Prove that square root of two is an irrational number.

Here’s the proof. We start by assuming the square root of two, shown as sqrt(2) here, is equal to some fraction m/n. We intend to contradict this assumption:

sqrt (2) = m/n   (a fraction, reduced to its lowest terms)
     2 = m²/n²   m² = 2n²   (So, m is a multiple of 2, call it 2q)
     4q² = 2n²   2q² = n²    (So, n is a multiple of 2)

So, both m and n are multiples of two, which is impossible, because m/n was reduced to its lowest terms. So, we have proved that the square root of two cannot be expressed as a fraction, i.e. it is irrational. via.Note:There are other methods of proof.

It follows that 2 = a²/b²,  or  a² = 2 * b². 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 b² is even, so we obtain the result.via