Prove that is an irrational number.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Linear Equations in One Variable

Question

Prove that $\sqrt{2}$ is an irrational number.

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Assume that $\sqrt{2}$ = $\frac{p}{q}$ (rational number)

squaring on both the sides,

2 =

\frac{p^{2}}{q^{2}}

p 2 = 2q 2

---(1)

p² is a perfect square number which can be even or odd.

But 2p² is only even number.

So, p² is not necessarily equal to 2q² which is contradictory to the equation (1).

So, our assumption was incorrect.

Hence, root is an irrational number.