Question

सिद्ध करे कि√3 एक आपरिमेय संख्या है?​

Answer 1

ANSWER

First of all ,

Let √3 is a rational no.

Let given number = p/q , where p and q co - primes.

So it can be written in the Form of P/Q

where p and Q be integers

$\implies \: \sqrt{3} = \frac{ \large \: p}{ \large \: q}$

Squaring Both side

$\implies \: {( \sqrt{3} )}^{2} = \frac{ { \large \: p}^{2} }{ { \large \: q}^{2} } \\ \\ \implies \: 3 {q}^{2} = {p}^{2}$

↪ Here 3 divides P² , Hence divide P with it we get 3 is the Factor of P .

Take P = 3a

$\implies \: 3 {q}^{2} \: = \: {(3a)}^{2} \\ \implies \: 3 {q}^{2} = 9 {a}^{2} \\ \implies \: {q}^{2} = 3 {a}^{2}$

↪ Here 3 Divides q² hence it divide q also .

↪ In both condition we get p and q both divided by 3 Hence 3 is the Common factor of P and Q ..

↪ Hence p and q have common factor 3 they are not co-primes so our assumption is wrong .

↪ To be a rational no. The common Factor should be 1 . Hence Our Contradiction proved Wrong .

°.° √3 is a Irrational No.

Proved ☑☑☑

Answer 2

Let us assume that √3 is a rational number

That is, we can find integers a and b (≠ 0) such that √3 = (a/b)

Suppose a and b have a common factor other than 1, then we can divide by the common factor, and assume that a and b are coprime.

√3b = a

⇒ 3b^2 = a^2 (Squaring on both sides) → (1)

Therefore, a2 is divisible by 3

Hence