Real Numbers Practice 3: Irrationality Proofs

Practise focused skills from Real Numbers

MathematicsGrade 10Mixed10 questions20 min

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Section A: Proof Structure · 5 questionsSection B: Misconceptions and Reasoning · 5 questions

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Section A: Proof Structure

Build and apply the key ideas for this part of the chapter.

Choose the best answer for each question.

Question 1

EasyQuestion

In a proof by contradiction, what is the first step when proving $\sqrt{2}$ is irrational?

Topic: Irrationality Proofs · Skill: Recognise proof by contradiction in irrationality

Question 2

MediumQuestion

In the proof that $\sqrt{2}$ is irrational, if $\sqrt{2}=\fracrac{p}{q}$ in lowest terms, then $p^2=2q^2$ . What must be true about $p$ ?

Topic: Irrationality Proofs · Skill: Prove √2 is irrational

Question 3

MediumQuestion

A proof reaches that both $p$ and $q$ are even, even though $\fracrac{p}{q}$ was assumed in lowest terms. What is the contradiction?

Topic: Irrationality Proofs · Skill: Prove √2 is irrational

Question 4

MediumQuestion

To adapt the proof of $\sqrt{2}$ for $\sqrt{3}$ , which fact is needed?

Topic: Irrationality Proofs · Skill: Prove √3 and √5 are irrational

Question 5

MediumQuestion

Which conclusion correctly supports the proof that $\sqrt{5}$ is irrational?

Topic: Irrationality Proofs · Skill: Prove √3 and √5 are irrational

What this assignment covers

Understand contradiction proofs and repair common mistakes in irrationality arguments.

Skills and sections

Recognise proof by contradiction in irrationality, Prove √2 is irrational, Prove √3 and √5 are irrational, Prove numbers like 3 + 2√5 are irrational, Identify common errors in irrationality arguments

Section A: Proof Structure: 5 questions
Section B: Misconceptions and Reasoning: 5 questions