If 7 is a prime number, then prove that \sqrt{7} is irrational.

Foundation

Mathematics Foundation

Properties of Real Number

Irrational Number

Law of Exponents

Question

7

is a prime number, then prove that

\sqrt{7}

is irrational.

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Let us assume,

\sqrt{7}

is a rational number.
Let,

\sqrt{7} = \frac{a}{b}

, where HCF

(a, b)

= 1; a ε N, b ε N

Now,

\sqrt{7} = \frac{a}{b}

\Rightarrow a = \sqrt{7} b

\Rightarrow a^{2} = 7 b^{2}

........ (1)

So,

a^{2}

is divisible by

7

∴ a

is divisible by

7

Let,

a = 7 k

, where

k

ε

N

From (1),

\Rightarrow (7 k)^{2} = 7 b^{2}

\Rightarrow 49 k^{2} = 7 b^{2}

\Rightarrow, b^{2} = 7 k^{2}

So,

b^{2}

is divisible by

7

∴ b

is divisible by

7

Thus,

a

and

b

both are divisible by

7

.
This is a contradiction, because HCF

(a, b) = 1

Thus, our assumption is wrong .
Hence,

\sqrt{7}

is irrational.