Theorem: Let a be rational number and b irrational. Then
i. \(a+b\) is irrational
ii. if \(a \neq 0\), then \(a b\) is also irrational.
Proof. i. Suppose that \(a+b\) is rational, so \(a+b:=\frac{m}{n} .\) Now, as \(a\) is rational, we can write it as \(a:=\frac{p}{q} . \mathrm{So}\)
\[
b=(a+b)-a=\frac{m}{n}-\frac{p}{q}=\frac{m q-p n}{n q}
\]
hence \(b\) is rational, which contradicts the assumption.
ii. left as an exercise :)