Prove that is an irrational number.
To prove is irrational, let's assume it is a rational number.
Hence, can be written is form of
Hence, are co-prime and
Squaring both sides
_____(a)
this means is divided by
By theorem = If is a prime number of divides then also divides
So we can say [hence k is integer]
_____(b)
Putting the value of (b) in (a)
Here we could also say that is divided by . By the theorem stated above, b should also be divisible by
hence a & b both are divisible by but as we know a&b are co-prime (hence they can't have a co-factor )
Hence by contradiction, is an irrational number.