Prove that is an irrational number.
Let us assume that is a rational;
squaring both sides, we get
Therefore is divisible by 5 and also a is divisible by 5.
Now we can write for some integer c.
Substituting this value of for a, in (1) we get
This means that is divisible by 5 and also b is divisible by 5. Therefore a and b have 5 as their common factor, but this contradicts the fact that a and b are co prime. The contradiction arises because of our wrong assumption that is a rational.
So, we conclude that is an irrational.
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