Proof by Contradiction

Proof by contradiction usually starts with an assumption that opposes the statement in the question. Going from there, we hope to meet a contradiction so as to rule out original assumption. In this question, we assume that $\sqrt{2}$ is rational. We express a rational number as $\frac{a}{b}$ where $a$ and $b$ are integers with no common factors. It follows that

$\sqrt{2}=\frac{a}{b}\hspace{10pt}\Rightarrow\hspace{10pt} 2=\frac{a^2}{b^2}\hspace{10pt}\Rightarrow\hspace{10pt}a^2=2b^2$

From this last expression we can see that 2 is a factor of $a^2$  and so it must be a factor of $a$ (the factors of $a$ will appear in factors of $a^2$ in pairs). We can write $a=2m$ and so $a^2=4m^2=2b^2$ or $b^2=2m^2$. It follows that 2 is also a factor of $b^2$ and also $b$.  So 2 is factor of both $a$ and $b$. The original assumption that $\sqrt{2}$ can be written as a rational number $a/b$ where $a$ and $b$ are integers with NO COMMON FACTORS has been violated. Our original assumption must be wrong. We may conclude that $\sqrt{2}$ is irrational.

Proving that there are infinitely many prime numbers is also a common proof that uses proof by contradiction. You will be required to know this proof (see Example 2) as well as the proof that $\sqrt{2}$ is irrational as above. Exam questions may also deviate slightly from these examples where you will have to apply this knowledge to prove other statements by contradiction.