Year 12s: what will be the first way you research your future options? >>

A level maths help please

Question:
A student makes the following statement:
If a + b is an rational number then at least one of a and b is a rational number. Show by mean of a counterexample that this statement is not true.
Mark scheme says: a= 2- root 2. B= root 2

Pls explain to me why this is?
Many thanks

Reply 1

wa17

Well it doesn't ask for a proof. Root 2 is irrational, so picking an irrational number in this case 2-√2, to make it rational the other number will have to cancel out the irrational part. So to cancel out -√2 you need √2. This example includes two irrational numbers so contradicts the statement.

Reply 2

Sinnoh

Moved to Maths :smile:

Both

\sqrt 2 \mathrm{ \ and \ } 2 - \sqrt 2

are irrational numbers. If you're doubtful you could even use proof by contradiction to show that the latter is. And yet, add them together and you get a rational number.