Two positive irrational numbers, xxx and yyy, have a rational sum and a rational product.
Jamie is trying to prove that x2+y2x^2 + y^2x2+y2 is rational.
Here is Jamie's proof:
Step 1: x2+y2=(x+y)2x^2 + y^2 = (x + y)^2x2+y2=(x+y)2
Step 2: x+yx + yx+y is rational, so (x+y)2(x + y)^2(x+y)2 is rational.
Step 3: Therefore, x2+y2x^2 + y^2x2+y2 is rational.
Identify Jamie's mistake.
Write down a correct version of the proof that x2+y2x^2 + y^2x2+y2 is rational.
Prove by contradiction that the sum of any rational number and any irrational number is irrational.
Practise AQA A Level Maths 1.4 A: Proof with exam-style questions for A Level Maths. 107 questions covering 1.4.1 Structure and methods of proof, matched to the AQA A Level Maths (7357) 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.
