Prove that the following numbers are irrational numbers:

Let us consider that 1/√2 is a rational number.

Let 1/√2 = a/b for b ≠ 0 ……………… (i)

Where a and b are co-prime integer numbers.

From (i) we can write as follows:

b = a√2

b² = 2a² …………………………….. (i)

Now since 2a² is divisible by 2, so b² has to be divisible by 2. From theorem 2.3 we can clearly states that if b² is divided by 2 so b is also divisible by 2. So we conclude that 2 divides b.

Now we can write the integer b in following format,

b = 2c

b² = 4c² …………………………… (ii)

By comparing (i) and (ii) we can state as follows:

4c² = 2a²

a² = 2c²

From the above equation we can conclude that a² is divisible by 2 and also by a.

From the values of a and b it is seen that a and b has a common factor and it is clearly indicates that 2 is a common factor. But it is assumed in the beginning that a and b has no common factors.

So our assumption made at the beginning of the problem is wrong.

Hence it is proved that 1/√2 is an irrational number.