What is a mathematical proof and why is it important in IB DP Mathematics?

A mathematical proof is a logical argument that demonstrates the truth of a given statement based on previously established statements, such as axioms and theorems. In IB DP Mathematics, proofs are crucial as they help students understand the underlying principles of mathematical concepts, ensuring a deeper comprehension beyond mere memorization. Proofs also develop critical thinking and problem-solving skills, which are essential for success in higher-level mathematics.

What are the different types of proofs commonly used in IB DP Mathematics?

In IB DP Mathematics, students encounter several types of proofs, including direct proofs, indirect proofs (such as proof by contradiction), and mathematical induction. Direct proofs involve straightforward logical deductions, while indirect proofs assume the opposite of the statement to derive a contradiction. Mathematical induction is used to prove statements about integers, particularly useful for sequences and series.

How can I effectively approach writing a proof in IB DP Mathematics?

To effectively write a proof in IB DP Mathematics, start by clearly understanding the statement you need to prove. Identify known information and relevant theorems. Plan your approach by deciding whether a direct, indirect, or induction proof is most suitable. Write your proof step-by-step, ensuring each step logically follows from the previous one. Finally, review your proof to ensure clarity and correctness, and check that it addresses all parts of the statement.

What is the difference between a theorem and a lemma in mathematical proofs?

In mathematical proofs, a theorem is a significant statement that has been proven based on previously established statements, such as other theorems, axioms, and lemmas. A lemma, on the other hand, is a minor result or an intermediate step used to prove a larger theorem. While theorems are often the main results, lemmas serve as building blocks that simplify the proof of more complex statements.

Why is proof by contradiction a powerful method in IB DP Mathematics?

Proof by contradiction is a powerful method in IB DP Mathematics because it allows students to prove statements by demonstrating that assuming the opposite leads to a logical inconsistency. This method is particularly useful when direct proof is challenging or when the statement involves negations or inequalities. By showing that the negation of a statement results in a contradiction, students can confidently conclude that the original statement must be true.

Why is "Doing the algebra but not stating 'as required' or '= RHS'." a common mistake in Proofs?

Why it happens: Trusting the reader to see it. How to avoid it: Always close with an explicit statement. Mark schemes award a mark for the conclusion line.

Why is "Using the conclusion to prove itself (circular)." a common mistake in Proofs?

Why it happens: Manipulating both sides to a common form without noting that the steps are reversible. How to avoid it: Manipulate ONE side. If you change both sides, justify why each step is reversible.

Why is "Stating a counterexample WITHOUT showing the calculation." a common mistake in Proofs?

Why it happens: Treating the example as obvious. How to avoid it: Always show that the example violates the claim. E.g. 'For n = 4: 2^4 - 1 = 15, which is composite (15 = 3 · 5).'

ProofsIB Maths AA SL

Proofs

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Proofs — IB Maths AA SL: direct algebraic proofs and proof by counterexample

AA SL is the IB DP's algebra/proof course. This note covers the core direct algebraic proofs, manipulation of identities, and the proof-by-counterexample method examiners love to ask on Paper 1.

What you’ll learn

Mapped to the IB DP Maths AA SL subject guide (2021 onwards (applies to 2026 exams)).

AO1 — Recall standard algebraic identities and definitions.
AO2 — Prove identities by manipulating one side until it matches the other.
AO2 — Disprove a universal claim with a single counterexample.
AO3 — Communicate each step with justification; conclude clearly.

Direct proof: LHS → RHS or RHS → LHS

Manipulate ONE side step by step.

Strategy.

Pick the side that's more complex (more algebra to do).
Manipulate it using identities, expansion, factoring, or known results.
Conclude when it equals the other side.

Worked example. Prove that $(x + y)^2 - (x - y)^2 = 4xy$ for all real $x, y$ .

LHS $= (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = 4xy =$ RHS. ✓

(Always state '= RHS' or '= so it follows' to close the proof clearly.)

Worked example (trig). Prove $\dfrac{\sin\theta}{1 + \cos\theta} + \dfrac{1 + \cos\theta}{\sin\theta} = \dfrac{2}{\sin\theta}$ .

LHS = combine: $\dfrac{\sin^2\theta + (1 + \cos\theta)^2}{\sin\theta(1 + \cos\theta)}$ .

Numerator: $\sin^2\theta + 1 + 2\cos\theta + \cos^2\theta = 1 + 1 + 2\cos\theta = 2 + 2\cos\theta = 2(1 + \cos\theta)$ .

So LHS $= \dfrac{2(1 + \cos\theta)}{\sin\theta(1 + \cos\theta)} = \dfrac{2}{\sin\theta} =$ RHS. ✓

Two-side method (when neither side is clearly simpler). Manipulate BOTH sides to a common form. Many students avoid this because it's harder to communicate — make sure each step is reversible.

Worked example (log). Prove $\log_a x = \dfrac{\ln x}{\ln a}$ for $a > 1$ , $x > 0$ .

Let $y = \log_a x$ , so $a^y = x$ by definition.

Take $\ln$ : $y \ln a = \ln x \Rightarrow y = \dfrac{\ln x}{\ln a}$ .

So $\log_a x = \dfrac{\ln x}{\ln a}$ . ✓

This is the change-of-base formula derived from first principles.

Pick the more complex side; manipulate to the simpler.
Each step needs a justification (rule used, identity invoked).
Close with '= RHS' explicitly.

Disproof by counterexample

One example is enough to demolish a universal claim.

A statement of the form 'for all $x$ , $P(x)$ ' is FALSE if there exists EVEN ONE $x$ for which $P(x)$ is false.

Worked example. Disprove: 'For all integers $n$ , $n^2 + n + 41$ is prime.'

Try $n = 40$ : $40^2 + 40 + 41 = 1681 = 41 \cdot 41$ . NOT prime.

(This is the famous Euler quadratic, prime for $n = 0, 1, \ldots, 39$ but composite at $n = 40$ .)

Worked example. Disprove: 'For all real $x$ , $\sqrt{x^2} = x$ .'

Try $x = -3$ : $\sqrt{(-3)^2} = \sqrt{9} = 3 \ne -3$ . Counterexample found.

(Correct statement: $\sqrt{x^2} = |x|$ .)

Worked example. Show that $\sin(A + B) \ne \sin A + \sin B$ in general.

Try $A = B = \dfrac{\pi}{2}$ : LHS $= \sin \pi = 0$ , RHS $= 1 + 1 = 2$ . So the statement fails.

Crucial point. A counterexample DISPROVES a universal statement but DOESN'T PROVE its negation. Many true statements are 'for all $x$ , $P(x)$ ' even though they fail at some special cases (e.g. log identities fail at $x \le 0$ — the statement should specify $x > 0$ ).

AO3 marks are explicitly awarded for stating the counterexample AND showing the calculation. Don't just write ' $n = 40$ '; SHOW that $n^2 + n + 41$ is composite there.

ONE valid counterexample disproves a 'for all' statement.
Show the calculation that demonstrates the failure.
Counterexample $\ne$ proof of opposite; just shows the original is too strong.

How it’s examined

Paper 1: 4–6-mark proofs of identities, especially trig and algebraic. Examiner reports stress that AO3 marks are lost when the conclusion isn't stated clearly. Counterexample questions test whether students can think critically — show the failing case explicitly.

Sources: IB Diploma Programme Mathematics: Analysis and Approaches subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Proofs

Step-by-step solutions to past-paper-style questions on proofs, written exactly the way a tutor would explain them at the board.

1Algebraic identity
Building confidence• algebra
Question
Prove that $(a + b)^3 - (a - b)^3 = 2b(3a^2 + b^2)$ for all real $a, b$ .
Step-by-step solution
1. Step 1
  Expand $(a+b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$ .
2. Step 2
  Expand $(a-b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3$ .
3. Step 3
  Subtract: $6a^2 b + 2b^3 = 2b(3a^2 + b^2)$ . = RHS ✓.
Answer
Proved.
2Trigonometric proof
Stretch• trig
Question
Prove $1 - 2\sin^2\theta = \cos^2\theta - \sin^2\theta$ for all $\theta$ .
Step-by-step solution
1. Step 1
  LHS $= 1 - 2\sin^2\theta$ .
2. Step 2
  Use Pythagoras: $1 = \sin^2\theta + \cos^2\theta$ .
3. Step 3
  Substitute: $\sin^2\theta + \cos^2\theta - 2\sin^2\theta = \cos^2\theta - \sin^2\theta$ = RHS ✓.
Answer
Proved using the Pythagorean identity.
3Counterexample — prime claim
Building confidence• counterexample
Question
Disprove: 'For all positive integers $n$ , $2^n - 1$ is prime.'
Step-by-step solution
1. Step 1
  Try small $n$ : $2^2 - 1 = 3$ ✓, $2^3 - 1 = 7$ ✓, $2^4 - 1 = 15 = 3 \cdot 5$ ✗.
2. Step 2
  Counterexample: $n = 4$ gives $2^4 - 1 = 15$ , which is composite (not prime).
Answer
Disproved by $n = 4$ since $2^4 - 1 = 15 = 3 \cdot 5$ .
4Counterexample — rational claim
Stretch• counterexample
Question
Disprove: 'The sum of two irrational numbers is always irrational.'
Step-by-step solution
1. Step 1
  Try $\sqrt{2}$ and $-\sqrt{2}$ , both irrational.
2. Step 2
  Sum: $\sqrt{2} + (-\sqrt{2}) = 0$ , which is RATIONAL.
Answer
Disproved by $\sqrt{2} + (-\sqrt{2}) = 0$ .

Key Definitions and Keywords — Proofs

Definitions to memorise and the exact keywords mark schemes credit for proofs answers — sharpened from recent examiner reports for the 2026 IB DP Maths AA SL sitting.

Direct proof
Examiner keyword
A proof that derives the conclusion from the assumption by a sequence of logical steps.
Counterexample
Examiner keyword
A specific case that satisfies the hypotheses of a universal claim but violates its conclusion.
Identity
An equation that holds for ALL values of the variables (e.g. $\sin^2 + \cos^2 = 1$ ).

Common Mistakes and Misconceptions — Proofs

The traps other students keep falling into on proofs questions — taken from recent IB DP Maths AA SL examiner reports and mark schemes — and how to avoid them.

✕Doing the algebra but not stating 'as required' or '= RHS'.
Why it happens
Trusting the reader to see it.
How to avoid it
Always close with an explicit statement. Mark schemes award a mark for the conclusion line.
✕Using the conclusion to prove itself (circular).
Why it happens
Manipulating both sides to a common form without noting that the steps are reversible.
How to avoid it
Manipulate ONE side. If you change both sides, justify why each step is reversible.
✕Stating a counterexample WITHOUT showing the calculation.
Why it happens
Treating the example as obvious.
How to avoid it
Always show that the example violates the claim. E.g. 'For $n = 4$ : $2^4 - 1 = 15$ , which is composite ( $15 = 3 \cdot 5$ ).'

Past paper style quiz

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Proofs — frequently asked questions

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Proofs

Short Study Notes in the form of Slides

Short Study Notes — Proofs

Detailed Notes

What you’ll learn

How it’s examined

1Algebraic identity

2Trigonometric proof

3Counterexample — prime claim

4Counterexample — rational claim

Direct proof

Counterexample

Identity

✕Doing the algebra but not stating 'as required' or '= RHS'.

✕Using the conclusion to prove itself (circular).

✕Stating a counterexample WITHOUT showing the calculation.

Past paper style quiz

Proofs — frequently asked questions