Question

Use strong induction to prove that √2 is irrational

Answer 1

$assume \: \: \sqrt{2} \: \: is \: \: \: \: rational$

a, b are Co primes where b is not equal to zero

$\sqrt{2} \: \: = \frac{a}{b}$

On squ both sides

$2 = \frac{ {a}^{2} }{ {b}^{2} }$

$2 {b}^{2} = {a}^{2}$

$2 \: \: divides \: {a}^{2} and \: a$

Suppose a=2c

$\sqrt{2} = \frac{2c}{b}$

On squ both sides

${b}^{2} = 2 {c}^{2}$

Thus

$2 \: divides \: \: {b}^{2} \: and \: b$

This contradict the fact that a, b has common factor 1

Thus

$\sqrt{2} \: \: is \: \: irrational$