Two irrational quantities, α\alphaα and β\betaβ, have a rational sum and a rational product.
An engineer is trying to prove that α2+β2\alpha^2 + \beta^2α2+β2 is rational.
Here is the engineer's proof:
Step 1: α2+β2=(α+β)2\alpha^2 + \beta^2 = (\alpha + \beta)^2α2+β2=(α+β)2
Step 2: α+β\alpha + \betaα+β is rational, so (α+β)2(\alpha + \beta)^2(α+β)2 is rational.
Step 3: Therefore, α2+β2\alpha^2 + \beta^2α2+β2 is rational.
(i) Identify the engineer's mistake.
(ii) Write down a correct version of the proof that α2+β2\alpha^2 + \beta^2α2+β2 is rational.
Prove by contradiction that the product of any non-zero rational number and any irrational number is irrational.
Practise Edexcel A Level Maths 1.1 Proof by Contradiction with exam-style questions for A Level Maths. 100 questions, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.