Proof (AA SL)
A mathematical proof is a chain of algebra that shows a statement is true for every case it claims to cover, not just the ones you happened to check. This topic covers how to lay out a simple deductive proof from a left-hand side to a right-hand side, how to represent general integers algebraically (even, odd, consecutive), and how a single counterexample is enough to disprove a claim that a proof would otherwise need to hold for infinitely many cases.
What the syllabus says
This topic maps onto one point in the official IB Analysis & Approaches syllabus.
| Code | Syllabus content |
|---|---|
| SL1.6 | Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof. The symbols and notation for equality and identity. |
This is a core AA syllabus point, and proof is expected to be woven throughout the course rather than tested only in isolation.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a deductive proof?
A deductive proof starts from a general algebraic representation of the situation and applies logical, justified steps until it reaches the stated result. It must hold for every possible case the statement covers, which is why specific numerical examples never count as a proof on their own.
e.g. To prove "the sum of two even numbers is even", start from \(2m+2n\), not from \(4+6=10\).
What does LHS to RHS mean?
LHS to RHS is the standard layout for an algebraic proof: you start on the left-hand side of the identity you're proving and manipulate it step by step until it matches the right-hand side exactly, showing each line of working.
e.g. Show \((x+1)^2-x^2=2x+1\): LHS \(=x^2+2x+1-x^2=2x+1=\) RHS.
How do you write a general even or odd number?
An even number is written \(2n\) for some integer \(n\); an odd number is written \(2n+1\) (or \(2n-1\)). Using this general algebraic form, rather than a specific value, is what allows the argument to cover every even or odd number at once.
e.g. \(n=4\) gives the even number \(2(4)=8\); \(n=4\) in \(2n+1\) gives the odd number \(9\).
What is a counterexample?
A counterexample is a single specific case where a claimed statement fails to hold. Because a general statement claims to be true for every case, finding just one case where it's false is enough to disprove it completely - no further checking is needed.
e.g. "\(n^2-n+11\) is always prime" fails at \(n=11\): \(11^2-11+11=121=11^2\), not prime.
What does "divisible by k" mean algebraically?
An expression is divisible by \(k\) if it can be written as \(k\) multiplied by an integer. Proving divisibility usually means factoring an expression until it has the form \(k\times(\text{something})\), where that something is shown to be an integer.
e.g. \(8n+4=4(2n+1)\) is divisible by 4, since \(2n+1\) is an integer whenever \(n\) is.
Key formulas
Proof doesn't use a formula in the way other topics do - it's a technique. The table below gives the standard algebraic representations you'll use as building blocks, and the sections after that walk through how they're combined.
Formula reference
None of these are formulas in the formula-booklet sense - they're conventions for representing integers algebraically, and you're expected to know and use them without prompting.
| Representation | Used for | Booklet? |
|---|---|---|
| \(2n\), \(n\in\mathbb{Z}\) | A general even number | Not in booklet |
| \(2n+1\), \(n\in\mathbb{Z}\) | A general odd number | Not in booklet |
| \(n,\ n+1,\ n+2,\ldots\) | Consecutive integers | Not in booklet |
| \(a=bq\), \(q\in\mathbb{Z}\) | "\(a\) is divisible by \(b\)" | Not in booklet - prior knowledge |
Proof vs disproof
Proving a general statement and disproving one require opposite amounts of work - this table sets out the contrast.
| Feature | To prove | To disprove |
|---|---|---|
| What's needed | Every case must hold | One failing case is enough |
| Method | General algebraic argument | A specific counterexample |
| Starting point | A general variable, e.g. \(n\) | A chosen numerical value |
| Conclusion phrase | "...for all \(n\), hence proven" | "...so the statement is false" |
Laying out a proof
Examiners are marking the logic as much as the algebra - a correct final line with unjustified steps in between loses marks.
Start general
Represent the objects in the statement algebraically before doing anything else - e.g. two odd numbers as \(2m+1\) and \(2n+1\), not as \(3\) and \(5\).
A proof about "any" number must start from a variable, not a value.
Show every step
Each line should follow clearly from the one before, usually by expanding, factoring, or substituting. Skipping algebra is where most marks are lost, even when the final answer is correct.
Method (M1) and accuracy (A1) marks are typically awarded separately for each step.
Conclude explicitly
Finish with a sentence linking the algebra back to the original claim - e.g. "since \(2n+1\) is an integer, \(4(2n+1)\) is a multiple of 4, so the difference is divisible by 4."
This reasoning line (R1) is often worth a separate mark from the algebra itself.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Consider \((a+b)^2\) and \((a-b)^2.\)
(a) Expand them.
(b) Hence prove that \((a+b)^2 - (a-b)^2 = 4ab.\)
Worked solution
(a) Expand both squares. \((a+b)^2 = a^2+2ab+b^2\) A1
\((a-b)^2 = a^2-2ab+b^2.\) A1
(b) Subtract to prove the identity. Using part (a): \((a^2+2ab+b^2)-(a^2-2ab+b^2) = 4ab.\) M1
The \(a^2\) and \(b^2\) terms cancel and the \(2ab\) terms add, giving \(4ab\). AG ∎
The odd numbers \(1, 3, 5, \dots\) form an arithmetic sequence. Prove that the sum of the first \(n\) odd numbers equals \(n^2\).
Worked solution
First term \(u_1 = 1\), common difference \(d = 2\), so the \(n\)th odd number is \(u_n = 1 + (n-1)\cdot 2 = 2n-1.\) M1
\(S_n = \tfrac{n}{2}(u_1+u_n)\): M1
\(S_n = \frac{n}{2}\big(1 + (2n-1)\big) = \frac{n}{2}(2n).\) A1
\(\dfrac{n}{2}(2n) = n^2.\) A1 The cancellation of \(2\) is exact, so the sum equals \(n^2\) for every \(n\). R1 Therefore the sum of the first \(n\) odd numbers is \(n^2\). AG ∎
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Proving with a specific example instead of a general one. Showing "\(4+6=10\) is even" checks one case; it doesn't prove that the sum of any two even numbers is even. Start from \(2m\) and \(2n\), not from chosen numbers.
- Skipping algebraic steps between the start and the stated result. Even when the final line is given (AG), every intermediate step still needs to be shown - jumping straight from the setup to the answer loses the method marks in between.
- Forgetting the reasoning line at the end. Reaching \(4(2n+1)\) isn't itself a proof of divisibility by 4 - you also need to state that \(2n+1\) is an integer, so the whole expression is a multiple of 4.
- Trying to "prove" a false statement instead of disproving it. If a question asks you to test or disprove a claim, look for a counterexample first - don't waste time attempting a general proof of something that isn't actually true for every case.
Using your GDC
Proof questions are set on Paper 1, without a calculator, and a GDC can't write or check the logic of an algebraic argument for you. Where it does help is exploring a conjecture before you commit to proving (or disproving) it - trying several values quickly to spot a pattern or hunt for a counterexample. Pick your model to filter down to just the steps that apply to you.
Useful for testing a claimed identity or equation against several values quickly, before you commit to writing out the general algebraic proof by hand.
- Graph \(f(x)\) first so you can see how many solutions exist and roughly where they are.
- Rearrange so everything is on one side: \(f(x) = 0\) - or graph both sides as separate functions and find intersections.
- MATH → Solver: enter the expression, type a starting guess close to one root, press ALPHA + ENTER. Move the guess to near a different root and repeat for each solution.TI-84
- Type nSolve(f(x)=0, x, guess) - include a guess or interval e.g. nSolve(f(x)=0, x, 2) or nSolve(f(x)=0, x, {1,5}) to target a specific root.Nspire
- Run-Matrix → SolveN(f(x), x) returns all real roots at once; or use the Equation app for a visual approach.Casio
- Always verify each solution by substituting back into the original equation.
Tip: This is exploratory only - the calculator can help you find a counterexample or check a pattern, but the final written proof must still be algebraic, since no calculator is allowed on Paper 1.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Proof questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What does "prove" actually require in an IB answer?
A logically complete chain of algebra from a general starting point to the stated result, with every step justified - not just a single numerical check. Testing one value only verifies a case; it never proves a general statement.
How do I represent "an even number" or "an odd number" algebraically?
An even number is \(2n\) for some integer \(n\), and an odd number is \(2n+1\) (or \(2n-1\)). Using these general forms - rather than a specific number like 4 or 7 - is what makes the argument a proof rather than an example.
What's the difference between AG and a normal mark?
AG ("answer given") appears next to a result you were told to show - no mark is awarded for stating it, because it was given in the question. The marks come from the algebra that gets you there, so the working must be complete even though the destination isn't in doubt.
How do I disprove a statement?
Find a single counterexample - one value for which the statement fails - and show the calculation clearly. A general claim only needs one failure to be false, unlike a proof, which needs every case to hold.
Sub-topics
Proof broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AA SL syllabus unit, in case you want to keep going.