Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4

# 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 *b *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, *a*^{2} is divisible by 5 and it can be said that *a* is divisible by 5.

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

(1)

This means that *b*^{2} 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