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What is an example of mathematical proof by contradiction?
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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 :)

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