3.1 Proof (A-level only)

Two positive irrational numbers, $x$ and $y$ , have a rational sum and a rational product.

Jamie is trying to prove that $x^2 + y^2$ is rational.

Here is Jamie's proof:

Step 1: $x^2 + y^2 = (x + y)^2$

Step 2: $x + y$ is rational, so $x + y)^2$ is rational.

Step 3: Therefore, $x^2 + y^2$ is rational.

ai.

Identify Jamie's mistake.

[1]

aii.

Write down a correct version of the proof that $x^2 + y^2$ is rational.

[3]

Prove by contradiction that the sum of any rational number and any irrational number is irrational.

[4]

3.1 Proof (A-level only) Questions