EXERCISE 1.3
Sol. Let us assume, to the contrary, that 
is rational.
Now, let 
, where a and b are coprime and 
Squaring on both sides, we get
=> 
… (1)
This shows that 
is divisible by 5.
It follows that a is divisible by 5 … (2)
=> a = 5m for some integer m.
Substituting a = 5m in (1), we get
or 
=>
is divisible by 5
and hence b is divisible by 5 … (3)
From (2) and (3), we can conclude that 5 is a common factor of both a and b.
But this contradicts our supposition that a and b are coprime.
Hence, is irrational.
2. Prove that 3 + 
is irrational.
Sol. Let us assume, to the contrary, that 3 + is a rational number.
Now, let 3 + 
, where a and b are coprime and
a and b are integers.
is a rational number
=> is a rational number.
But is an irrational number.
This shows that our assumption is incorrect.
So, 3 + 2 is an irrational number.
3. Prove that the following are irrationals:
Sol. (i) Let us assume, to the contrary, that 
is rational.
That is, we can find co-prime integers p and q 
such that
Since p and q are integers, 
is rational, and so
is rational.
But this contradicts the fact that is irrational.
So, we conclude that 
is irrational.
(ii) Let us assume, to the contrary, that 
is rational.
That is, we can find co-prime integers p and q 
such that 
Since p and q are integers, 
is rational and so is
But this contradicts the fact that is irrational.
So, we conclude that 
is irrational.
(iii) Let us assume, to the contrary, that 
is rational.
That is, we can find integers p and q such that
Since p and q are integers, we get 6 – = 
is rational, and so is rational.
But this contradicts the fact that is irrational.
So, we conclude that 6+ is irrational.