Exercise 1.3

Question 1:

Prove that
is irrational.

Let

is a rational number.

Therefore, we can find two integers a, b (b ≠ 0) 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.

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

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

Let

is rational.
Therefore, we can find two integers a, b (b ≠ 0) 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:

Let

 is rational.
Therefore, we can find two integers a, b (b ≠ 0) such that

a and b are integers.

is rational as
Therefore,

 is rational which contradicts to the fact that
 

is irrational.
Hence, our assumption is false and

is irrational.

Let

is rational.
Therefore, we can find two integers a, b (b ≠ 0) such that

for some integers a and b

is rational as a and b are integers.
Therefore,

 should be rational.
This contradicts the fact that

is irrational. Therefore, our assumption that

 is rational is false. Hence,

  is irrational.

Let

  be rational.
Therefore, we can find two integers a, b (b ≠ 0) such that

Since a and b are integers,

  is also rational and hence,

 should be rational. This contradicts the fact that

is irrational. Therefore, our assumption is false and hence,

is irrational.