Show that the following numbers are irrational. 1 2 - Vidyadip

Question

Show that the following numbers are irrational.
$\frac{1}{\sqrt{2}}$

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Answer

Let, us assume that $\frac{1}{\sqrt{2}}$ is rational.
Then, there exist positive co primes a and b such that,
$\frac{1}{\sqrt{2}}=\frac{\text{a}}{\text{b}}$
$\frac{1}{\sqrt{2}}=\Big(\frac{\text{a}}{\text{b}}\Big)^2$
$\Rightarrow\ \frac{1}{2}=\frac{\text{a}^2}{\text{b}^2}$
$\Rightarrow\ \text{b}^2=2\text{a}^2$
$\Rightarrow\ 2|\text{b}^2(\because2|2\text{a}^2)$
$\Rightarrow\ 2|\text{b}$
$\Rightarrow\ \text{b}=2\text{c}$ for some positive integer c
$\Rightarrow\ 2\text{a}^2=\text{b}^2$
$\Rightarrow\ 2\text{a}^2=4\text{c}^2\ (\because\text{a}=\text{pc})$
$\Rightarrow\ \text{a}^2=2\text{c}^2$
$\Rightarrow\ 2|\text{a}^2\ (\because2|2\text{c}^2)$
$\Rightarrow\ 2|\text{b}$
$\Rightarrow\ 2|\text{b and 2|b}$
This contradicts the fact that a and b are co-primes.
Hence $\frac{1}{\sqrt{2}}$ is irrational

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