Question

Sum or difference (H). Let a and b be any two irrational numbers. Show that either a+b or a-b must be irrational

Answer 1

i think you can do this by assuming that \(a+b\) is rational, say \(a+b=r\) and then showing that \(a-b\) is irrational

Answer 2

or maybe easier by contradiction. suppose both are rational say
if \(a+b=r_2\) and \(a-b=r_2\) where \(r_1,r_2\) are rational
then \[a+b+a-b=2a=r_1+r_2\] so \(2a\) is rational as the sum of two rational numbers is also rational
therefore \(a=\frac{r_1+r_2}{2}\) is also rational
contradicting the assumption that \(a\) is irrational