Prove that 7\sqrt{5} is irrationals number.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Linear Equations in One Variable

Question

Prove that 7

\sqrt{5}

is irrationals number.

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Let us assume that

7 \sqrt{5}

is rational number

Hence

7 \sqrt{5}

can be written in the form of

\frac{a}{b}

where

a, b (b \neq 0)

are co-prime

⟹ 7 \sqrt{5} = \frac{a}{b}

⟹ \sqrt{5} = \frac{a}{7 b}

But here

\sqrt{5}

is irrational and

\frac{a}{7 b}

is rational

R a t i o n a l \neq I r r a t i o n a l

This is a contradiction

7 \sqrt{5}

is a irrational number