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Let √2 be a rational number p/q, where p and q are integers. p and q are relatively prime to each other, coprime. no common factors among them.

√2 = p / q

p² = 2 q²

p * p = 2 * q * q

p and q are co-prime. no common factors. Then 2 on RHS must be a factor of p.

then p = 2 k

2k * 2k = 2 * q * q

2 k * k = q * q

2 on LHS must be a factor of q on RHS. So q = 2 * m.

But we have started √2 = p/q with p and q such that they are co-prime. They have no common factors. But if √2 is rational, they have a common factor 2.

So the assumption is wrong.

√2 cannot be a rational number.

√2 = p / q

p² = 2 q²

p * p = 2 * q * q

p and q are co-prime. no common factors. Then 2 on RHS must be a factor of p.

then p = 2 k

2k * 2k = 2 * q * q

2 k * k = q * q

2 on LHS must be a factor of q on RHS. So q = 2 * m.

But we have started √2 = p/q with p and q such that they are co-prime. They have no common factors. But if √2 is rational, they have a common factor 2.

So the assumption is wrong.

√2 cannot be a rational number.

:. √2/1 =a/b

a=√2b

squaring on both sides

a²= 2b²

a²/2=b²

2 divides a²⇒2 divides a

:. as 2 divides 'a' if we multiply 2 by a number we will get a

let that number be c

:. a=2c

(2c)²=2b² (a²=2b²)

4c²= 2b²

b²=4c²/2

b²=2c²

c²=b²/2

⇒2 divides b²⇒2 divides b

this is a contradiction that they are co-prime

hence our assumption is wrong

:. √2 is irrational