law and reaction order lesson
Why use the initial rate and not a later one?
At the start, the concentrations are known exactly and almost no product has formed. Read the full answer in the Method of initial rates lesson
How do I find the order if the rate does not change when I double a concentration?
If doubling a concentration leaves the rate unchanged, the order in that reactant is zero. Read the full answer in the Method of initial rates lesson
How do I tell the order from a graph?
Plot [A] against t, ln[A] against t, and 1/[A] against t. Read the full answer in the First-order integrated rate law lesson
What does the rate constant k tell me?
The rate constant measures how fast a reaction goes at a given temperature once the concentrations are accounted for. Read the full answer in the First-order integrated rate law lesson
Why is first-order half-life constant?
In first-order decay, the rate is proportional to the amount left. Read the full answer in the Half-life lesson
How many half-lives until a reaction is basically over?
After 5 half-lives, about 3 percent of the reactant remains, and after 7, less than 1 percent. Read the full answer in the Half-life lesson
Why does raising the temperature speed up reactions so much?
A small temperature rise increases the fraction of molecules that have at least the activation energy, and that fraction grows exponentially. Read the full answer in the Collision theory lesson
What is the steric factor?
The steric factor accounts for orientation. Read the full answer in the Collision theory lesson
How do I find Ea from experimental data?
Measure k at several temperatures. Read the full answer in the Activation energy and the Arrhenius equation lesson
Is activation energy the same as ΔH?
No. Read the full answer in the Activation energy and the Arrhenius equation lesson
What is the molecularity of an elementary step?
Molecularity is the number of molecules colliding in a single elementary step. Read the full answer in the Reaction mechanism lesson
What is an intermediate and how is it different from a catalyst?
An intermediate is made in one step and used up in a later step, so it does not appear in the overall equation. Read the full answer in the Reaction mechanism lesson
Do enzymes work the same way as other catalysts?
What is the difference between homogeneous and heterogeneous catalysts?
A homogeneous catalyst is in the same phase as the reactants, such as an acid dissolved in a liquid reaction. Read the full answer in the Catalyst lesson
PART 07Chemical equilibrium and solubility
Does equilibrium mean the concentrations are equal?
Can a reaction reach equilibrium in an open container?
Usually not. Read the full answer in the Dynamic equilibrium lesson
Why are solids and pure liquids left out of K?
Their concentration does not change in a meaningful way, because it is fixed by their density. Read the full answer in the Equilibrium constant lesson
What happens to K if I reverse the equation or multiply it?
If you reverse the reaction, the new K is 1/K. Read the full answer in the Equilibrium constant lesson
Why does Q < K mean the reaction goes forward?
If Q is smaller than K, there are too few products compared with equilibrium. Read the full answer in the Reaction quotient lesson
Can Q be calculated before any reaction has started?
When is Kp equal to Kc?
When the number of gas moles is the same on both sides, Δn = 0, and (RT) raised to the power zero is 1. Read the full answer in the Kp and Kc lesson
Which value of R do I use for Kp?
Use R = 0.08206 L·atm/(mol·K) when Kc uses mol/L and Kp uses atm, with T in kelvin. Read the full answer in the Kp and Kc lesson
When can I ignore x in an ICE table?
If K is very small (below about 10⁻³) compared with the starting concentration, then x is tiny and 1.0 − x is almost 1.0. Read the full answer in the ICE table lesson
What if the ICE equation needs a quadratic?
If the shortcut isn't allowed, expand the equation into ax² + bx + c = 0 and solve with the quadratic formula. Read the full answer in the ICE table lesson
Does a catalyst shift the equilibrium?
No. Read the full answer in the Le Chatelier's principle lesson
How does temperature change K?
Heating an endothermic reaction increases K, because heat acts like a reactant and shifts the equilibrium to products. Read the full answer in the Le Chatelier's principle lesson
What is the difference between ΔG and ΔG°?
ΔG° applies when all species are at standard conditions, 1 M or 1 bar. ΔG applies to the actual mixture at any time, and ΔG = ΔG° + RT ln Q. Read the full answer in the Free energy and the equilibrium constant lesson
If ΔG° is positive, does the reaction never happen?
No. Read the full answer in the Free energy and the equilibrium constant lesson
How do I tell if a precipitate will form?
Calculate the ion product Q using the concentrations in the mixture, and compare it with Ksp. Read the full answer in the Solubility product lesson
Can I compare Ksp values to rank solubility?
Only for salts that give the same number of ions, such as AgCl and AgBr. Read the full answer in the Solubility product lesson
Why does the common ion lower solubility?
Dissolving AgCl produces Ag⁺ and Cl⁻. Read the full answer in the Common ion effect lesson
Can a different ion increase solubility?
PART 08Acids, bases and electrochemistry
What makes an acid strong?
A strong acid ionizes almost completely in water, so nearly every molecule gives up its proton. Read the full answer in the Brønsted-Lowry acids and bases lesson
What is an amphoteric substance?
An amphoteric substance can act as an acid or a base depending on what it meets. Read the full answer in the Brønsted-Lowry acids and bases lesson
Why is neutral water pH 7 only at 25 °C?
Water's ionization is endothermic, so Kw rises with temperature. Read the full answer in the pH and Kw lesson
Can pH be negative or above 14?
Is a weak acid the same as a dilute acid?
How are Ka and Kb of a conjugate pair related?
For a conjugate acid-base pair, Ka × Kb = Kw = 1.0×10⁻¹⁴ at 25 °C. Read the full answer in the Weak acids and Ka lesson
What makes a good buffer?
A good buffer has similar amounts of weak acid and conjugate base, and a pKa within about 1 unit of the target pH. Read the full answer in the Buffer solution lesson
When does a buffer stop working?
A buffer fails once one of its two components is used up. Read the full answer in the Buffer solution lesson
What is the difference between the equivalence point and the endpoint?
The equivalence point is when moles of acid and base match exactly. Read the full answer in the Acid-base titration lesson
Why is the equivalence point above pH 7 for a weak acid with a strong base?
At equivalence, all the weak acid has become its conjugate base. Read the full answer in the Acid-base titration lesson
What does the salt bridge do?
The salt bridge lets ions move between the two half-cells. Read the full answer in the Galvanic cell lesson
How can I remember anode and cathode?
The mnemonic is AN OX and RED CAT: oxidation happens at the anode, and reduction happens at the cathode. Read the full answer in the Galvanic cell lesson
Do I multiply E° by the number of electrons?
No. Read the full answer in the Standard cell potential lesson
How do I know which half-cell is the cathode?
The half-cell with the higher (more positive) reduction potential is the cathode, because it is more strongly pulled toward being reduced. Read the full answer in the Standard cell potential lesson
Why does a battery go flat?
As the cell reacts, reactants are used up and products build up, so Q increases. Read the full answer in the Nernst equation lesson
How are E° and K related?
At equilibrium, E = 0 and Q = K, so E° = (RT/nF) ln K. Read the full answer in the Nernst equation lesson
What is the difference between a galvanic cell and an electrolytic cell?
A galvanic cell runs a spontaneous reaction and produces current. Read the full answer in the Electrolysis lesson
Why is molten NaCl electrolysed differently from salt water?
In molten NaCl, only Na⁺ and Cl⁻ are present, so sodium metal forms at the cathode and chlorine at the anode. Read the full answer in the Electrolysis lesson
PART 09Organic structure and stereochemistry
How do I find the functional group in a molecule?
Look for atoms other than carbon and hydrogen, and for double or triple bonds. Read the full answer in the Functional group lesson
Why does the functional group matter more than the carbon chain?
The carbon chain is mostly unreactive C–C and C–H bonds. Read the full answer in the Functional group lesson
Why do we number to give the lowest numbers to branches?
The rules need one agreed name for each molecule. Read the full answer in the IUPAC naming of alkanes lesson
What if there are two of the same branch?
Use di, tri or tetra before the branch name and give each position. Read the full answer in the IUPAC naming of alkanes lesson
Do isomers have the same properties?
No. Read the full answer in the Constitutional isomers lesson
How do I draw all the isomers of a formula?
Start with the longest straight chain, then shorten it by one carbon and add that carbon as a branch in each possible position. Read the full answer in the Constitutional isomers lesson
What is a Newman projection?
A Newman projection shows a molecule looking straight down one carbon-carbon bond. Read the full answer in the Conformations and torsional strain lesson
Are conformations the same as isomers?
Conformations of a molecule normally interconvert rapidly at room temperature by rotating about single bonds, so they can't be separated. Read the full answer in the Conformations and torsional strain lesson
What is a ring flip?
A ring flip is the quick change of one chair form into the other. Read the full answer in the Cyclohexane chair lesson
Why is a large group more stable when equatorial?
An axial group sits close to the two axial hydrogens on the same side of the ring, causing steric crowding called 1,3-diaxial strain. Read the full answer in the Cyclohexane chair lesson
Why can't double bonds rotate?
A double bond has a sigma bond and a pi bond. Read the full answer in the E/Z isomerism lesson
What is the difference between cis-trans and E/Z?
Cis-trans names work when each carbon has one identical or obvious group, such as the two methyl groups in but-2-ene. Read the full answer in the E/Z isomerism lesson
How do I spot a stereocentre?
Look for a carbon with four different groups attached, as in 2-butanol or lactic acid. Read the full answer in the Chirality and stereocentres lesson
Do chiral molecules have different physical properties?
Mirror-image molecules (enantiomers) have identical melting points, boiling points and solubility. Read the full answer in the Chirality and stereocentres lesson
What if the lowest-priority group points toward me?
If the lowest-priority group faces you, trace 1 to 2 to 3 as normal and then reverse the answer. Read the full answer in the R/S configuration lesson
How do I rank groups when the first atoms are the same?
Move outward to the next atoms and compare them in order, comparing the highest first. Read the full answer in the R/S configuration lesson
Why does a meso compound not rotate plane-polarised light?
A meso compound has an internal mirror plane, so one half of the molecule cancels the other half's effect on light. Read the full answer in the Enantiomers, diastereomers and meso compounds lesson
Do diastereomers have the same physical properties?
No. Read the full answer in the Enantiomers, diastereomers and meso compounds lesson
PART 10Organic reactions and mechanisms
How can I tell which species is the nucleophile?
Look for lone pairs, negative charges or pi bonds, because these are electron-rich. Read the full answer in the Nucleophiles, electrophiles and curved arrows lesson
Where should a curved arrow start and end?
The arrow starts where the electrons are, at a lone pair or a bond, and points to where they go, an atom or a new bond. Read the full answer in the Nucleophiles, electrophiles and curved arrows lesson
Why does SN2 work poorly on tertiary halides?
The nucleophile must reach the back of the carbon. Read the full answer in the SN2 reaction lesson
What makes a good leaving group?
A good leaving group is stable once it has left with the bonding electrons. Read the full answer in the SN2 reaction lesson
Why does SN1 give a racemic mixture?
The carbocation intermediate is flat. Read the full answer in the SN1 reaction lesson
How do I choose between SN1 and SN2?
Look at the substrate first. Read the full answer in the SN1 reaction lesson
What is a carbocation rearrangement?
A carbocation can shift a hydrogen or alkyl group from a neighbouring carbon to become more stable, for example turning a secondary cation into a tertiary one. Read the full answer in the Carbocation stability lesson
Why are allylic and benzylic cations extra stable?
In an allylic or benzylic cation, the positive charge sits next to a pi system and spreads over several atoms by resonance. Read the full answer in the Carbocation stability lesson
How do I decide between substitution and elimination?
Strong, bulky bases and heat favour elimination. Read the full answer in the E2 elimination lesson
What is Zaitsev's rule?
Zaitsev's rule says elimination gives mainly the most substituted alkene, which is the most stable. Read the full answer in the E2 elimination lesson
Why does Markovnikov addition happen?
The first step forms the more stable carbocation. Read the full answer in the Electrophilic addition to alkenes lesson
What changes when peroxides are present?
With peroxides, HBr adds by a radical path instead, and bromine ends up on the less substituted carbon. Read the full answer in the Electrophilic addition to alkenes lesson
Why doesn't benzene react like an alkene?
Addition to benzene would destroy the aromatic ring, losing its extra stability of about 150 kJ/mol. Read the full answer in the Electrophilic aromatic substitution lesson
What does an activating or directing group do?
Groups on the ring change both the speed and the position of the next substitution. Read the full answer in the Electrophilic aromatic substitution lesson
Why are aldehydes more reactive than ketones?
A ketone has two alkyl groups that donate electron density to the carbonyl carbon and make it less positive. Read the full answer in the Nucleophilic addition to carbonyls lesson
What do LiAlH₄ and NaBH₄ do?
Both are sources of hydride (H⁻) that reduce carbonyl compounds to alcohols. Read the full answer in the Nucleophilic addition to carbonyls lesson
Why do acid chlorides react faster than amides?
Chloride is a good leaving group and is weakly basic. Read the full answer in the Nucleophilic acyl substitution lesson
Why is Fischer esterification reversible?
Ester formation produces water, and the reverse reaction, hydrolysis, uses water to break the ester. Read the full answer in the Nucleophilic acyl substitution lesson
How can I stop at the aldehyde and not go on to the acid?
Use a mild oxidant such as PCC in dry solvent, or distil the aldehyde out as it forms, since it boils lower than the alcohol. Read the full answer in the Oxidation of alcohols lesson
Why can't tertiary alcohols be oxidised easily?
Oxidation needs a hydrogen on the carbon that holds the OH, so a C=O can form. Read the full answer in the Oxidation of alcohols lesson
Physics (school)
PART 01Motion and measurement
Why do we use SI units in physics?
Physics formulas only work when every quantity uses matching units. Read the full answer in the SI units lesson
What is the difference between a base unit and a derived unit?
A base unit stands alone, such as the metre, kilogram or second. Read the full answer in the SI units lesson
What is the difference between distance and displacement?
Distance is the total length of the path you travelled, and it has no direction. Read the full answer in the Scalar and vector lesson
How do you show a vector on a diagram?
Draw an arrow. Read the full answer in the Scalar and vector lesson
How do you calculate average speed?
Divide the total distance by the total time taken. Read the full answer in the Speed lesson
How do you change km/h into m/s?
Divide by 3.6. Read the full answer in the Speed lesson
Can an object have constant speed but changing velocity?
Why can velocity be negative?
The sign shows direction. Read the full answer in the Velocity lesson
What does a flat line on a distance-time graph mean?
The distance is not changing, so the object is not moving. Read the full answer in the Distance-time graph lesson
What does the gradient of a distance-time graph tell you?
Work out the gradient. Read the full answer in the Distance-time graph lesson
What does an acceleration of 2 m/s squared mean?
It means the velocity goes up by 2 m/s every second. Read the full answer in the Acceleration lesson
Is slowing down also acceleration?
What does the area under a velocity-time graph show?
The area is the distance travelled. Read the full answer in the Velocity-time graph lesson
What does a horizontal line on a velocity-time graph mean?
The velocity is not changing, so the acceleration is zero. Read the full answer in the Velocity-time graph lesson
When can I use v squared = u squared + 2as?
Use it only when acceleration stays constant, and the motion is in a straight line. Read the full answer in the Equations of uniform acceleration lesson
How do I choose the right motion equation?
Write down what you know and what you want. Read the full answer in the Equations of uniform acceleration lesson
PART 02Forces and Newton's laws
What is a newton?
A newton (N) is the unit of force. Read the full answer in the Force lesson
Can a force change the shape of an object?
How do you work out a resultant force?
Add forces that point the same way, and subtract forces that point opposite ways. Read the full answer in the Resultant force lesson
What does a zero resultant force mean?
The forces are balanced. Read the full answer in the Resultant force lesson
Why do I lurch forward when a bus brakes?
Your body wants to keep moving at the same speed, as Newton's first law says. Read the full answer in the Newton's first law lesson
Why do moving objects on Earth eventually stop?
Friction and air resistance are forces that act against the motion. Read the full answer in the Newton's first law lesson
How do I use F = ma?
Write down the values you know, with SI units. Read the full answer in the Newton's second law lesson
Why does a heavier object accelerate more slowly?
Mass measures how hard an object is to speed up. Read the full answer in the Newton's second law lesson
If forces come in equal pairs, why does anything move?
The two forces in a pair act on different objects, so they cannot cancel each other. Read the full answer in the Newton's third law lesson
Is the weight of a book and the table's push on it a third law pair?
What is the difference between mass and weight?
Mass is how much matter an object has, measured in kilograms. Read the full answer in the Weight and mass lesson
What value of g should I use?
On Earth, gravitational field strength is about 9.8 N/kg. Read the full answer in the Weight and mass lesson
Is friction always bad?
What makes air resistance bigger?
Higher speed and a larger, less streamlined shape both increase it. Read the full answer in the Friction and drag lesson
Why does a falling skydiver stop accelerating?
As speed rises, air resistance rises too. Read the full answer in the Terminal velocity lesson
Why does a parachute slow a skydiver down?
The open parachute has a huge area, so drag becomes much bigger than the weight. Read the full answer in the Terminal velocity lesson
PART 03Energy, work and power
What are the main energy stores?
The main stores are kinetic (moving), gravitational potential (height), elastic potential (stretched or squashed), chemical (food, fuel, batteries), thermal (heat in a material) and nuclear. Read the full answer in the Energy stores and transfers lesson
What is the unit of energy?
The joule (J). Read the full answer in the Energy stores and transfers lesson
If energy is never lost, why do things run out of energy?
Energy is not destroyed, but it spreads out into the surroundings as heat, which is hard to use again. Read the full answer in the Conservation of energy lesson
Where does the energy go when a ball stops bouncing?
It ends up as thermal energy in the ball, the floor and the air, plus a little sound. Read the full answer in the Conservation of energy lesson
Why does doubling speed quadruple kinetic energy?
Speed is squared in the formula KE = 0.5 x m x v squared. Read the full answer in the Kinetic energy lesson
What are the units in the kinetic energy formula?
Use mass in kilograms and speed in metres per second. Read the full answer in the Kinetic energy lesson
What does GPE depend on?
Three things: the mass of the object, the gravitational field strength, and how high the object is lifted. Read the full answer in the Gravitational potential energy lesson
How does GPE turn into kinetic energy?
As an object falls, its height drops, so gravitational energy falls. Read the full answer in the Gravitational potential energy lesson
Why is holding a heavy bag still not work in physics?
Work is force multiplied by distance moved. Read the full answer in the Work done lesson
How is work done linked to energy?
Work done is the energy transferred. Read the full answer in the Work done lesson
What is the difference between energy and power?
Energy is how much is transferred, in joules. Read the full answer in the Power lesson
How do I calculate power from work done?
Divide work done by the time it took: P = W / t. Read the full answer in the Power lesson
Can a machine be 100% efficient?
No real machine can. Read the full answer in the Efficiency lesson
How can you make a machine more efficient?
Reduce the energy wasted. Read the full answer in the Efficiency lesson
What is the difference between renewable and non-renewable resources?
Renewable resources, such as wind and sunlight, are replaced naturally as fast as we use them. Read the full answer in the Energy resources lesson
Why do we still use fossil fuels?
They are cheap, reliable and available at any time of day, and power stations that use them can be switched on quickly. Read the full answer in the Energy resources lesson
PART 04Particles and heat
Why does a steel ship float?
The ship is hollow and full of air, so its overall density, with the air included, is less than the density of water. Read the full answer in the Density lesson
How do you measure the density of an irregular object?
Find its mass using a balance. Read the full answer in the Density lesson
Why can you pour a liquid but not a solid?
In a liquid the particles are close but not locked in place, so they slide past each other and the liquid flows to take the shape of its container. Read the full answer in the States of matter lesson
Why can gases be squashed but liquids cannot?
In a gas the particles are far apart with lots of empty space, so pushing them closer is easy. Read the full answer in the States of matter lesson
Why does the temperature stay constant while ice melts?
The energy you add is used to break the bonds holding particles in the solid, not to make them move faster. Read the full answer in the Changes of state lesson
Is a change of state a chemical change?
No, it is a physical change. Read the full answer in the Changes of state lesson
What does a high specific heat capacity mean?
It means the material needs a lot of energy to warm up by one degree, and also releases a lot as it cools. Read the full answer in the Specific heat capacity lesson
How do I use E = mc(delta T)?
Write mass in kilograms, specific heat capacity in J per kg per degree, and the temperature change in degrees C. Read the full answer in the Specific heat capacity lesson
Why are metals good conductors of heat?
Metals contain free electrons that can move through the whole solid and carry energy quickly. Read the full answer in the Conduction lesson
What is an insulator?
An insulator is a material that conducts heat poorly. Read the full answer in the Conduction lesson
Why does hot air rise?
When air is heated its particles move faster and spread out, so the same mass fills a bigger volume. Read the full answer in the Convection lesson
Why can convection not happen in a solid?
Convection needs particles that can move from place to place carrying energy with them. Read the full answer in the Convection lesson
Why can heat reach us from the Sun through space?
Space is almost empty, so conduction and convection cannot carry the energy there. Read the full answer in the Thermal radiation lesson
Why are cars painted white in hot countries?
Light, shiny surfaces reflect radiation instead of absorbing it, so the car stays cooler. Read the full answer in the Thermal radiation lesson
Why does a gas's pressure rise when it is heated?
The particles gain energy and move faster. Read the full answer in the Gas pressure lesson
Why does squeezing a gas increase its pressure?
In a smaller volume the same particles hit the walls more often, because they have less room to travel between hits. Read the full answer in the Gas pressure lesson
PART 05Waves, sound and light
What is the difference between transverse and longitudinal waves?
In a transverse wave the vibration is at right angles to the direction the wave travels, like ripples on water. Read the full answer in the Waves lesson
Do waves carry matter?
What is the difference between amplitude and frequency?
Amplitude is how big the wave is, measured from the middle to a peak. Read the full answer in the Amplitude, wavelength and frequency lesson
What is the unit of frequency?
The hertz (Hz). Read the full answer in the Amplitude, wavelength and frequency lesson
How do I use v = f x wavelength?
Put frequency in hertz and wavelength in metres, then multiply to get speed in m/s. Read the full answer in the Wave speed equation lesson
If frequency increases, what happens to wavelength?
In a given material the wave speed is fixed. Read the full answer in the Wave speed equation lesson
What is the normal in a ray diagram?
The normal is an imaginary line drawn at 90 degrees to the surface at the point where the ray hits. Read the full answer in the Reflection lesson
Why can I see myself in a mirror but not on a wall?
A mirror is smooth, so parallel rays all bounce off at the same angle and form a clear image. Read the full answer in the Reflection lesson
Why does light bend when it enters glass?
Light travels more slowly in glass than in air. Read the full answer in the Refraction lesson
Why does a swimming pool look shallower than it is?
Light from the bottom bends away from the normal as it leaves the water into air. Read the full answer in the Refraction lesson
Which electromagnetic waves are dangerous?
Ultraviolet, X-rays and gamma rays carry enough energy to damage living cells, which can cause sunburn, skin cancer or other harm. Read the full answer in the Electromagnetic spectrum lesson
How fast do electromagnetic waves travel?
All electromagnetic waves travel at about 300,000,000 m/s in a vacuum, the speed of light. Read the full answer in the Electromagnetic spectrum lesson
Why can sound not travel through a vacuum?
Sound is made by particles bumping into each other and passing on a vibration. Read the full answer in the Sound lesson
Does sound travel faster in solids than in air?
What is ultrasound?
Ultrasound is sound with a frequency higher than 20,000 Hz, which humans cannot hear. Read the full answer in the Pitch and loudness lesson
Why do some animals hear sounds we cannot?
The range of hearing differs between species. Read the full answer in the Pitch and loudness lesson
PART 06Electric circuits
What is the difference between current and charge?
Charge is the amount of electricity that moves, measured in coulombs (C). Read the full answer in the Electric current lesson
Does current get used up in a circuit?
What does a voltmeter measure?
A voltmeter measures the potential difference across a component, which is how much energy each coulomb of charge gives to it. Read the full answer in the Potential difference lesson
What is the difference between voltage and current?
Voltage is the push that makes charge move, and current is how much charge actually flows each second. Read the full answer in the Potential difference lesson
What affects the resistance of a wire?
Length, thickness and material all matter. Read the full answer in the Resistance lesson
Why do wires get hot when current flows?
Moving electrons collide with the atoms of the wire and give them energy, so the wire warms up. Read the full answer in the Resistance lesson
How do I use V = IR?
Cover the quantity you want in the V-I-R triangle. Read the full answer in the Ohm's law lesson
What does a straight line through the origin on a V-I graph mean?
It shows the component obeys Ohm's law: the current is directly proportional to the voltage, so resistance is constant. Read the full answer in the Ohm's law lesson
What happens to a series circuit when one bulb breaks?
The loop is broken, so there is no path for charge to flow and every bulb goes out. Read the full answer in the Series circuits lesson
Why does adding more bulbs in series make them dimmer?
Each extra bulb adds more resistance, so total resistance goes up and the current falls. Read the full answer in the Series circuits lesson
Why does adding a branch in parallel lower total resistance?
A new branch gives the current another route to take. Read the full answer in the Parallel circuits lesson
Why are household appliances wired in parallel?
Each appliance gets the full supply voltage and can be switched on or off independently. Read the full answer in the Parallel circuits lesson
How do I calculate the energy used by an appliance?
Multiply its power by the time it is on. Read the full answer in the Electrical power lesson
Why do power cables carry electricity at high voltage?
For a given power, a higher voltage means a smaller current, since P = VI. Read the full answer in the Electrical power lesson
What does an LDR do?
A light-dependent resistor (LDR) has a high resistance in the dark and a low resistance in bright light. Read the full answer in the Circuit symbols and components lesson
Why is a diode useful?
A diode lets current flow in one direction only. Read the full answer in the Circuit symbols and components lesson
PART 07Mains electricity and magnetism
Which particles move when something becomes charged?
Only electrons move. Read the full answer in the Static electricity lesson
Why do charged objects attract small bits of paper?
The charged object pushes or pulls electrons within the paper, leaving the nearer side with the opposite charge. Read the full answer in the Static electricity lesson
What is the difference between a.c. and d.c.?
Direct current (d.c.) flows in one direction, as it does from a battery. Read the full answer in the Mains electricity lesson
Why is mains electricity dangerous?
A voltage of 230 V can push a current through the body that is high enough to stop the heart or cause burns. Read the full answer in the Mains electricity lesson
What does a fuse do?
A fuse contains a thin wire that heats up and melts if the current goes above its rating. Read the full answer in the Plug wires and fuses lesson
What colour is each wire in a UK plug?
The live wire is brown, the neutral wire is blue, and the earth wire is green and yellow stripes. Read the full answer in the Plug wires and fuses lesson
Why do some appliances not need an earth wire?
Appliances with plastic cases are double insulated, so the live wire cannot touch any metal a person might handle. Read the full answer in the Earth wire lesson
What happens if the live wire touches a metal case?
Without an earth wire, the case becomes live and a person touching it could be shocked. Read the full answer in the Earth wire lesson
How can I draw a magnetic field?
Draw lines leaving the north pole and entering the south pole, with arrows showing the direction. Read the full answer in the Magnetic fields lesson
Do like poles attract or repel?
Like poles repel and unlike poles attract. Read the full answer in the Magnetic fields lesson
How can I make an electromagnet stronger?
Wrap more turns of wire around the core, increase the current, or put a soft iron core inside the coil. Read the full answer in the Electromagnet lesson
What is the advantage of an electromagnet over a permanent magnet?
An electromagnet can be switched on and off, and its strength can be changed by altering the current. Read the full answer in the Electromagnet lesson
How can the force on a wire in a magnetic field be made bigger?
Increase the current, use a stronger magnet, or use a longer length of wire inside the field. Read the full answer in the Motor effect lesson
How does an electric motor keep spinning?
A coil of wire sits between two magnets. Read the full answer in the Motor effect lesson
How does a generator make electricity?
A coil spins in a magnetic field, or a magnet spins inside a coil. Read the full answer in the Electromagnetic induction lesson
How could you make a larger induced voltage?
Move the magnet or coil faster, use a stronger magnet, or add more turns to the coil. Read the full answer in the Electromagnetic induction lesson
PART 08Atoms and radioactivity
Why are atoms neutral overall?
A normal atom has the same number of protons as electrons. Read the full answer in the Structure of the atom lesson
Where is most of an atom's mass?
Almost all of an atom's mass is in the nucleus. Read the full answer in the Structure of the atom lesson
How do you find the number of neutrons?
Subtract the atomic number from the mass number. Read the full answer in the Atomic number and mass number lesson
What is an isotope?
Isotopes are atoms of the same element with the same number of protons but different numbers of neutrons. Read the full answer in the Atomic number and mass number lesson
Can you predict when an atom will decay?
What is background radiation?
Background radiation is low-level radiation that is always around us. Read the full answer in the Radioactive decay lesson
Which type of radiation is most ionising?
Alpha is the most ionising, because it is large and strongly charged, so it knocks electrons off atoms easily. Read the full answer in the Alpha, beta and gamma lesson
How does the nucleus change in alpha and beta decay?
In alpha decay the nucleus loses 2 protons and 2 neutrons, so its mass number drops by 4 and its atomic number by 2. Read the full answer in the Alpha, beta and gamma lesson
How do I do a half-life calculation?
Work out how many half-lives have passed by dividing the total time by the half-life. Read the full answer in the Half-life lesson
Does a radioactive sample ever disappear completely?
In theory the number of unstable nuclei keeps halving and never reaches exactly zero. Read the full answer in the Half-life lesson
How can people protect themselves from radiation?
Reduce the time exposed, stay as far away as possible and use shielding such as lead or thick concrete. Read the full answer in the Ionising radiation and safety lesson
What is the difference between irradiation and contamination?
Irradiation is exposure to radiation from a source outside you, and it stops when the source is removed. Read the full answer in the Ionising radiation and safety lesson
What is a chain reaction?
In a chain reaction, each fission releases neutrons that go on to split other nuclei, which release more neutrons. Read the full answer in the Nuclear fission lesson
What are the pros and cons of nuclear power?
Nuclear power makes lots of electricity from a small amount of fuel and releases no carbon dioxide while running. Read the full answer in the Nuclear fission lesson
Why is fusion so difficult on Earth?
Nuclei are positive, so they repel each other strongly. Read the full answer in the Nuclear fusion lesson
Why would fusion be better than fission?
Fusion fuel is hydrogen, which is plentiful, and its waste is far less radioactive and shorter lived. Read the full answer in the Nuclear fusion lesson
Physics (university)
PART 01Motion and vectors
Is displacement a vector or a scalar?
Displacement is a vector. Read the full answer in the Displacement lesson
Can displacement be zero if I walked a long way?
What is the difference between average and instantaneous velocity?
Average velocity is total displacement divided by total time for a whole trip. Read the full answer in the Velocity lesson
Why is velocity measured in metres per second?
Velocity is a change in position, measured in metres, divided by the time it takes, measured in seconds. Read the full answer in the Velocity lesson
Can an object have zero velocity and non-zero acceleration?
What does negative acceleration mean?
Negative acceleration means the acceleration points in the negative direction you chose. Read the full answer in the Acceleration lesson
When can I not use the constant-acceleration equations?
They only work when acceleration stays the same the whole time. Read the full answer in the Constant-acceleration equations lesson
How do I choose which kinematic equation to use?
List what you know and what you want. Read the full answer in the Constant-acceleration equations lesson
Do heavier objects fall faster?
Without air, no. Read the full answer in the Free fall lesson
Why do we use 9.8 m/s^2 for gravity?
It is the average acceleration of falling objects near Earth's surface. Read the full answer in the Free fall lesson
Why do we split vectors into components?
Perpendicular components do not affect each other, so you can solve each direction separately with simple algebra. Read the full answer in the Vector components lesson
How do I add two vectors that are at an angle?
Break each vector into x and y components. Read the full answer in the Vector components lesson
What launch angle gives the longest range?
On level ground with no air, 45 degrees gives the longest range. Read the full answer in the Projectile motion lesson
Does air resistance matter in projectile problems?
In real life yes, but most first-year problems ignore it so the maths stays simple. Read the full answer in the Projectile motion lesson
How do I find the velocity of A relative to B?
Subtract B's velocity from A's velocity as vectors. Read the full answer in the Relative velocity lesson
Why does rain look slanted from a moving car?
The rain falls straight down relative to the ground, but you also move forward. Read the full answer in the Relative velocity lesson
PART 02Newton's laws and forces
Why do objects need no force to keep moving?
In space or on very smooth ice, little pushes against motion, so objects drift on. Read the full answer in the Newton's first law lesson
What is inertia?
Inertia is an object's resistance to changes in its motion. Read the full answer in the Newton's first law lesson
Does zero net force mean no forces act?
How do I add forces that point in different directions?
Split each force into x and y components, then add the x parts and the y parts separately. Read the full answer in the Net force lesson
Does your mass change on the Moon?
Mass is the amount of matter in an object, measured in kilograms, and it never changes. Read the full answer in the Newton's second law lesson
Why does F = ma use net force?
Acceleration depends on the combined effect of all forces, not on any single one. Read the full answer in the Newton's second law lesson
If forces are equal and opposite, how does anything move?
The two forces act on different objects. Read the full answer in the Newton's third law lesson
Does a truck hitting a car exert more force on the car?
Why is weight different on the Moon?
Weight is mass times local gravity. Read the full answer in the Weight lesson
Is weight measured in kilograms?
In physics, no. Read the full answer in the Weight lesson
Is the normal force always equal to weight?
Why is it called normal?
In maths, normal means at right angles. Read the full answer in the Normal force lesson
What is the difference between static and kinetic friction?
Static friction holds an object that is not sliding. Read the full answer in the Friction lesson
Does friction depend on contact area?
In the simple model used in first-year courses, no. Read the full answer in the Friction lesson
What should I include in a free-body diagram?
Include only the forces acting on the chosen object, such as weight, normal force, tension, friction and applied pushes. Read the full answer in the Free-body diagram lesson
How do I pick axes for a slope problem?
Tilt the axes so one runs along the slope and one runs perpendicular to it. Read the full answer in the Free-body diagram lesson
Is the tension the same on both sides of a pulley?
For an ideal pulley with no mass and no friction, yes. Read the full answer in the Tension lesson
Can a rope push?
Why does the normal force on a slope equal mg cos theta?
The surface supports only the part of weight pressing into it. Read the full answer in the Inclined plane lesson
What angle makes a block start sliding?
A block begins to slide when the slope part of weight, mg sin theta, exceeds the maximum friction, coefficient times mg cos theta. Read the full answer in the Inclined plane lesson
PART 03Work, energy and momentum
Can a force do zero work?
Can work be negative?
Why is kinetic energy proportional to speed squared?
Kinetic energy equals the work needed to reach a speed. Read the full answer in the Kinetic energy lesson
Can kinetic energy be negative?
When should I use the work-energy theorem instead of F = ma?
Use it when you know distances and speeds but not time, or when the force changes along the path. Read the full answer in the Work-energy theorem lesson
Does the theorem work with friction?
Where do I put the zero of potential energy?
Anywhere you like. Read the full answer in the Gravitational potential energy lesson
Does the path matter for potential energy?
No. Read the full answer in the Gravitational potential energy lesson
Where does energy go when friction acts?
It turns into thermal energy, which warms the surfaces and the air a little. Read the full answer in the Conservation of energy lesson
Is a perpetual motion machine possible?
No. Read the full answer in the Conservation of energy lesson
How many joules does a 100 W bulb use in an hour?
Energy is the total amount transferred, in joules. Read the full answer in the Power lesson
How do I convert watts to horsepower?
One mechanical horsepower is about 746 watts. Read the full answer in the Power lesson
Why is momentum conserved?
It follows from Newton's third law. Read the full answer in the Momentum lesson
Can two objects have the same momentum but different kinetic energy?
Why do you bend your knees when you land?
Bending your knees makes the stopping time longer. Read the full answer in the Impulse lesson
What is the area under a force-time graph?
It equals the impulse, which is the change in momentum. Read the full answer in the Impulse lesson
What is the difference between elastic and inelastic collisions?
In both, momentum is conserved. Read the full answer in the Collisions lesson
How do I solve a two-object collision?
Choose a positive direction. Read the full answer in the Collisions lesson
PART 04Rotation and gravity
How do I convert rpm to rad/s?
One revolution is 2 pi radians, and one minute is 60 seconds. Read the full answer in the Angular velocity lesson
Why do physicists use radians instead of degrees?
Radians link angle directly to arc length: arc length equals radius times angle. Read the full answer in the Angular velocity lesson
Is centripetal acceleration different from centrifugal force?
Centripetal means centre-seeking and describes the real inward acceleration. Read the full answer in the Centripetal acceleration lesson
Does constant speed on a circle mean zero acceleration?
No. Read the full answer in the Centripetal acceleration lesson
Is centripetal force a new type of force?
What happens if the centripetal force suddenly disappears?
The object flies off in a straight line along the tangent to the circle, not outward along the radius. Read the full answer in the Centripetal force lesson
Why is a door handle far from the hinge?
Torque equals force times distance from the pivot. Read the full answer in the Torque lesson
Is torque a vector?
Why does a hollow cylinder roll slower than a solid one?
A hollow cylinder has its mass farther from the axis, so it has a bigger moment of inertia. Read the full answer in the Moment of inertia lesson
Does moment of inertia depend on the axis?
How is tau = I alpha like F = ma?
They are the same idea. Read the full answer in the Rotational second law lesson
How do I solve a pulley with a hanging mass?
Write F = ma for the hanging mass, with tension and weight. Read the full answer in the Rotational second law lesson
Why does a spinning top stay upright?
Its spin gives it large angular momentum. Read the full answer in the Angular momentum lesson
Why do cats land on their feet?
A falling cat has no outside torque, so total angular momentum stays near zero. Read the full answer in the Angular momentum lesson
Why is gravity weak between everyday objects?
The constant G is tiny, about 6.67e-11 in SI units. Read the full answer in the Universal gravitation lesson
How do I get g = 9.8 from Newton's law?
Set weight mg equal to GMm over R squared and cancel m. Read the full answer in the Universal gravitation lesson
Why don't satellites fall down?
They do fall, but they also move sideways so fast that the ground curves away beneath them. Read the full answer in the Orbital speed lesson
Does a higher orbit move faster or slower?
Is escape velocity the speed rockets must reach?
Not exactly. Read the full answer in the Escape velocity lesson
Why is a black hole's escape velocity special?
A black hole is so dense that its escape velocity at the surface called the event horizon exceeds the speed of light. Read the full answer in the Escape velocity lesson
Does Kepler's third law work for moons too?
Yes, for any set of objects orbiting the same central body. Read the full answer in the Kepler's third law lesson
What does Kepler's third law tell us about distance?
Outer planets are not just slower, they travel farther too. Read the full answer in the Kepler's third law lesson
PART 05Oscillations and waves
What makes motion simple harmonic?
The restoring force must be proportional to displacement and point back toward the centre, like F = -kx. Read the full answer in the Simple harmonic motion lesson
Where is the speed greatest in SHM?
The speed is greatest at the centre, where the mass passes through the equilibrium point. Read the full answer in the Simple harmonic motion lesson
What is the difference between frequency and angular frequency?
Frequency f counts cycles per second in hertz. Read the full answer in the Period and frequency lesson
How do I find frequency from a graph?
Read the time between two matching points, such as two peaks. Read the full answer in the Period and frequency lesson
Does gravity change the period of a vertical spring?
What is the spring constant?
The spring constant k measures stiffness in newtons per metre. Read the full answer in the Mass on a spring lesson
Why does the pendulum formula only work for small angles?
For small angles, the restoring force is nearly proportional to the angle, which gives SHM. Read the full answer in the Simple pendulum lesson
How can a pendulum measure g?
Time many swings of a pendulum of known length, find the period T, then rearrange T = 2 pi sqrt(L/g) to get g = 4 pi squared L over T squared. Read the full answer in the Simple pendulum lesson
Why can an opera singer break a glass?
Every glass has a natural frequency. Read the full answer in the Resonance lesson
What is damping?
Damping is energy loss that makes oscillations die away, usually through friction or air drag. Read the full answer in the Resonance lesson
Does a wave's speed depend on its frequency?
For most waves in a given medium, no. Read the full answer in the Wave speed lesson
How fast does sound travel?
In air at 20 degrees Celsius, sound moves at about 343 m/s. Read the full answer in the Wave speed lesson
Why can light travel through space but sound cannot?
Light is an electromagnetic wave. Read the full answer in the Transverse and longitudinal waves lesson
What is a compression?
A compression is a region in a longitudinal wave where particles are pushed close together, so the pressure is higher. Read the full answer in the Transverse and longitudinal waves lesson
What is the difference between constructive and destructive interference?
Constructive interference happens when waves meet in step, so their displacements add to a bigger wave. Read the full answer in the Superposition lesson
What is path difference?
Path difference is how much farther one wave travelled than another to reach the same point. Read the full answer in the Superposition lesson
What are nodes and antinodes?
Nodes are points on a standing wave that stay still, where the two waves always cancel. Read the full answer in the Standing waves lesson
Why do different instruments sound different at the same note?
Each instrument produces the main frequency plus a particular mix of higher harmonics. Read the full answer in the Standing waves lesson
Does the Doppler effect work for light?
Why does pitch change with the Doppler effect but loudness does not directly?
Pitch depends on how many wave crests reach your ear each second. Read the full answer in the Doppler effect lesson
Why do we use decibels?
Our ears respond to sound intensity on a roughly logarithmic scale. Read the full answer in the Sound intensity level lesson
How does intensity fall with distance?
For a small source spreading evenly, intensity equals power divided by 4 pi r squared. Read the full answer in the Sound intensity level lesson
PART 06Thermodynamics
What is the difference between heat and temperature?
Temperature is a measure of the average kinetic energy of particles. Read the full answer in the Temperature lesson
Why do we use kelvin in gas law problems?
The gas law needs a scale that starts at true zero. Read the full answer in the Temperature lesson
Why do bridges have expansion joints?
Materials get longer when the temperature rises. Read the full answer in the Thermal expansion lesson
Why does water behave unusually near freezing?
Most liquids shrink as they cool, but water is densest at about 4 degrees Celsius and then expands as it approaches freezing. Read the full answer in the Thermal expansion lesson
Why does sand get hot faster than water?
Sand has a specific heat of about 800 J/(kg K), roughly five times lower than water at 4186. Read the full answer in the Specific heat lesson
How do I solve a mixing problem?
Heat lost by the hot object equals heat gained by the cold one if nothing escapes. Read the full answer in the Specific heat lesson
Why does boiling water stay at 100 degrees Celsius?
During boiling, extra heat goes into breaking bonds between molecules, not into raising the speed of molecules. Read the full answer in the Latent heat lesson
Why does steam burn more than boiling water?
Steam at 100 degrees Celsius gives out its latent heat as it condenses on your skin. Read the full answer in the Latent heat lesson
Why do metals feel colder than wood at the same temperature?
Metal conducts heat away from your hand faster than wood does. Read the full answer in the Heat transfer lesson
How does a thermos flask keep drinks hot?
A vacuum layer between two walls stops conduction and convection. Read the full answer in the Heat transfer lesson
What makes a gas ideal?
An ideal gas has tiny particles that do not attract each other and collide perfectly elastically. Read the full answer in the Ideal gas law lesson
How do I use the ideal gas law with a changing gas?
For a fixed amount of gas, nR is constant, so P1V1/T1 equals P2V2/T2. Read the full answer in the Ideal gas law lesson
What is the speed of air molecules?
The root mean square speed is sqrt(3kT/m). Read the full answer in the Kinetic theory lesson
Why does a hot balloon push harder on its walls?
Hotter gas molecules move faster, so they hit the wall more often and with more force. Read the full answer in the Kinetic theory lesson
What is internal energy?
Internal energy is the total energy of the particles inside a system, from their motion and from forces between them. Read the full answer in the First law of thermodynamics lesson
What sign should work have in the first law?
In the form change in U = Q - W, work W is positive when the system does work on its surroundings, such as a gas expanding. Read the full answer in the First law of thermodynamics lesson
Why can't an engine be 100 percent efficient?
The second law says some heat must always flow to a cold reservoir. Read the full answer in the Heat engines lesson
What does a refrigerator do?
A refrigerator is a heat engine run backwards. Read the full answer in the Heat engines lesson
Does entropy ever decrease?
Entropy of a part of a system can decrease if it sends entropy elsewhere. Read the full answer in the Entropy lesson
Why does the second law give time a direction?
Events such as a dropped cup shattering can happen, but the reverse never does by itself, because mixing and spreading are far more probable than un-mixing. Read the full answer in the Entropy lesson
PART 07Electric fields and potential
How do objects become charged?
Objects become charged when electrons move from one to another, since protons stay in the nucleus. Read the full answer in the Electric charge lesson
What does conservation of charge mean?
The total charge in an isolated system never changes. Read the full answer in the Electric charge lesson
How does Coulomb's law compare with gravity?
Both fall off with 1/r squared. Read the full answer in the Coulomb's law lesson
How do I add forces from several charges?
The force on a charge is the vector sum of the forces from each other charge, which is the superposition principle. Read the full answer in the Coulomb's law lesson
What is a test charge?
A test charge is a tiny positive charge used to probe a field. Read the full answer in the Electric field lesson
What does the field of a dipole look like?
A dipole is a positive and a negative charge close together. Read the full answer in the Electric field lesson
Why can't field lines cross?
If two lines crossed, the field at the crossing point would have two directions at once. Read the full answer in the Field lines lesson
What do equipotential lines show?
Equipotential lines connect points at the same voltage. Read the full answer in the Field lines lesson
When is Gauss's law useful?
It is easiest when charges are spread in a symmetric shape, such as a sphere, a long line or a flat sheet. Read the full answer in the Gauss's law lesson
What is electric flux?
Flux measures how much field passes through a surface, calculated as E times area times cosine of the angle between the field and the surface's normal. Read the full answer in the Gauss's law lesson
Why is the field zero inside a conductor?
Free charges inside a metal move until they cancel any field. Read the full answer in the Conductors in equilibrium lesson
How does a Faraday cage work?
A metal cage lets charges rearrange on its surface so the field inside cancels. Read the full answer in the Conductors in equilibrium lesson
What is the difference between potential and potential energy?
Potential energy belongs to a particular charge in a field, in joules. Read the full answer in the Electric potential lesson
Can potential be zero where the field is not?
What is an electron-volt?
An electron-volt is the energy an electron gains going through one volt: 1.6e-19 J. Read the full answer in the Potential difference lesson
How do I find the speed of a charge accelerated through a voltage?
Set the energy gained equal to kinetic energy: qV = 1/2 m v squared. Read the full answer in the Potential difference lesson
Why is the field uniform between plates?
Each large flat sheet makes a field that does not change with distance, and the fields from two opposite sheets add between them and cancel outside. Read the full answer in the Uniform field between plates lesson
How do I find an electron's path between plates?
The force qE is constant, so the electron has constant acceleration a = eE/m toward the positive plate. Read the full answer in the Uniform field between plates lesson
Why do capacitors store energy?
Putting charge on plates takes work against the repulsion from charge already there. Read the full answer in the Capacitance lesson
What does a dielectric do?
A dielectric is an insulator placed between the plates. Read the full answer in the Capacitance lesson
PART 08Circuits and magnetism
Which way does current flow?
By convention, current is the direction positive charge would move, from the positive terminal to the negative. Read the full answer in the Electric current lesson
What is the speed of electrons in a wire?
Electrons drift slowly, often less than a millimetre per second. Read the full answer in the Electric current lesson
Do all materials obey Ohm's law?
How do length and thickness change a wire's resistance?
Resistance equals resistivity times length divided by cross-sectional area. Read the full answer in the Ohm's law lesson
Why is total resistance in parallel always smaller?
Adding a parallel branch gives current an extra path, like adding a lane to a road. Read the full answer in the Series and parallel resistors lesson
Why are home sockets wired in parallel?
Each device then gets the full mains voltage, and one switched off or broken does not stop the others. Read the full answer in the Series and parallel resistors lesson
How do I work out the cost of running a device?
Energy in kilowatt-hours equals power in kilowatts times hours. Read the full answer in the Electric power lesson
Why are power lines high voltage?
Power lost in a cable is I squared R. Read the full answer in the Electric power lesson
Why does the junction rule always hold?
Charge cannot pile up at a junction in steady current, and it cannot be created or destroyed. Read the full answer in the Kirchhoff's junction rule lesson
How do I use it with unknown currents?
Pick a direction for each unknown current, even if you guess wrong. Read the full answer in the Kirchhoff's junction rule lesson
What sign do I give a voltage drop?
Choose a direction around the loop. Read the full answer in the Kirchhoff's loop rule lesson
Why does the loop rule work?
The electric field of a steady circuit is conservative, so moving a charge once around a closed loop gives zero net energy change. Read the full answer in the Kirchhoff's loop rule lesson
What happens when a capacitor charges?
At first the capacitor is empty, so the current is large. Read the full answer in the RC circuit lesson
How long does a capacitor take to discharge?
The voltage falls as e to the power minus t over tau. Read the full answer in the RC circuit lesson
Can you have a magnet with only one pole?
No one has ever found one. Read the full answer in the Magnetic field lesson
What makes a magnet magnetic?
Spinning electrons inside atoms act like tiny magnets. Read the full answer in the Magnetic field lesson
How do I find the field direction around a wire?
Use the right-hand rule: point your right thumb along the current and your curled fingers show the field direction. Read the full answer in the Field of a long straight wire lesson
What is mu zero?
Mu zero is the permeability of free space, 4 pi times 10 to the minus 7 tesla metres per ampere. Read the full answer in the Field of a long straight wire lesson
Why does a magnetic force do no work?
The force always acts at right angles to the velocity, so it never pushes forward or back along the motion. Read the full answer in the Magnetic force on a charge lesson
What is the right-hand rule for force?
For a positive charge, point your fingers along the velocity, curl them toward the field direction, and your thumb shows the force. Read the full answer in the Magnetic force on a charge lesson
How does a mass spectrometer work?
Ions are sped up through a voltage, then enter a magnetic field and follow a circle with radius r = mv/(qB). Read the full answer in the Circular motion in a field lesson
Why do auroras form near the poles?
Charged particles from the Sun spiral along Earth's magnetic field lines. Read the full answer in the Circular motion in a field lesson
How does an electric motor use this force?
A coil of wire carrying current sits in a magnetic field. Read the full answer in the Force on a current-carrying wire lesson
How is this force related to the force on a single charge?
A wire holds many moving charges. Read the full answer in the Force on a current-carrying wire lesson
PART 09Induction, light and optics
Why does the angle matter in flux?
Only the part of the field perpendicular to the surface passes through it. Read the full answer in the Magnetic flux lesson
What changes flux in a loop?
Three things: the field strength B, the loop's area A, and the angle between the field and the loop. Read the full answer in the Magnetic flux lesson
Why does a static magnet not make current?
A steady flux gives zero rate of change, so no emf. Read the full answer in the Faraday's law lesson
How do induction cookers work?
An alternating current in a coil under the pan creates a changing magnetic field. Read the full answer in the Faraday's law lesson
Why does Lenz's law have a minus sign?
The minus sign shows that the induced emf opposes the change in flux. Read the full answer in the Lenz's law lesson
How do magnetic brakes work?
A moving magnet or metal disc induces eddy currents in nearby metal. Read the full answer in the Lenz's law lesson
Why does an inductor resist sudden changes in current?
A changing current changes its own magnetic flux, which induces a voltage that opposes the change, as Lenz's law says. Read the full answer in the Inductance lesson
Where is energy stored in an inductor?
In the magnetic field inside the coil, with energy 1/2 L I squared. Read the full answer in the Inductance lesson
Why do transformers only work with AC?
A transformer needs changing flux to induce a voltage in the second coil. Read the full answer in the Transformer lesson
Is a transformer 100 percent efficient?
Not quite, but large ones reach over 98 percent. Read the full answer in the Transformer lesson
How did Maxwell predict light?
Maxwell's equations showed that a changing electric field makes a magnetic field and vice versa. Read the full answer in the Electromagnetic wave lesson
Do EM waves need a medium?
What is the difference between radio waves and X-rays?
They are the same kind of wave with different frequency. Read the full answer in the Electromagnetic spectrum lesson
How is photon energy related to frequency?
Each photon carries energy E = h f, where h is Planck's constant, 6.63e-34 J s. Read the full answer in the Electromagnetic spectrum lesson
What is the refractive index?
The refractive index n is the ratio of the speed of light in a vacuum to the speed in the material, n = c/v. Read the full answer in the Snell's law lesson
Why does a straw look bent in water?
Light from the part under water bends as it leaves into air, so rays seem to come from a different place. Read the full answer in the Snell's law lesson
How does an optical fibre work?
Light enters a thin glass core and strikes the core wall beyond the critical angle, so it reflects completely each time. Read the full answer in the Total internal reflection lesson
Why can't total internal reflection happen from air to glass?
It needs light to move from a material with higher refractive index to a lower one. Read the full answer in the Total internal reflection lesson
What is the sign convention for the lens equation?
In the common textbook convention, real object and image distances are positive, and a positive focal length means a converging lens. Read the full answer in the Thin lens equation lesson
What is magnification?
Magnification m is the image height over the object height, and it equals minus di over do. Read the full answer in the Thin lens equation lesson
Why did the double-slit experiment matter?
In 1801 Thomas Young used it to show that light makes interference fringes, which only waves can do. Read the full answer in the Double-slit interference lesson
What happens to the fringes if I use red light instead of blue?
Red light has a longer wavelength than blue, and fringe spacing is lambda L over d. Read the full answer in the Double-slit interference lesson
PART 10Relativity and quantum physics
Why can nothing go faster than light?
As an object with mass speeds up, its energy grows as gamma times mc squared. Read the full answer in the Postulates of relativity lesson
Was Newton wrong?
Newton's laws are an excellent approximation at everyday speeds. Read the full answer in the Postulates of relativity lesson
Has time dilation been measured?
Does the moving observer notice their own clock running slow?
Does the ship really get shorter?
Observers measuring the ship moving at speed find a shorter length along the motion, and that is a real measurement in their frame. Read the full answer in the Length contraction lesson
Does length contraction happen sideways?
Where does the Sun get its energy?
In the Sun's core, four hydrogen nuclei fuse into one helium nucleus. Read the full answer in the Mass-energy equivalence lesson
Does a hot object weigh more than a cold one?
Technically yes, because energy has mass. Read the full answer in the Mass-energy equivalence lesson
Is light a wave or a particle?
It behaves as both, depending on the experiment. Read the full answer in the Photon lesson
What is an electron-volt for photons?
The electron-volt is a convenient energy unit for atoms, equal to 1.6e-19 J. Read the full answer in the Photon lesson
Why does brightness not matter below threshold?
Each electron absorbs a single photon in one go. Read the full answer in the Photoelectric effect lesson
What is the work function?
The work function is the minimum energy needed to free an electron from a metal's surface. Read the full answer in the Photoelectric effect lesson
Why don't we see everyday objects behave like waves?
Their wavelength is absurdly small. Read the full answer in the de Broglie wavelength lesson
How is it used in electron microscopes?
Electrons accelerated through thousands of volts have wavelengths below 0.01 nm, much shorter than visible light at 400 to 700 nm. Read the full answer in the de Broglie wavelength lesson
Is uncertainty caused by bad measuring tools?
No. Read the full answer in the Uncertainty principle lesson
Why do atoms not collapse?
If an electron fell into the nucleus, its position would be known very precisely, so its momentum and energy would be huge and uncertain. Read the full answer in the Uncertainty principle lesson
What does the wave function mean?
The wave function psi by itself is not directly measured. Read the full answer in the Wave function lesson
What is Schrodinger's equation?
It is the equation that tells you how the wave function changes with time. Read the full answer in the Wave function lesson
Why do elements give off specific colours?
Electrons can only have certain energy levels in an atom. Read the full answer in the Bohr energy levels lesson
What happens at the ionisation energy?
Hydrogen's ground state is -13.6 eV, so an electron needs 13.6 eV to be freed completely. Read the full answer in the Bohr energy levels lesson
Can we predict when one atom will decay?
No. Read the full answer in the Radioactive half-life lesson
How does carbon dating work?
Living things take in carbon-14, which decays with a half-life of about 5730 years. Read the full answer in the Radioactive half-life lesson
Maths (school)
PART 01Number and ratio
Do multiplication or division come first in BIDMAS?
Neither one is first. Read the full answer in the BIDMAS lesson
What does the I in BIDMAS stand for?
The I stands for indices, which means powers such as squares and cubes. Read the full answer in the BIDMAS lesson
How do you find a common denominator?
Find a number that both bottom numbers divide into. Read the full answer in the Adding fractions lesson
Do you add the denominators when adding fractions?
How do you find 10% of a number quickly?
Divide the number by 10. Read the full answer in the Percentage of an amount lesson
How do you turn a percentage into a decimal?
Divide by 100, which moves the decimal point two places left. Read the full answer in the Percentage of an amount lesson
What is the percentage change formula?
Work out the change (new minus old), divide it by the original value, then multiply by 100. Read the full answer in the Percentage change lesson
Why divide by the original amount and not the new one?
The percentage tells you how big the change is compared with where you started. Read the full answer in the Percentage change lesson
How do you share an amount in a ratio?
Add the ratio numbers to get the total parts, divide the amount by that total to find one part, then multiply by each ratio number. Read the full answer in the Ratio lesson
How do you simplify a ratio?
Divide every number in the ratio by the same number, just like simplifying a fraction. Read the full answer in the Ratio lesson
What is the unitary method?
It means finding the value of one item first, then scaling up. Read the full answer in the Direct proportion lesson
How can you tell if two things are in direct proportion?
Divide one amount by the other for each pair. Read the full answer in the Direct proportion lesson
What does a negative power mean?
A negative power means 'one over' the positive power. Read the full answer in the Laws of indices lesson
Why is any number to the power 0 equal to 1?
Use the division rule. Read the full answer in the Laws of indices lesson
How do you write a small number in standard form?
Move the decimal point right until the number is between 1 and 10, then count the moves. Read the full answer in the Standard form lesson
Why must the first number be between 1 and 10?
The rule keeps every number written the same way, so they are easy to compare. Read the full answer in the Standard form lesson
How do you find the HCF using prime factors?
Write each number as a product of primes, then multiply the primes they share, using the smaller power each time. Read the full answer in the HCF and LCM lesson
What is the difference between a factor and a multiple?
A factor divides exactly into a number: the factors of 12 include 3 and 4. Read the full answer in the HCF and LCM lesson
PART 02Algebra basics
What are like terms in algebra?
Like terms have the same letters raised to the same powers, so only the numbers in front differ. Read the full answer in the Simplifying expressions lesson
Can you add x and x^2?
No. Read the full answer in the Simplifying expressions lesson
Why do you square before multiplying in 2x^2?
Powers only apply to the thing directly next to them. Read the full answer in the Substitution lesson
What happens when you substitute a negative number?
Put the negative number in brackets so the sign is not lost. Read the full answer in the Substitution lesson
Do you multiply the number by everything inside the bracket?
What if there is a minus sign in front of the bracket?
A minus outside flips the sign of every term inside. Read the full answer in the Expanding brackets lesson
What does FOIL mean?
FOIL stands for First, Outer, Inner, Last. Read the full answer in the Double brackets lesson
Why do double brackets give four terms?
Each of the two terms in the first bracket multiplies each of the two terms in the second, so 2 x 2 = 4 products. Read the full answer in the Double brackets lesson
How do you check a factorised answer?
Expand it again. Read the full answer in the Factorising lesson
How do you factorise with letters, like 4x^2 + 6x?
Take out the biggest number and the most letters shared by both terms. Read the full answer in the Factorising lesson
What do the two numbers need to do?
They must multiply to give the last number and add to give the middle number. Read the full answer in the Factorising quadratics lesson
What if the last number is negative?
Then the two numbers have opposite signs, one positive and one negative. Read the full answer in the Factorising quadratics lesson
Why is there no middle term?
When you expand (x - 7)(x + 7), you get -7x and +7x. Read the full answer in the Difference of two squares lesson
Does it work for x^2 + 49?
No. Read the full answer in the Difference of two squares lesson
How do you find the nth term of a linear sequence?
Find the common difference between terms. Read the full answer in the nth term of a sequence lesson
How do you check if a number is in a sequence?
Put it equal to the nth term and solve for n. Read the full answer in the nth term of a sequence lesson
PART 03Equations and formulae
What is the first step in solving an equation?
Undo the addition or subtraction first, then undo the multiplication or division. Read the full answer in the Solving linear equations lesson
How do you check your answer to an equation?
Put your answer back into the original equation. Read the full answer in the Solving linear equations lesson
Which side should the x terms go on?
Either side works. Read the full answer in the Unknowns on both sides lesson
What if the equation has brackets?
Expand the brackets first, then collect the letter terms. Read the full answer in the Unknowns on both sides lesson
What does 'make x the subject' mean?
It means rearrange the formula so x is alone on one side, with everything else on the other. Read the full answer in the Changing the subject lesson
Do the rules for solving equations work on formulae?
Yes. Read the full answer in the Changing the subject lesson
When do you flip the inequality sign?
Flip the sign when you multiply or divide both sides by a negative number. Read the full answer in the Inequalities lesson
How do you show an inequality on a number line?
Draw an open circle for < or >, because the end number is not included. Read the full answer in the Inequalities lesson
What is the elimination method?
You add or subtract the two equations so that one letter disappears. Read the full answer in the Simultaneous equations lesson
How do you know the answers are correct?
Put both values into both original equations. Read the full answer in the Simultaneous equations lesson
Why must the equation equal zero first?
Because if two numbers multiply to zero, one of them must be zero. Read the full answer in the Solving quadratics by factorising lesson
Why does a quadratic have two answers?
The graph of a quadratic is a curve that can cross the x-axis in two places. Read the full answer in the Solving quadratics by factorising lesson
What is the quadratic formula?
It is x = (-b +/- sqrt(b^2 - 4ac)) / 2a. Read the full answer in the The quadratic formula lesson
What does the part under the square root tell you?
The expression b^2 - 4ac is called the discriminant. Read the full answer in the The quadratic formula lesson
How do you start a worded algebra problem?
Choose a letter for the unknown and write down what it stands for. Read the full answer in the Forming equations lesson
Why should you check the answer in the words?
Checking in the original wording catches mistakes in forming the equation, not just in solving it. Read the full answer in the Forming equations lesson
PART 04Graphs
Which number comes first in a coordinate pair?
The x value always comes first. Read the full answer in the Coordinates lesson
What are the four quadrants of a graph?
The x-axis and y-axis split the grid into four quadrants. Read the full answer in the Coordinates lesson
How do you calculate gradient from two points?
Divide the change in y by the change in x. Read the full answer in the Gradient lesson
What does a negative gradient mean?
A negative gradient means the line goes downhill when you read it from left to right. Read the full answer in the Gradient lesson
What is the y-intercept?
The y-intercept is the point where a line crosses the y-axis. Read the full answer in the y = mx + c lesson
How do you find the equation of a line from a graph?
Read the y-intercept for c. Read the full answer in the y = mx + c lesson
How do you find a perpendicular gradient?
Flip the fraction upside down and change the sign. Read the full answer in the Parallel and perpendicular lines lesson
Why do parallel lines have the same gradient?
The gradient measures steepness. Read the full answer in the Parallel and perpendicular lines lesson
What is the midpoint formula?
Add the two x values and divide by 2, then add the two y values and divide by 2. Read the full answer in the Midpoint of a line lesson
Can a midpoint have decimals or negative numbers?
What is the turning point of a quadratic graph?
The turning point is where the curve changes direction. Read the full answer in the Quadratic graphs lesson
What decides if the parabola is a U or an upside-down U?
The sign of the x^2 term. Read the full answer in the Quadratic graphs lesson
Why does y = 1/x never touch the axes?
You cannot divide by zero, so x = 0 has no answer. Read the full answer in the Cubic and reciprocal graphs lesson
What does a cubic graph look like?
A basic cubic such as y = x^3 passes through the origin and rises to the right while falling to the left, forming an S-like shape. Read the full answer in the Cubic and reciprocal graphs lesson
What does a flat line mean on a distance-time graph?
A flat horizontal line means the distance is not changing, so the object is stationary. Read the full answer in the Distance-time graphs lesson
How do you find speed from a distance-time graph?
Find the gradient of the line. Read the full answer in the Distance-time graphs lesson
PART 05Angles and area
Why do the angles in a triangle add to 180 degrees?
If you tear off the three corners of any triangle and line them up, they form a straight line, which is 180 degrees. Read the full answer in the Angle rules lesson
What are vertically opposite angles?
When two straight lines cross, they make four angles. Read the full answer in the Angle rules lesson
How do you spot alternate angles?
Look for a Z shape (or an S shape) made by the crossing line and the two parallel lines. Read the full answer in the Angles in parallel lines lesson
What is the difference between corresponding and co-interior angles?
Corresponding angles sit in the same position at each crossing, like an F shape, and are equal. Read the full answer in the Angles in parallel lines lesson
Why is the sum (n - 2) x 180 degrees?
From one corner, you can draw lines to split any polygon into n - 2 triangles. Read the full answer in the Angles in polygons lesson
What is a regular polygon?
A regular polygon has all sides the same length and all angles the same size. Read the full answer in the Angles in polygons lesson
What is the area of a triangle?
Half the base times the perpendicular height: area = 1/2 x b x h. Read the full answer in the Area of 2D shapes lesson
Why are areas measured in square units?
Area counts how many squares of a standard size fit inside a shape. Read the full answer in the Area of 2D shapes lesson
What is the difference between radius and diameter?
The radius goes from the centre to the edge. Read the full answer in the Circles lesson
What is pi?
Pi is the number you get when you divide a circle's circumference by its diameter, for every circle. Read the full answer in the Circles lesson
How do you find the area of a sector?
Work out the fraction of the circle: the angle divided by 360. Read the full answer in the Arcs and sectors lesson
What is the difference between an arc and a sector?
An arc is only the curved line along the edge, so its length is part of the circumference. Read the full answer in the Arcs and sectors lesson
What is the formula for volume of a cylinder?
Volume = pi x r^2 x h. Read the full answer in the Volume of prisms and cylinders lesson
Why are volumes in cubic units?
Volume counts how many unit cubes fit inside a solid, so you multiply three lengths. Read the full answer in the Volume of prisms and cylinders lesson
What is the angle in a semicircle?
It is always 90 degrees. Read the full answer in the Circle theorems lesson
What does the angle at the centre theorem say?
The angle at the centre of a circle is twice the angle at the edge when both are drawn from the same two points. Read the full answer in the Circle theorems lesson
PART 06Pythagoras and trigonometry
Which side is the hypotenuse?
The hypotenuse is the longest side of a right-angled triangle. Read the full answer in the Pythagoras' theorem lesson
When do you subtract in Pythagoras?
You subtract when you are finding one of the shorter sides and you know the hypotenuse. Read the full answer in the Pythagoras' theorem lesson
How do you find the distance between two points?
Subtract the x values to get the horizontal gap and subtract the y values for the vertical gap. Read the full answer in the Distance between two points lesson
Does it matter which point you subtract first?
No. Read the full answer in the Distance between two points lesson
How do you choose between sin, cos and tan?
Label the sides: hypotenuse, opposite and adjacent. Read the full answer in the Sine, cosine and tangent lesson
Does trigonometry only work for right-angled triangles?
The basic SOH CAH TOA ratios only work in right-angled triangles. Read the full answer in the Sine, cosine and tangent lesson
Why should your calculator be in degree mode?
Calculators can measure angles in degrees or radians. Read the full answer in the Finding a missing side lesson
What if the unknown side is on the bottom of the fraction?
Rearrange so the unknown is on top. Read the full answer in the Finding a missing side lesson
What does tan^-1 mean?
It is the inverse tangent. Read the full answer in the Finding a missing angle lesson
How do you know whether to use sin^-1, cos^-1 or tan^-1?
Pick the ratio using SOH CAH TOA, based on the two sides you know. Read the full answer in the Finding a missing angle lesson
Why do sin 30 and cos 60 give the same value?
In a right-angled triangle with angles of 30 and 60 degrees, the side opposite 30 is the side next to 60. Read the full answer in the Exact trigonometry values lesson
How can you remember tan 45 = 1?
A right-angled triangle with a 45 degree angle also has another 45 degree angle, so it is isosceles. Read the full answer in the Exact trigonometry values lesson
Where is the angle of elevation measured from?
From the horizontal line, up to the line of sight. Read the full answer in the Angles of elevation and depression lesson
Are the angle of elevation and depression equal?
Yes, when two people look at each other. Read the full answer in the Angles of elevation and depression lesson
When do you use the sine rule?
Use it when you know an angle and the side opposite it, plus one more angle or side. Read the full answer in the Sine rule lesson
How does the sine rule go when finding an angle?
Flip the rule so the sines are on top: sin A / a = sin B / b. Read the full answer in the Sine rule lesson
When do you use the cosine rule?
Use it when you know two sides and the angle between them, and you want the third side. Read the full answer in the Cosine rule lesson
What happens to the cosine rule when C is 90 degrees?
Since cos 90 = 0, the last part of the formula disappears. Read the full answer in the Cosine rule lesson
PART 07Shape and space
What stays the same in a reflection?
The shape keeps its size and angles. Read the full answer in the Reflection lesson
How do you describe a reflection fully?
Say it is a reflection and give the equation of the mirror line, such as 'reflection in the y-axis' or 'reflection in the line x = 2'. Read the full answer in the Reflection lesson
What three things describe a rotation?
You need the angle, the direction (clockwise or anticlockwise) and the centre of rotation. Read the full answer in the Rotation lesson
Does rotation change the size of a shape?
How do you write a translation?
Use a column vector with two numbers. Read the full answer in the Translation lesson
Does a translation turn the shape?
What does a scale factor less than 1 do?
It makes the shape smaller, so the enlargement is really a reduction. Read the full answer in the Enlargement lesson
How does enlargement change area?
If lengths are multiplied by a scale factor k, the area is multiplied by k squared. Read the full answer in the Enlargement lesson
How can you prove two triangles are congruent?
Show that one of the four rules holds: SSS, SAS, ASA, or RHS for right-angled triangles. Read the full answer in the Congruent shapes lesson
Why is AAA not enough to prove congruence?
Three equal angles only guarantee the same shape, not the same size. Read the full answer in the Congruent shapes lesson
How do you find a missing side in similar shapes?
Find the scale factor by dividing a known pair of matching sides (new over old). Read the full answer in the Similar shapes lesson
Why does area change by the scale factor squared?
Area has two dimensions, length and width, and both grow by the scale factor. Read the full answer in the Similar shapes lesson
How do you find a back bearing?
Add 180 if the original bearing is less than 180. Read the full answer in the Bearings lesson
Why do bearings have three digits?
Writing three digits keeps all bearings the same format and prevents confusion. Read the full answer in the Bearings lesson
How do you add two vectors?
Add the top numbers together and add the bottom numbers together. Read the full answer in the Vectors lesson
What does multiplying a vector by a number do?
It stretches or shrinks the vector, but keeps its direction. Read the full answer in the Vectors lesson
Why must you keep the compass width the same?
The arcs must be equal distances from each end, and this equal distance is what makes the new line a perfect bisector. Read the full answer in the Constructions lesson
Why should you leave your construction lines in?
Construction arcs show the examiner your method. Read the full answer in the Constructions lesson
PART 08Probability and statistics
How do you work out a probability?
Divide the number of ways your event can happen by the total number of equally likely outcomes. Read the full answer in the Probability scale lesson
Can a probability be bigger than 1?
Why do more trials give a better estimate?
With few trials, luck can distort the results. Read the full answer in the Relative frequency lesson
How do you predict how many times something will happen?
Multiply the probability by the number of trials. Read the full answer in the Relative frequency lesson
What does P(not A) equal?
It equals 1 - P(A). Read the full answer in the Mutually exclusive events lesson
When do you add probabilities?
Add when you want the chance of one event OR another and they cannot both happen. Read the full answer in the Mutually exclusive events lesson
Why do you multiply for independent events?
You want the chance of the first thing and then the second. Read the full answer in the Independent events lesson
What does 'and' mean in probability questions?
'And' means both events must happen, so you multiply. 'Or' means either can happen, so you add (if they cannot overlap). Read the full answer in the Independent events lesson
What changes when there is no replacement?
The total number of items goes down after the first pick, so the second probabilities change. Read the full answer in the Tree diagrams lesson
How do you find the chance of several outcomes on a tree diagram?
Multiply along each branch that leads to a wanted outcome, then add the results of the different branches. Read the full answer in the Tree diagrams lesson
How do you find the median with an even number of values?
Put the values in order, then take the mean of the two middle values. Read the full answer in the Averages lesson
Why might the median be better than the mean?
Extreme values pull the mean up or down, but they barely move the median. Read the full answer in the Averages lesson
Why do you multiply value by frequency?
Each value appears several times, so you count it that many times in the total. Read the full answer in the Mean from a frequency table lesson
What is a common mistake with frequency tables?
Dividing by the number of rows instead of the total frequency. Read the full answer in the Mean from a frequency table lesson
What is a line of best fit?
A straight line drawn through the middle of the points, with roughly equal points on each side. Read the full answer in the Scatter graphs and correlation lesson
Does correlation mean one thing causes the other?
No. Read the full answer in the Scatter graphs and correlation lesson
Maths (university)
PART 01Logic, sets and proof
What is the difference between a statement and an open sentence?
A statement has a fixed truth value: true or false. Read the full answer in the Statement lesson
Can a statement be both true and false in maths?
Why is 'if P then Q' true when P is false?
Because the only way to break the promise is P true with Q false. Read the full answer in the Implication lesson
How do you disprove an implication?
Find one counterexample: a case where P is true and Q is false. Read the full answer in the Implication lesson
What is the difference between the contrapositive and the converse?
The contrapositive of 'if P then Q' is 'if not Q then not P' and always has the same truth value as the original. Read the full answer in the Contrapositive lesson
When should I prove the contrapositive instead of the original?
Use it when the conclusion's negation gives you something concrete to work with. Read the full answer in the Contrapositive lesson
How do you negate a statement with 'for all'?
Change 'for all' to 'there exists' and negate what follows. Read the full answer in the Quantifiers lesson
Does the order of quantifiers matter?
Yes, a lot. 'For every person there is a birthday' is fine, but 'there is one day that is every person's birthday' is false. Read the full answer in the Quantifiers lesson
How do I start a direct proof?
Write down what you assume and what you must show. Read the full answer in the Direct proof lesson
Does checking examples count as a proof?
Why does proof by contradiction work?
A statement and its opposite cannot both be true. Read the full answer in the Proof by contradiction lesson
When should I use contradiction instead of a direct proof?
Choose it when the claim says something does not exist or cannot happen, since assuming it does gives you an object to work with. Read the full answer in the Proof by contradiction lesson
Why do I need the base case in induction?
The step only says 'if it works for k, it works for k + 1'. Read the full answer in the Induction lesson
What is the inductive hypothesis?
It is the assumption that the claim is true for n = k, for one fixed but arbitrary k. Read the full answer in the Induction lesson
What is the difference between a union and an intersection?
A union contains everything that is in at least one of the sets. Read the full answer in the Set operations lesson
What is the empty set and why does it matter?
The empty set has no elements and is written with a slashed circle or empty braces. Read the full answer in the Set operations lesson
How do I test whether a function is injective?
Assume f(a) = f(b) and try to show a = b. Read the full answer in the Injective and surjective lesson
Why does a bijection have an inverse?
Each output comes from exactly one input, so you can run the function backwards without any guesswork. Read the full answer in the Injective and surjective lesson
PART 02Limits and continuity
Does a limit have to equal the function's value?
What does it mean when a limit does not exist?
The values of f(x) near a do not settle on one number. Read the full answer in the Limit lesson
How do I write a one-sided limit?
Put a small plus or minus after the number. Read the full answer in the One-sided limit lesson
When do I need one-sided limits?
Use them for piecewise functions, absolute values, floor functions and square roots near the edge of their domain. Read the full answer in the One-sided limit lesson
Can I just substitute the value into the function?
For polynomials and other continuous functions, yes. Read the full answer in the Limit laws lesson
What can I do when substitution gives 0/0?
The form 0/0 is indeterminate, meaning the limit could be anything. Read the full answer in the Limit laws lesson
When should I use the squeeze theorem?
Use it when a function has a bounded wobbling part, such as sine or cosine, multiplied by something tending to zero. Read the full answer in the Squeeze theorem lesson
Why does the squeeze theorem work?
If g is below f and h is above f, and both g and h close in on the same number L, then f has nowhere else to go. Read the full answer in the Squeeze theorem lesson
How do I find the limit of a fraction as x goes to infinity?
Divide every term by the highest power of x in the denominator. Read the full answer in the Limit at infinity lesson
What is a horizontal asymptote?
It is a horizontal line y = L that the graph approaches as x goes to plus or minus infinity. Read the full answer in the Limit at infinity lesson
What are the three conditions for continuity at a point?
First, f(a) must be defined. Read the full answer in the Continuity lesson
Are polynomials continuous everywhere?
Can a removable discontinuity be fixed?
Yes. Read the full answer in the Types of discontinuity lesson
What is a jump discontinuity?
A jump happens when the left-hand and right-hand limits both exist but are different numbers. Read the full answer in the Types of discontinuity lesson
Why does the intermediate value theorem need continuity?
Without continuity the graph can jump over a value. Read the full answer in the Intermediate value theorem lesson
Does the theorem tell me where the root is?
No, it only says a root exists inside the interval. Read the full answer in the Intermediate value theorem lesson
How do I choose delta in an epsilon-delta proof?
Start from |f(x) - L| and rewrite it in terms of |x - a|. Read the full answer in the Epsilon-delta definition lesson
Why do we need such a formal definition of limit?
Words like 'gets close' are vague and can fail on tricky functions. Read the full answer in the Epsilon-delta definition lesson
PART 03Differentiation
What does the derivative actually measure?
It measures how quickly the output changes when the input changes by a tiny amount. Read the full answer in the Derivative lesson
Why is h in the definition of the derivative?
The number h is a small step from a to a + h. Read the full answer in the Derivative lesson
How do I find the equation of a tangent line?
Find the point (a, f(a)) and the slope f'(a). Read the full answer in the Tangent line lesson
What is a normal line?
The normal line passes through the same point as the tangent but at right angles to it. Read the full answer in the Tangent line lesson
How are position, velocity and acceleration related by derivatives?
Velocity is the derivative of position with respect to time. Read the full answer in the Rate of change lesson
What do the units of a derivative tell me?
The derivative's unit is the output unit divided by the input unit. Read the full answer in the Rate of change lesson
Does the power rule work for fractional powers?
Why does the derivative of a constant equal zero?
A constant does not change, so its rate of change is zero. Read the full answer in the Power rule lesson
Why is the derivative of a product not the product of derivatives?
Both factors change at once, and each change affects the total. Read the full answer in the Product rule lesson
How do I remember the product rule?
Say it in words: derivative of the first times the second, plus the first times derivative of the second. Read the full answer in the Product rule lesson
Do I have to use the quotient rule for every fraction?
Why is there a minus sign in the quotient rule?
Increasing the denominator makes the fraction smaller, so its change counts against the result. Read the full answer in the Quotient rule lesson
How do I spot when to use the chain rule?
Look for a function inside a bracket, a root, an exponent or a trig function, such as sin of x squared or e to the power of 3x. Read the full answer in the Chain rule lesson
What is the most common chain rule mistake?
Forgetting to multiply by the derivative of the inside. Read the full answer in the Chain rule lesson
Why is e to the x its own derivative?
The number e is defined so that e to the x grows at a rate equal to its current size. Read the full answer in the Standard derivatives lesson
What is the derivative of the natural logarithm?
The derivative of ln x is 1 over x for x greater than 0. Read the full answer in the Standard derivatives lesson
Why does y get a dy/dx when I differentiate implicitly?
Because y depends on x, so it is a function inside the equation. Read the full answer in the Implicit differentiation lesson
Can the slope formula contain both x and y?
Yes. Read the full answer in the Implicit differentiation lesson
Does continuity imply differentiability?
How can I tell where a function is not differentiable?
Look for corners, cusps, jumps and vertical tangents. Read the full answer in the Differentiability lesson
PART 04Applications of derivatives
Is every critical point a maximum or minimum?
Why do maxima and minima occur at critical points?
At a smooth peak or valley the tangent line is flat, so the slope is zero. Read the full answer in the Critical point lesson
What if f' does not change sign at a critical point?
Then it is neither a maximum nor a minimum. Read the full answer in the First derivative test lesson
How do I check the sign of f' between critical points?
Pick one test value in each interval and put it into f'. Read the full answer in the First derivative test lesson
What does the second derivative tell me?
It tells you how the slope is changing. Read the full answer in the Second derivative test lesson
What if f'' equals zero at a critical point?
The test gives no answer. Read the full answer in the Second derivative test lesson
What are the steps of an optimisation problem?
First, draw a picture and name the variables. Read the full answer in the Optimisation lesson
Why must I check the endpoints?
The best value may sit at the edge of the allowed range, where the derivative is not zero. Read the full answer in the Optimisation lesson
What does the mean value theorem say in plain words?
Somewhere between two points on a smooth curve, the tangent line is parallel to the straight line joining the two ends. Read the full answer in the Mean value theorem lesson
Why does the theorem need differentiability?
If the graph has a sharp corner, the tangent may never match the average slope. Read the full answer in the Mean value theorem lesson
How do I start a related rates problem?
Draw a diagram and name every changing quantity. Read the full answer in the Related rates lesson
Why not substitute the numbers before differentiating?
Quantities like the ladder's height change with time, so substituting a single value turns them into constants and removes their rates. Read the full answer in the Related rates lesson
Do I use the quotient rule in L'Hopital's rule?
When is L'Hopital's rule not allowed?
Only use it when the limit is 0/0 or infinity over infinity. Read the full answer in the L'Hopital's rule lesson
How accurate is linear approximation?
It is very good very close to a and worsens as you move away. Read the full answer in the Linear approximation lesson
Does the approximation overestimate or underestimate?
It depends on the bend. Read the full answer in the Linear approximation lesson
Why does Newton's method sometimes fail?
It can fail if f' is zero at a guess, since you divide by zero, or if the starting point is far from the root and the tangent sends you somewhere strange. Read the full answer in the Newton's method lesson
How fast does Newton's method converge?
Near a simple root it roughly doubles the number of correct digits each step. Read the full answer in the Newton's method lesson
PART 05Integration
Why do we add + C to an indefinite integral?
Many functions share the same derivative, because constants disappear when you differentiate. x squared, x squared plus 5 and x squared minus 3 all give 2x. Read the full answer in the Antiderivative lesson
How can I check an antiderivative is right?
Differentiate your answer. Read the full answer in the Antiderivative lesson
Why does the power rule for integrals fail when n is -1?
Dividing by n + 1 would mean dividing by zero. Read the full answer in the Power rule for integrals lesson
How do I integrate roots and fractions?
Rewrite them as powers first. Read the full answer in the Power rule for integrals lesson
Why does a Riemann sum overestimate here?
On [0, 3] the curve x squared is increasing, so the right-hand corner of each rectangle sits above the curve. Read the full answer in the Riemann sum lesson
How do I get the exact area from Riemann sums?
Let the number of rectangles grow without bound, so each width shrinks to zero. Read the full answer in the Riemann sum lesson
What does a negative definite integral mean?
It means the signed area below the x-axis is larger than that above. Read the full answer in the Definite integral lesson
What are the limits a and b?
They are the start and end of the interval you are adding up over. Read the full answer in the Definite integral lesson
Why is the fundamental theorem so important?
It shows that integration and differentiation undo each other. Read the full answer in the Fundamental theorem of calculus lesson
Do I need + C in a definite integral?
No. Read the full answer in the Fundamental theorem of calculus lesson
Why does the integral of sin x have a minus sign?
Because the derivative of cos x is minus sin x. Read the full answer in the Standard integrals lesson
Why is the integral of 1/x written with absolute value signs?
The natural logarithm only accepts positive inputs, but 1/x exists for negative x too. Read the full answer in the Standard integrals lesson
How do I choose u in a substitution?
Choose the inside of a bracket, root or exponent, especially if its derivative appears elsewhere in the integral. Read the full answer in the Integration by substitution lesson
What happens to the limits in a definite integral?
If you change variable, change the limits to match. Read the full answer in the Integration by substitution lesson
How do I know which curve is on top?
Pick a test value between the crossing points and compare the two functions there. Read the full answer in the Area between curves lesson
What if the curves cross more than once?
Split the integral at every crossing point. Read the full answer in the Area between curves lesson
Why divide by b minus a?
The integral is a total, like total distance. Read the full answer in the Average value lesson
Does the average value always occur at some point?
Yes, if f is continuous. Read the full answer in the Average value lesson
How do I find the true area when a curve crosses the axis?
Find where the curve crosses the x-axis, split the integral there, and take the absolute value of each piece before adding. Read the full answer in the Signed area and net change lesson
What does net change mean in practice?
It is the overall difference between the start and the end. Read the full answer in the Signed area and net change lesson
PART 06Integration techniques and uses
How is integration by parts related to the product rule?
Differentiating u times v gives u prime v plus u v prime. Read the full answer in the Integration by parts lesson
How do I decide what to call u?
Pick u as the part that gets simpler when differentiated, such as x or a logarithm. Read the full answer in the Integration by parts lesson
Why can't I integrate ln x directly by a table?
The natural log has no simple antiderivative from the standard list. Read the full answer in the Choosing u and dv lesson
What if integrating by parts makes things worse?
Swap your choice of u and dv. Read the full answer in the Choosing u and dv lesson
How do I find the numbers in a partial fraction split?
Write the split with unknown constants, such as A over x minus 1 plus B over x plus 1. Read the full answer in the Partial fractions lesson
When do I use partial fractions?
Use them when the integrand is a polynomial divided by a polynomial that factorises, and substitution does not work. Read the full answer in the Partial fractions lesson
Which identity helps with sine squared and cosine squared?
The double-angle identities: sin squared x equals one half times one minus cos 2x, and cos squared x equals one half times one plus cos 2x. Read the full answer in the Trigonometric integrals lesson
Why is the integral of sine squared not just sine cubed over 3?
That rule only works when the derivative of the inner function also appears, as in substitution. Read the full answer in the Trigonometric integrals lesson
How can an infinite area be finite?
The curve gets close to the x-axis fast enough that the thin tail adds up to a small amount. Read the full answer in the Improper integral lesson
What does it mean for an improper integral to diverge?
The limit either does not exist or is infinite, so the area has no finite value. Read the full answer in the Improper integral lesson
Why is the disc formula pi times f squared?
Each thin slice is a circle (disc) of radius f(x), and the area of a circle is pi times radius squared. Read the full answer in the Volume of revolution lesson
What if the region has a hole in the middle?
Use washers: discs with a hole. Read the full answer in the Volume of revolution lesson
Why use numerical integration when I have a formula?
Many functions, such as e to the minus x squared, have no antiderivative in terms of standard functions. Read the full answer in the Numerical integration lesson
Which is more accurate, the trapezoid rule or Simpson's rule?
Simpson's rule, usually. Read the full answer in the Numerical integration lesson
What is a differential equation?
It is an equation that links a function to its derivatives. Read the full answer in the Separable differential equation lesson
What is an initial condition for?
The general solution contains a constant, so there are infinitely many solutions. Read the full answer in the Separable differential equation lesson
Why isn't work just force times distance here?
That formula needs a constant force. Read the full answer in the Work as an integral lesson
What does Hooke's law say?
It says a spring's force is proportional to how far it is stretched or compressed, F equals k times x. Read the full answer in the Work as an integral lesson
PART 07Sequences and series
What does it mean for a sequence to converge?
A sequence converges if its terms approach one fixed number as n grows. Read the full answer in the Sequence limit lesson
How is a sequence different from a series?
A sequence is a list of numbers. Read the full answer in the Sequence limit lesson
How can infinitely many positive numbers add to a finite total?
If the terms shrink quickly enough, the running total stays below a ceiling. Read the full answer in the Partial sum lesson
What does it mean if the partial sums diverge?
It means the running total never settles on one number. Read the full answer in the Partial sum lesson
How do I find the ratio of a geometric series?
Divide any term by the one before it. Read the full answer in the Geometric series lesson
What happens if the ratio is 1 or bigger?
The terms do not shrink, so the total cannot settle. Read the full answer in the Geometric series lesson
If the terms go to zero, must the series converge?
Why must the terms of a convergent series go to zero?
If the running total settles on a number, consecutive totals get very close together. Read the full answer in the Divergence test lesson
Why does the harmonic series diverge if its terms go to zero?
The terms shrink too slowly. Read the full answer in the p-series lesson
How does the integral relate to p-series?
The integral test links the sum of 1 over n to the power p with the integral of 1 over x to the power p. Read the full answer in the p-series lesson
What if the ratio test gives L = 1?
The test is inconclusive. Read the full answer in the Ratio test lesson
Which series suit the ratio test best?
Those with factorials, such as 2 to the n over n factorial, or with exponentials like n over 3 to the n. Read the full answer in the Ratio test lesson
What is the difference between absolute and conditional convergence?
A series converges absolutely if the sum of the absolute values converges. Read the full answer in the Alternating series lesson
How accurate is a partial sum of an alternating series?
The error is smaller than the first term you left out, and the true sum sits between consecutive partial sums. Read the full answer in the Alternating series lesson
How do I find the radius of convergence?
Apply the ratio test to the terms, including the powers of x. Read the full answer in the Radius of convergence lesson
What happens at the endpoints of the interval?
The ratio test says nothing at the ends, so test them separately by substituting the endpoint values. Read the full answer in the Radius of convergence lesson
Where does the Taylor series formula come from?
Match a polynomial to the function at one point: its value, slope, and bend. Read the full answer in the Taylor series lesson
What is the difference between Taylor and Maclaurin series?
A Maclaurin series is a Taylor series centred at 0. Read the full answer in the Taylor series lesson
PART 08Vectors, matrices and linear systems
How do I add and scale vectors?
Add vectors entry by entry, so (1, 2) plus (3, 4) is (4, 6). Read the full answer in the Vector lesson
What is a unit vector?
A unit vector has length 1 and shows pure direction. Read the full answer in the Vector lesson
How do I use the dot product to find an angle?
Rearrange a dot b equals length of a times length of b times cos of the angle. Read the full answer in the Dot product lesson
What does a dot product of zero mean?
The two vectors are perpendicular (orthogonal), assuming neither is the zero vector. Read the full answer in the Dot product lesson
Is the cross product commutative?
Why does the cross product only work in 3D?
The standard cross product needs a unique direction perpendicular to two vectors. Read the full answer in the Cross product lesson
When can two matrices be added?
Only when they have the same size, since you add matching entries. Read the full answer in the Matrix lesson
What is the identity matrix?
A square matrix with ones down the diagonal and zeros elsewhere. Read the full answer in the Matrix lesson
Is matrix multiplication commutative?
No. Read the full answer in the Matrix multiplication lesson
When can I multiply two matrices?
The number of columns in the first matrix must equal the number of rows in the second. Read the full answer in the Matrix multiplication lesson
What does linear mean in a linear system?
Each unknown appears on its own, to the first power, multiplied only by a number. Read the full answer in the Linear system lesson
How do I solve a system by elimination?
Combine equations to remove one unknown. Read the full answer in the Linear system lesson
Which row operations are allowed?
You may swap two rows, multiply a row by a non-zero number, or add a multiple of one row to another. Read the full answer in the Augmented matrix lesson
Why do row operations keep the solutions the same?
Each operation matches a step you would use on the equations themselves, such as swapping their order or adding two equations. Read the full answer in the Augmented matrix lesson
What is back-substitution?
After elimination, the last row has only one unknown, so solve it first. Read the full answer in the Gaussian elimination lesson
Why use Gaussian elimination instead of guessing?
It always works and follows fixed steps, so a computer can do it for hundreds of equations. Read the full answer in the Gaussian elimination lesson
How can I tell there is no solution?
During elimination you reach a row of zeros on the left with a non-zero number on the right, such as 0 = 1. Read the full answer in the Number of solutions lesson
What is a free variable?
When elimination leaves fewer useful equations than unknowns, some unknowns can take any value, and the others depend on them. Read the full answer in the Number of solutions lesson
PART 09Determinants, inverses and vector spaces
What does a negative determinant mean?
The matrix flips space over, like looking in a mirror, as well as scaling it. Read the full answer in the Determinant lesson
What does a determinant of zero tell me?
The matrix squashes the plane onto a line or a point, so some area is lost and cannot be recovered. Read the full answer in the Determinant lesson
Which row should I expand along?
Choose the row or column with the most zeros, since each zero removes a whole smaller determinant from the work. Read the full answer in the Cofactor expansion lesson
How do the signs in cofactor expansion work?
The signs follow a checkerboard pattern starting with plus in the top left corner. Read the full answer in the Cofactor expansion lesson
When does a matrix have an inverse?
Exactly when it is square and its determinant is not zero. Read the full answer in the Inverse matrix lesson
How do I check my inverse is correct?
Multiply the matrix by the inverse you found. Read the full answer in the Inverse matrix lesson
Why multiply on the left by the inverse?
Matrix multiplication depends on order. Read the full answer in the Solving with the inverse lesson
Is using the inverse better than elimination?
For small systems it is fine, but for large ones elimination is faster and more accurate, because finding an inverse costs more work. Read the full answer in the Solving with the inverse lesson
Why must a subspace contain the zero vector?
Scaling any vector in the subspace by zero must stay inside it, and that gives the zero vector. Read the full answer in the Vector space lesson
Are polynomials a vector space?
How do I test linear independence?
Write a combination of the vectors equal to zero and solve for the coefficients. Read the full answer in the Linear independence lesson
Can three vectors in the plane be independent?
Is a basis for a space unique?
How do I find coordinates in a new basis?
Write the vector as a combination of the basis vectors and solve for the coefficients. Read the full answer in the Span and basis lesson
What is the rank-nullity theorem?
For a matrix with n columns, the rank plus the nullity equals n. Read the full answer in the Rank and dimension lesson
How do I find the rank of a matrix?
Do row operations until the matrix is in echelon form, then count the non-zero rows. Read the full answer in the Rank and dimension lesson
How do I find the matrix of a linear transformation?
See where the transformation sends each basis vector, and use those results as the columns. Read the full answer in the Linear transformation lesson
Why are translations not linear transformations?
A linear transformation must send the zero vector to itself. Read the full answer in the Linear transformation lesson
PART 10Eigenvalues and eigenvectors
Can the zero vector be an eigenvector?
No. Read the full answer in the Eigenvector and eigenvalue lesson
Does an eigenvector keep its direction?
Yes, up to a flip. Read the full answer in the Eigenvector and eigenvalue lesson
Why does det(A minus lambda I) equal zero?
We need A v equals lambda v for a non-zero v, which means A minus lambda I sends v to zero. Read the full answer in the Characteristic equation lesson
How many eigenvalues does an n by n matrix have?
Up to n, counting repeats, and allowing complex numbers. Read the full answer in the Characteristic equation lesson
Why are the rows of A minus lambda I dependent?
Because the determinant is zero by design. Read the full answer in the Finding eigenvectors lesson
Is an eigenvector unique?
Can a matrix have a zero eigenvalue?
Yes. Read the full answer in the Trace and determinant check lesson
How do I get a 2 by 2 matrix's eigenvalues from trace and determinant?
Solve L squared minus trace times L plus determinant equals zero. Read the full answer in the Trace and determinant check lesson
Which matrices can be diagonalised?
Those with a full set of independent eigenvectors, meaning n of them for an n by n matrix. Read the full answer in the Diagonalisation lesson
Why is the diagonal form useful?
A diagonal matrix is easy to work with: to square it, just square each diagonal entry. Read the full answer in the Diagonalisation lesson
What happens to A to the n as n gets large?
The eigenvalue of largest size dominates. Read the full answer in the Matrix powers lesson
Why is computing A to the n directly hard?
Multiplying a matrix by itself n times is slow and the numbers grow large. Read the full answer in the Matrix powers lesson
Why is the steady state an eigenvector with eigenvalue 1?
A steady state does not change when you apply the transition matrix once more, so P v equals v. Read the full answer in the Steady state lesson
Does every Markov chain have a steady state?
Every one has an eigenvalue of 1, so a steady distribution exists. Read the full answer in the Steady state lesson
Why do symmetric matrices have real eigenvalues?
This is a proven theorem, the spectral theorem. Read the full answer in the Symmetric matrix lesson
What does orthogonal eigenvectors mean in practice?
The eigenvector directions are at right angles, like the x and y axes. Read the full answer in the Symmetric matrix lesson
Why does a rotation have no real eigenvectors?
A real eigenvector would stay on the same line after the transformation. Read the full answer in the Complex eigenvalues lesson
What do complex eigenvalues tell me about a system?
They show that the system turns or oscillates instead of simply growing or shrinking. Read the full answer in the Complex eigenvalues lesson
Statistics (university)
PART 01Describing data
What is the difference between discrete and continuous data?
Discrete data takes separate values you can count, such as the number of siblings. Read the full answer in the Types of data lesson
Is a rating from 1 to 5 stars numerical or categorical?
It is usually called ordinal: categorical data with a natural order. Read the full answer in the Types of data lesson
How do you calculate the mean of a data set?
Add up every value, then divide by the number of values. Read the full answer in the Mean lesson
Why is the mean sensitive to extreme values?
Every value contributes its full size to the sum, so one huge value drags the total up. Read the full answer in the Mean lesson
How do you find the median when there is an even number of values?
Sort the data, then take the two middle values and find their mean. Read the full answer in the Median and mode lesson
Can a data set have more than one mode?
How can you tell skew from the mean and median?
Compare them. Read the full answer in the Skewness lesson
Which average should I report for skewed data?
Report the median, because it stays near the typical value while the mean is dragged toward the tail. Read the full answer in the Skewness lesson
How do you calculate standard deviation step by step?
Find the mean. Read the full answer in the Standard deviation lesson
Why do we square the deviations instead of just adding them?
Deviations from the mean always add up to zero, because positives cancel negatives. Read the full answer in the Standard deviation lesson
Why do we divide by n minus 1 for a sample?
The sample mean is calculated from the same data, so values sit closer to it than to the true mean. Read the full answer in the Sample variance lesson
What does n minus 1 mean in degrees of freedom?
Once the sample mean is known, only n minus 1 values are free to vary, because the last one must make the total match. Read the full answer in the Sample variance lesson
What does the interquartile range tell you?
It is the width of the middle 50 percent of the data, from Q1 to Q3. Read the full answer in the Quartiles and IQR lesson
Why might different software give slightly different quartiles?
There are several accepted rules for placing quartiles, especially when the count does not split evenly. Read the full answer in the Quartiles and IQR lesson
Should you always remove outliers from your data?
How does the 1.5 times IQR rule work?
Multiply the IQR by 1.5 and subtract that from Q1 to get the lower fence, then add it to Q3 for the upper fence. Read the full answer in the Outliers lesson
What does a z-score of 2 mean?
The value sits 2 standard deviations above the mean. Read the full answer in the z-score lesson
How do z-scores let you compare different tests?
Raw marks depend on each test's average and spread. Read the full answer in the z-score lesson
PART 02Probability
What is the range of a probability?
A probability is always between 0 and 1. Read the full answer in the Sample space and events lesson
What does equally likely outcomes mean?
It means each outcome has the same chance, like the six faces of a fair die. Read the full answer in the Sample space and events lesson
Why do you subtract P(A and B) in the addition rule?
Outcomes in both events get counted once in P(A) and again in P(B). Read the full answer in the Addition rule lesson
What is the complement rule used for?
It is a shortcut for questions with the words at least one. Read the full answer in the Addition rule lesson
Are mutually exclusive events independent?
No, usually the opposite. Read the full answer in the Mutually exclusive events lesson
How can I tell if two events are mutually exclusive?
Ask whether both can happen in the same trial. Read the full answer in the Mutually exclusive events lesson
If I flip heads five times, is tails more likely next?
How do you test whether two events are independent?
Check whether P(A and B) equals P(A) x P(B). Read the full answer in the Independent events lesson
Is P(A given B) the same as P(B given A)?
No, and mixing them up is a classic error. Read the full answer in the Conditional probability lesson
How does conditional probability change the sample space?
Once you know B happened, only outcomes inside B remain possible. Read the full answer in the Conditional probability lesson
When do I use the law of total probability?
Use it when the overall chance of an event is hard to find directly but easy within separate cases. Read the full answer in the Law of total probability lesson
Do the case probabilities have to add up to 1?
Yes. Read the full answer in the Law of total probability lesson
Why is a positive test often wrong for rare diseases?
Because healthy people vastly outnumber sick ones. Read the full answer in the Bayes' theorem lesson
What is a base rate and why does it matter?
The base rate is how common something is before any evidence. Read the full answer in the Bayes' theorem lesson
How do I decide between a permutation and a combination?
Ask whether swapping two chosen items gives a different result. Read the full answer in the Permutations and combinations lesson
Why divide by r factorial in the combination formula?
A permutation counts each group once for every way of ordering it. Read the full answer in the Permutations and combinations lesson
PART 03Random variables and discrete distributions
What is the difference between a random variable and a normal variable?
In algebra a variable stands for one unknown value. Read the full answer in the Random variable lesson
What is the difference between a discrete and a continuous random variable?
A discrete random variable takes countable values, such as 0, 1, 2 goals. Read the full answer in the Random variable lesson
Does the expected value have to be a possible outcome?
How is expected value different from the mean of a sample?
The expected value is a fixed property of the whole distribution, worked out from probabilities. Read the full answer in the Expected value lesson
Why is variance measured in squared units?
Variance averages squared gaps, so the units are squared too: metres become square metres. Read the full answer in the Variance of a random variable lesson
What does a variance of zero mean?
It means the variable never changes: every outcome equals the expected value. Read the full answer in the Variance of a random variable lesson
Why does adding a constant not change the variance?
Adding the same number to every value moves the whole distribution along the line, but the gaps between values stay the same. Read the full answer in the Linear rules for mean and variance lesson
Do variances add when you add random variables?
Only when the variables are independent. Read the full answer in the Linear rules for mean and variance lesson
What conditions must hold for a binomial model?
There is a fixed number of trials n. Read the full answer in the Binomial distribution lesson
Where does the nCk come from in the binomial formula?
It counts the different orders in which k successes can occur among n trials. Read the full answer in the Binomial distribution lesson
When should I use Poisson instead of binomial?
Use Poisson when you count events over time or space and there is no fixed number of trials, only an average rate. Read the full answer in the Poisson distribution lesson
What happens to lambda if I change the time window?
Lambda scales with the window. Read the full answer in the Poisson distribution lesson
Why is the mean of a geometric distribution 1 divided by p?
A rare success takes longer to arrive. Read the full answer in the Geometric distribution lesson
Does the geometric distribution have a memory?
No. Read the full answer in the Geometric distribution lesson
How do I find 'at least' probabilities from a CDF?
Use the complement. Read the full answer in the Cumulative distribution function lesson
Is P(X < 3) the same as P(X <= 3) for discrete variables?
No. Read the full answer in the Cumulative distribution function lesson
PART 04Continuous distributions and the normal curve
Why is the probability of an exact value zero for a continuous variable?
There are infinitely many possible values, so a single point has no width and no area under the curve. Read the full answer in the Probability density function lesson
What is the difference between a PDF and a probability?
The height of a PDF is a density, not a probability, and it can even exceed 1. Read the full answer in the Probability density function lesson
How do I find probabilities for a uniform distribution?
Find the width of the interval you want and divide by the total width b minus a. Read the full answer in the Uniform distribution lesson
Is the mean of a uniform distribution always the midpoint?
Yes. Read the full answer in the Uniform distribution lesson
How are the exponential and Poisson distributions related?
If events occur at random with rate lambda, the number in a fixed time is Poisson, and the gap between consecutive events is exponential. Read the full answer in the Exponential distribution lesson
Why is the exponential distribution called memoryless?
If you have already waited 5 minutes, the extra time you still must wait has the same distribution as a fresh wait. Read the full answer in the Exponential distribution lesson
Why does the bell curve appear so often?
When many small independent effects add together, the total tends to be bell-shaped, a result called the central limit theorem. Read the full answer in the Normal distribution lesson
What does it mean to write X ~ N(170, 64)?
It says X follows a normal distribution with mean 170 and variance 64. Read the full answer in the Normal distribution lesson
How do I use a z table?
Convert your value to a z-score, then look up that z in the table to read the area to its left. Read the full answer in the Standard normal and z-scores lesson
Why standardise before using normal tables?
There are infinitely many normal curves, one for each mean and SD, and no one can print them all. Read the full answer in the Standard normal and z-scores lesson
What is a percentile?
The kth percentile is the value below which k percent of the data falls. Read the full answer in the Normal probabilities and percentiles lesson
How do I find the middle 95 percent of a normal distribution?
Use z values of -1.96 and +1.96, which leave 2.5 percent in each tail. Read the full answer in the Normal probabilities and percentiles lesson
What is a continuity correction?
Binomial counts are whole numbers, but the normal curve is continuous. Read the full answer in the Normal approximation to the binomial lesson
When is the normal approximation to the binomial reliable?
Check that both np and n(1-p) are at least 10. Read the full answer in the Normal approximation to the binomial lesson
Why do variances add but not standard deviations?
For independent variables, the squared spreads combine like the sides of a right-angled triangle. Read the full answer in the Sums of normal variables lesson
Does the sum of two normal variables stay normal?
Yes, if they are independent. Read the full answer in the Sums of normal variables lesson
PART 05Sampling and estimation
What is sampling bias?
Sampling bias happens when the way you pick people makes some groups more likely to be included than others. Read the full answer in the Population and sample lesson
Why does a larger sample not fix a biased sample?
Size reduces random noise but not systematic error. Read the full answer in the Population and sample lesson
Does the law of large numbers mean results must balance out?
How is the law of large numbers different from the central limit theorem?
The law of large numbers says the sample mean approaches the true mean. Read the full answer in the Law of large numbers lesson
What is the difference between standard deviation and standard error?
Standard deviation measures how spread out individual values are. Read the full answer in the Sampling distribution and standard error lesson
Why do I need four times the data to halve the error?
Standard error shrinks with the square root of n, not with n itself. Read the full answer in the Sampling distribution and standard error lesson
How large must n be for the central limit theorem?
A common rule is 30 or more, but it depends on the shape. Read the full answer in the Central limit theorem lesson
Does the central limit theorem make the data normal?
No. Read the full answer in the Central limit theorem lesson
Is an unbiased estimator always the best?
Not always. Read the full answer in the Point estimate and bias lesson
Why is a point estimate not enough?
A single number gives no sense of how far off it might be. Read the full answer in the Point estimate and bias lesson
What does 95 percent confidence actually mean?
If you repeated the sampling many times and built an interval each time, about 95 percent of those intervals would contain the true mean. Read the full answer in the Confidence interval for a mean lesson
How can I make a confidence interval narrower?
Increase the sample size, which shrinks the standard error, or accept lower confidence, such as 90 percent instead of 95. Read the full answer in the Confidence interval for a mean lesson
Why is the t distribution wider than the normal?
The sample SD varies from sample to sample, which adds uncertainty beyond the sample mean. Read the full answer in the t distribution lesson
When does the t distribution become the same as the normal?
As degrees of freedom grow, the estimate s becomes very reliable, and the t curve approaches the normal curve. Read the full answer in the t distribution lesson
What is the margin of error in a poll?
It is the plus-or-minus part of a confidence interval, half its width. Read the full answer in the Confidence interval for a proportion lesson
How do I find the sample size needed for a given margin of error?
For a proportion, use n = z squared x p(1-p) / E squared, with p = 0.5 if unknown, as it gives the largest n. Read the full answer in the Confidence interval for a proportion lesson
PART 06Hypothesis testing
Why do we test the null instead of proving the alternative?
A single unusual result cannot prove a general claim, but data can be shown to clash with a precise default. Read the full answer in the Null and alternative hypotheses lesson
When should I use a one-sided test?
Use it only when you decided, before seeing the data, that you care about one direction, such as a new method being better. Read the full answer in the Null and alternative hypotheses lesson
What does a p-value of 0.03 mean?
If the null hypothesis were true, there would be a 3 percent chance of seeing data at least this extreme. Read the full answer in the p-value lesson
Does a small p-value mean the effect is large?
What happens when I run many tests at once?
Each test at alpha 0.05 carries a 5 percent false alarm chance, and the chances pile up. Read the full answer in the Significance level and decision lesson
Why say fail to reject instead of accept the null?
A non-significant result may simply mean the sample was too small to detect a real effect. Read the full answer in the Significance level and decision lesson
Can I reduce both types of error at the same time?
Not by changing alpha alone, since lowering alpha makes Type II errors more likely. Read the full answer in the Type I and Type II errors lesson
Which error is worse, Type I or Type II?
It depends on the cost. Read the full answer in the Type I and Type II errors lesson
How can I increase the power of a study?
Collect more data, since this shrinks the standard error. Read the full answer in the Power of a test lesson
What is the link between power and Type II error?
Power equals 1 minus the Type II error probability beta. Read the full answer in the Power of a test lesson
What assumptions does a t test make?
The data should be independent observations, and roughly normal when the sample is small. Read the full answer in the One-sample t test lesson
How does the t statistic relate to the confidence interval?
They give matching answers. Read the full answer in the One-sample t test lesson
When is a two-sample test the right choice?
Use it when the two groups contain different, unrelated people or items, such as patients given a drug versus patients given a placebo. Read the full answer in the Two-sample t test lesson
Why use Welch's test instead of assuming equal variances?
Welch's test works whether or not the groups have equal spread, and loses almost nothing when they do. Read the full answer in the Two-sample t test lesson
Why is a paired test more powerful than an unpaired one?
Pairing removes the variation between people, because each person acts as their own control. Read the full answer in the Paired t test lesson
How do I decide between paired and two-sample tests?
Ask whether each value in one group is naturally linked to one value in the other. Read the full answer in the Paired t test lesson
Why use p0 and not p-hat in the standard error?
A test assumes the null is true, so we measure the wobble that would occur if the share really were p0. Read the full answer in the Test for a proportion lesson
What conditions are needed for the z test for a proportion?
Observations should be independent, from a random sample. Read the full answer in the Test for a proportion lesson
PART 07Comparing categories and many groups
How do I read the chi-square statistic?
Each term measures how far a count is from its expected value, scaled by that expected value. Read the full answer in the Chi-square goodness of fit lesson
Can the chi-square statistic be negative?
No. Read the full answer in the Chi-square goodness of fit lesson
Why do expected counts need to be at least 5?
The chi-square test relies on a smooth curve approximating discrete counts. Read the full answer in the Expected counts lesson
What is the difference between observed and expected counts?
Observed counts are what you actually recorded. Read the full answer in the Expected counts lesson
How do I find the degrees of freedom for a table?
Multiply the number of rows minus 1 by the number of columns minus 1. Read the full answer in the Chi-square test of independence lesson
Does a significant chi-square show that one variable causes the other?
No. Read the full answer in the Chi-square test of independence lesson
Why not just run several t tests instead of ANOVA?
Each t test carries a 5 percent false alarm chance, and the chances add up. Read the full answer in the One-way ANOVA lesson
What are the null and alternative hypotheses in ANOVA?
The null says all group means are equal. Read the full answer in the One-way ANOVA lesson
What does an F value near 1 mean?
It means the variation between group means is about the same as the variation expected from noise within groups. Read the full answer in the F statistic lesson
What assumptions does ANOVA need?
Observations should be independent, each group roughly normal, and the groups should have similar variances. Read the full answer in the F statistic lesson
What is the multiple comparisons problem?
Each test has a chance of a false alarm, and running many tests multiplies the chances. Read the full answer in the Post-hoc comparisons lesson
What is the Bonferroni correction?
Divide the significance level by the number of tests. Read the full answer in the Post-hoc comparisons lesson
When should I use a nonparametric test?
Use one when your data is badly skewed, has strong outliers, is ordinal, or the sample is too small to trust normality. Read the full answer in the Rank-based tests lesson
Why does ranking make a test less sensitive to outliers?
A ranking records only order, not size. Read the full answer in the Rank-based tests lesson
Why report effect size as well as the p-value?
A p-value tells you whether a difference is unlikely to be chance, but not whether it is large enough to care about. Read the full answer in the Effect size lesson
How do I choose between a small, medium and large effect?
Cohen suggested 0.2, 0.5 and 0.8 as rough labels, but these are only guides. Read the full answer in the Effect size lesson
PART 08Correlation and regression
What does a correlation of 0.77 mean?
It means a fairly strong uphill straight-line link: as x goes up, y tends to go up. Read the full answer in the Correlation coefficient lesson
Can the correlation be zero when two variables are strongly related?
Yes. Read the full answer in the Correlation coefficient lesson
Why does the line minimise squared gaps?
Squaring stops positive and negative gaps from cancelling, and it punishes big misses more. Read the full answer in the Least squares regression line lesson
Why is extrapolating beyond the data risky?
The line only describes the range you observed. Read the full answer in the Least squares regression line lesson
How do I interpret the slope in words?
Say: for each one-unit increase in x, y is predicted to change by the slope, on average. Read the full answer in the Slope and intercept lesson
What if the intercept makes no sense?
Sometimes x = 0 is impossible or far outside the data, like a person with zero height. Read the full answer in the Slope and intercept lesson
What should a good residual plot look like?
A shapeless cloud of points around zero with roughly constant height. Read the full answer in the Residuals lesson
Why do residuals sum to zero?
The least squares line passes through the point of means, and the best-fitting intercept balances the points above against those below. Read the full answer in the Residuals lesson
Does a high R-squared mean the model is good?
Not necessarily. Read the full answer in the R-squared lesson
What is a good R-squared value?
It depends on the field. Read the full answer in the R-squared lesson
What is an influential point in regression?
It is a point that changes the fitted line a lot if removed, usually one far from the others in x. Read the full answer in the Regression assumptions lesson
What can I do if the assumptions are violated?
Try transforming the variables, for example with a logarithm, to straighten curves and stabilise spread. Read the full answer in the Regression assumptions lesson
Why does the slope test use n minus 2 degrees of freedom?
Fitting the line uses two estimated numbers, the intercept and the slope, so two degrees of freedom are used up. Read the full answer in the Inference for the slope lesson
How do I build a confidence interval for the slope?
Take the slope plus or minus the critical t value times the standard error, using n minus 2 degrees of freedom. Read the full answer in the Inference for the slope lesson
What does holding the other variables fixed mean?
Each coefficient shows the change linked to one predictor when the others stay constant. Read the full answer in the Multiple regression lesson
What is multicollinearity?
It happens when predictors are strongly correlated with each other, such as house area and number of rooms. Read the full answer in the Multiple regression lesson
How can I tell if a correlation is causal?
Look for a randomised experiment, a plausible mechanism, a consistent result across studies, and a cause that comes before the effect. Read the full answer in the Correlation and causation lesson
What is a confounding variable?
A confounder is a third variable linked to both the supposed cause and the effect, creating a misleading link. Read the full answer in the Correlation and causation lesson