Q. 2 C

# Prove that the following numbers are irrational numbers:

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

Let 3√2 = a/b for b ≠ 0 ……………… (i)

Where a and b are co-prime integer numbers.

From (i) we can write as follows:

a = 3b√2

a2 = 18b2 …………………………….. (i)

Now since 18b2 is divisible by 18, so a2 has to be divisible by 18. From theorem 2.3 we can clearly states that if a2 is divided by 18 so a is also divisible by 18. So we conclude that a divides 18.

Now we can write the integer b in following format,

a = 18c

a2 = 324c2 …………………………… (ii)

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

324c2 = 18b2

b2 = 18c2

From the above equation we can conclude that b2 is divisible by 18 and also by 18.

From the values of a and b it is seen that a and b has a common factor and it is clearly indicates that 18 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 3√2 is an irrational number.

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