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Prove that is irrational.

Solutions: Let us assume, that is rational number.
i.e. (where, and are co-primes)
Squaring both the sides, we get,

   

   

Thus, is divisible by 5 , so is also divisible by 5 .
Let us say, , for some value of and substituting the value of in equation (1), we get,
(a is divisible by 5 it means is divisible by 5 .
Clearly, and are not co-primes. Thus, our assumption about is rational is incorrect.
Hence, is irrational number.