Question

Prove that√2 is an irrational number​

Answer 1

There are two methods to prove √2 is an irrational number

• Long division method
• Wrong assumption method

•Wrong assumption method

Let us assume that √2 is a rational number

So it can be expressed in p/q form

√2=p/q

Squaring on both the sides

2=(p/q)²

2q²=p². equation 1

p²/2=q²

So 2 divides p and p is a multiple of 2

p=2m

p²=4m²

From equations 1&2 we get

2q²=4m²

q²=2m²

q² is a multiple of of 2

q is a multiple of 2

Hence p, q have a common factor. This contradicts our assumption that they are co-primes. Therefore p/q is not a rational number.

Hence √2 is an irrational nu`mber

Answer 2

Given

If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get:

2 = (2k)2/b2

2 = 4k2/b2

2*b2 = 4k2

b2 = 2k2

This means that b2 is even, from which follows again that b itself is even. And that is a contradiction!!!