Question

prove ✓5 is irrational?​

Answer 1

Here we go with the proof;

Let us assume that √5 is rational

So, √5 is a rational number

All rational numbers are in p/q form

So, √5 = p/q , where p≠0 and p&q are integers

Squaring on both sides

(√5)²=(p/q)²

5=p²/q²

5q²=p²–(1)

5 divides p [p is any prime number and a is any positive integer and p divides

a² then p divides a]

So, p is a positive integer—(2)

Then, p=5r, r is a positive integer

Squaring on both sides

p²=(5r)²—(3)

By 1&3

(5r)²=5q²

25r²=5q²

5r²=q²

So, 5 divides q²

Then, 5 divides q [by above reason]

Then,q is a positive integer—(4)

By 2&4

p and q have common factor 5.

This is a contradiction since p and q are co-primes

So, our assumption is wrong

Therefore √5 is an irrational number.

HOPE THIS HELPS YOU:)