Topic 1.15 · HL only

Proof by contradiction: notes and practice questions

Summary

Proof by contradiction assumes the negation of a statement is true.
This assumption is used to derive a mathematical impossibility (a contradiction).
The contradiction proves the initial assumption false, thus validating the original statement.
Common applications include proving irrationality (e.g., $\sqrt{2}$ ), properties of logarithms (e.g., $\log_2 3$ ), and number theory statements (e.g., if $n^2$ is even, then $n$ is even).
A key step is often defining a number as a fraction in simplest form, then showing it leads to a shared factor, contradicting the coprime assumption.
Euclid's proof for the infinitude of primes uses this method by constructing a number that must have a prime factor outside the assumed finite list.

How it is examined

Induction is marked on structure, not just on algebra: state the proposition, prove the base case, assume for $n = k$ , prove for $n = k+1$ , and write a conclusion that names the base case and the inductive step. Examiners award a mark for that final sentence and students routinely drop it. 6 to 8 marks, Paper 1.

Key ideas

Proof by mathematical induction.
Proof by contradiction.
Use of a counterexample to show that a statement is not always true.

Linking questions

Other contexts: the four-colour theorem.
TOK: what is the difference between the inductive method in science and proof by induction in mathematics?

Practice questions

7 questions · 6 medium · 1 hard

Showing 7 of 7

Question 1

MediumPaper 1 · no calculator6 marks

Consider an integer $n$ . Prove by contradiction that if $n^2$ is divisible by 3, then $n$ is also divisible by 3.

Question 2

HardPaper 1 · no calculator7 marks

Use the method of proof by contradiction to prove that for any integer $n$ , if $n^2$ is a multiple of 3, then $n$ is a multiple of 3.

Question 3

MediumPaper 2 · calculator6 marks

A cryptographer is investigating a proposed encryption algorithm. The algorithm relies on the property that for any two integers, $p$ and $q$ , the expression $p^2 - 4q - 7$ can never be equal to zero.

Prove this property by contradiction.

Question 4

MediumPaper 1 · no calculator8 marks

(a)

Give a counterexample to prove that each of the following statements is false:

(a) If $x > y$ , then $x^2 > y^2$ for all $x, y \in \mathbb{R}$ .

[2]

(b)

(b) The expression $n^2+n+1$ generates a prime number for all $n \in \mathbb{Z}^+$ .

[2]

(c)

[2]

(d)

(d) If a positive integer $p$ is prime, then $p+1$ is composite.

[2]

Question 5

MediumPaper 1 · no calculator5 marks

Prove by contradiction that the product of a non-zero rational number and an irrational number is irrational.

Question 6

MediumPaper 1 · no calculator4 marks

Use the method of proof by contradiction to prove that for any integer $n$ , if $n^2$ is a multiple of 3, then $n$ is also a multiple of 3.

Question 7

MediumPaper 2 · calculator6 marks

Prove by contradiction that $log_{3}(10)$ is an irrational number.

Every Proof by contradiction question, marked for you

Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.

Where marks are lost

Rounding an intermediate value and then using it.
Answering to the wrong accuracy. Two significant figures, or six, where the rule says exactly or three.
Writing the answer and nothing else.

FAQ

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What does Proof by contradiction cover in IB Maths AA?

Proof by contradiction assumes the negation of a statement is true. This assumption is used to derive a mathematical impossibility (a contradiction). The contradiction proves the initial assumption false, thus validating the original statement.

Is Proof by contradiction SL or HL?

Proof by contradiction is HL only. SL students are not examined on it.

How do I revise Proof by contradiction for IB Maths AA?

Start from the core idea: proof by contradiction assumes the negation of a statement is true. In the exam: induction is marked on structure, not just on algebra: state the proposition, prove the base case, assume for n = k, prove for n = k+1, and write a conclusion that names the base case and the inductive step. Examiners award a mark for that final sentence and students routinely drop it. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Proof by contradiction?

FourtyFive has 7 Proof by contradiction questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Proof by contradiction practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Proof by contradiction answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Proof by contradiction: notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Every Proof by contradiction question, marked for you

Where marks are lost

Practise this topic as a session

Questions,
answered.

Start with the IB question
bank built for you.

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Every Proof by contradiction question, marked for you

Where marks are lost

Questions,answered.

Start with the IB questionbank built for you.

Questions,
answered.

Start with the IB question
bank built for you.