1.4.1 Structure and methods of proof

Two irrational quantities, $α\alpha$ and $β\beta$ , have a rational sum and a rational product.

An engineer is trying to prove that $α2+β2\alpha^2 + \beta^2$ is rational.

Here is the engineer's proof:

Step 1: $α2+β2=(α+β)2\alpha^2 + \beta^2 = (\alpha + \beta)^2$

Step 2: $α+β\alpha + \beta$ is rational, so $(α+β)2(\alpha + \beta)^2$ is rational.

Step 3: Therefore, $α2+β2\alpha^2 + \beta^2$ is rational.

(i) Identify the engineer's mistake.

(ii) Write down a correct version of the proof that $α2+β2\alpha^2 + \beta^2$ is rational.

[3]

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

[4]

1.4.1 Structure and methods of proof Questions