Exercise 1.3

Question 1

Prove that is irrational .

Sol :

Let is a rational number.

Therefore, we can find two integers a , b such that

Let a and have a common factor other than 1 . Then we can divide them by the common factor and assume that a and b are co-prime .

…..eq-(1)

Therefore, a2 is divisible by 5 and it can be said that a is divisible by 5.

Let a = 5k, where k is an integer and putting this value in eq-(1) , we get

(1)

This means that b2 is divisible by 5 and hence, b is divisible by 5.

And also this implies that a and b have 5 as a common factor .

And this is a contradiction to the fact that a and b are co-prime .

Hence, cannot be expressed as or it can be said that is irrational .

Question 2

Prove that is irrational

Sol :

Let  is rational .

Therefore, we can find two integers a , b such that

Since a and b are integers, will also be rational and therefore is rational.

This contradicts the fact that  is irrational. Hence, our assumption that  is rational is false . Therefore ,  is irrational

Question 3

Prove that the following are irrationals:

(i)

Sol :

Let is rational .

Therefore, we can find two integers a , b such that

is rational as a and b are integers.

will also be rational and this contradicts the fact that  is irrational.

Hence, our assumption that  is rational is false . Therefore ,  is irrational

(ii)

Sol :

Let is rational .

Therefore, we can find two integers a , b such that

is rational as a and b are integers.

will also be rational and this contradicts the fact that  is irrational.

Hence, our assumption that  is rational is false . Therefore ,  is irrational

(iii)

Sol :

Let is rational .

Therefore, we can find two integers a , b such that

is rational as a and b are integers.

will also be rational and this contradicts the fact that  is irrational.

Hence, our assumption that  is rational is false . Therefore ,  is irrational

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