Proof by Contradiction Math Example 1

Solution

1. 1
   Assume for contradiction that $\sqrt{2}$ is rational. Then $\sqrt{2} = \frac{a}{b}$ where $a, b$ are integers with $\gcd(a,b)=1$.
2. 2
   Squaring: $2 = \frac{a^2}{b^2}$, so $a^2 = 2b^2$. This means $a^2$ is even, so $a$ is even. Write $a = 2c$.
3. 3
   Then $4c^2 = 2b^2$, so $b^2 = 2c^2$, meaning $b$ is also even.
4. 4
   But if both $a$ and $b$ are even, $\gcd(a,b) \ge 2$, contradicting $\gcd(a,b)=1$.