or
for some integer q. Show that 
is an irrational number.
Answer:
Let 
be any odd positive integer.
On squaring both sides, we get
(1/2)
(1)
(1/2)
where
Any square of odd positive integer is of the form 
Similarly, for 
Let 
be any odd positive integer.
On squaring both sides, we get
where
So, any square of odd positive integer is also of the form 
OR
Let 
is a rational number and 
where a and b are coprime numbers, 
(1)
?(i)
is a multiple of 3.
a will also be a multiple of 3.
Let 
Eq. (i) becomes
(1)
is a multiple of 3.
Hence, b will also be a multiple of 3.
and b have a common factor.
But a and b are coprime numbers.
Our assumption is wrong.
Hence, is irrational. (1)
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