PROOF · HL

Proof by Contradiction

To prove a statement is true prove the opposite is false.

Section 1 of 1

Prove $\sqrt{2}$ is irrational

To prove a statement is true prove the opposite is false.

Note:

Even number $= 2n$, $n \in \mathbb{N}$

$=$ Can be divided by 2.

$=$ 2 is a factor.

Irrational $\Rightarrow$ cannot be written as a fraction.

$\sqrt{2}$ is rational

$\Rightarrow \quad \sqrt{2} = \dfrac{a}{b}$ where $a$ and $b$ have no common factor

$2 = \dfrac{a^2}{b^2}$

$2b^2 = a^2$

$2b^2$ is even $\quad \Rightarrow \quad a^2$ must be even.

$\Rightarrow \quad a$ must be even

$\Rightarrow \quad a = 2K$

$2b^2 = (2k)^2$

$2b^2 = 4k^2$

$b^2 = 2k^2$

$\Rightarrow \quad b^2$ must be even

$\Rightarrow \quad b$ must be even

$a$ and $b$ are both so a common factor of 2. This is a contradiction to statement above

$\sqrt{2}$ is not rational $\Rightarrow$ irrational.

SUM

Key facts

1.To prove a statement is true prove the opposite is false.

2.Even number $= 2n$, $n \in \mathbb{N}$.

3.$\sqrt{2}$ is not rational $\Rightarrow$ irrational.

Proof by Contradiction — HL · Mathslive.ie