Question

Prove that $5+\surd 2$ is an irrational number,

Answer 1

To prove $5+\sqrt{2}$ is irrational,

Let us assume $\sqrt{2}$ is rational

$\Rightarrow \sqrt{2}=\frac{a}{b}$ $(a$ and $b$ has no common factors except $1$ )

$\Rightarrow$ $2=\frac{a^2}{b^2}$

$\Rightarrow$ $2b^2=a^2$

Hence $a$ is even ...(1)

Let $a=2k$

$\Rightarrow 2b^2={\left(2k\right)}^2$

$\Rightarrow b^2=2k^2$

Hence $b$ is also even ...(2)

This contradicts our assumptions as both $a$ and $b$ are even because they have $2$ as the common factor.

Hence, $\sqrt{2}$ is irrational.

Hence, proved.