Question

prove that root 3 is irrational ​

Answer 1

Answer:

Let us assume that √3 is a rational number.

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

⇒√3=p/q

On squaring both the sides we get,

⇒3=p²/q²

⇒3q²=p² —————–(i)

p²/3= q²

So 3 divides p

p is a multiple of 3

⇒p=3m

⇒p²=9m² ————-(ii)

From equations (i) and (ii), we get,

3q²=9m²

⇒q²=3m²

⇒q² is a multiple of 3

⇒q is a multiple of 3

Hence, p,q have a common factor 3. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

√3 is an irrational number

Hence proved.

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