Question

proof=√5, √3 irrational​

Answer 1

Answer:

use calculator and you will find non terminating non-repeating answer like

$\pi$

this is irrational no. but 22/7 is rational no.

because pie gives 3.1452763738so on

and 22/7 is p/q form and it gives 3.142857142857

here after decimal 142857 is repeating nos

Answer 2

Answer:

Given: Number 5

To Prove: Root 5 is irrational

Proof: Let us assume that square root 5 is rational.

Thus we can write, √5 = p/q, where p, q are the integers, and q is not equal to 0.

The integers p and q are coprime numbers thus, HCF (p,q) = 1.

√5 = p/q

⇒ p = √5 q ------- (1)

On squaring both sides we get,

⇒ p2 = 5 q2

⇒ p2/5 = q2 ------- (2)

Assuming if p was a prime number and p divides a2, then p divides a, where a is any positive integer.

Hence, 5 is a factor of p2.

This implies that 5 is a factor of p.

Thus we can write p = 5a (where a is a constant)

Substituting p = 5a in (2), we get

(5a)2/5 = q2

⇒ 25a2/5 = q2

⇒ 5a2 = q2

⇒ a2 = q2/5 ------- (3)

Hence 5 is a factor of q (from 3)

(2) indicates that 5 is a factor of p and (3) indicates that 5 is a factor of q. This contradicts our assumption that √5 = p/q.

Therefore, the square root of 5 is irrational.