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Prove that√2 is a irrational number ?​

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Answer:

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Explanation:

GIVEN √2

To prove: √2 is an irrational number.

Proof:

LETUS assume that √2 is a rational number.

So it t can be expressed in the form p/q where p,q are co-prime integers and q≠0

√2 = p/q

Here p and q are coprime numbers and q ≠ 0

Solving

√2 = p/q

On squaring both the SIDE we get,

=>2 = (p/q)2

=> 2Q2 = p2……………………………..(1)

p2/2 = q2

So 2 divides p and p is a multiple of 2.

⇒ p = 2m

⇒ p² = 4m² ………………………………..(2)

From EQUATIONS (1) and (2), we get,

2q² = 4m²

⇒ q² = 2m²

⇒ q² is a multiple of 2

⇒ q is a multiple of 2

Hence, p,q have a common factor 2. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√2 is an irrational number.


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