Question

prove that √2 is irrational​

Answer 1

I hope this will help you

Answer 2

$\huge\sf\blue{Given}$

✭ We have been given a number √2.

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$\huge\sf\gray{To \:Find}$

☆ We need to prove that √2 is irrational.

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$\huge\sf\purple{Steps}$

Let us assume that √2 is rational.

Therefore, it can be written in the form of p/q where p and q are coprime.

➝ √2 = p/q

On squaring both sides, we have

(√2)² = (p/q)²

➝ p²= 2q² _______(1)

2 divides p² => 2 also divides p.

Therefore, p is a Multiple of 2.

p = 2a [where a is any integer]

Putting p = 2a in equation 1, we have

(2a)² = 2q²

4a² = 2q²

2a² = q² __________(2)

2 divides q² => 2 also divides q.

Therefore, q is a Multiple of 2.

➝ q = 2b

From equation 1 and 2, we have

p and q have 2 as a common factor. But this contradicts the fact that p and q are coprime.

Therefore, our assumption was wrong.

Hence, √2 is irrational.

Hence proved!!

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