Question

proof root 2 is irrational​

Answer 1

Answer:

Given :- √2

=> To prove :- √2 is an irrational no.

Proof :-

Let us assume that √2 is a rational number.

So it 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

On Solving,

√2 = p/q

On squaring both the side we get,

= > 2 = (p/q)^{2}=>2=(p/q)

2

= > 2q^{2} = p ^{2} ……...(1)=>2q

2

=p

2

……...(1)

= > p^{2} /2 = q^{2}=>p

2

/2=q

2

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

⇒ p = 2m⇒p=2m

⇒ p^{2} = 4m² ………..(2)⇒p

2

=4m²………..(2)

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

= 2q² = 4m²

⇒ q² = 2m²⇒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.

Therefore,

= √2 is an irrational number.

I hope it help you