Question

Prove that the following are irrational.
1/✓2

Answer 1

GIVEN:

• 1/√2

TO FIND:

• Prove that 1/√2 is irrational.

SOLUTION:

Let us assume, to the contrary, that 1/√2 is rational. That is, we can find co - prime integers p and q (q ≠ 0) such that

$\rm{\dashrightarrow \dfrac{1}{\sqrt{2}} = \dfrac{p}{q}}$

$\rm{\dashrightarrow \dfrac{ 1 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \dfrac{p}{q}}$

$\rm{\dashrightarrow \dfrac{\sqrt{2}}{2} = \dfrac{p}{q}}$

$\rm{\dashrightarrow \sqrt{2} = \dfrac{2p}{q} }$

Since, p and q are integers so 2p/q is rational, and so √2 is rational.

But this contradicts the fact that √2 is irrational.

So, we conclude that √2 is an irrational.

Hence, 1/√2 is an irrational number.

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