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  • Prove that the following are irrationals : 1/root 2

Q3  Prove that the following are irrationals : 

(i) 1/ \sqrt 2 

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Let us assume \frac{1}{\sqrt{2}} is rational.

This means \frac{1}{\sqrt{2}} can be written in the form \frac{p}{q} where p and q are co-prime integers.

\\\frac{1}{\sqrt{2}}=\frac{p}{q}\\ \sqrt{2}=\frac{q}{p}

Since p and q are co-prime integers \frac{q}{p} will be rational, this contradicts the fact that  \sqrt{2} is irrational. This contradiction arises because our initial assumption that \frac{1}{\sqrt{2}}  is rational was wrong. Therefore \frac{1}{\sqrt{2}} is irrational.

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