Question

Prove that 5+√2 is irrational

Answer 1

Let us assume on the contrary that 5+​​ root 2 is rational.
Then, there exist coprime a and b(b not equal to 0), such that
5+ root 2= a by b
= root 2= a by b--2
= root 2 =a--2b by b
= root 2 is rational [Therefore, 5, a & b are integers, a--2b by b is a rational number]
This contadicts the fact the root 2 is irrational.
The contradiction has arisen bacause of our incorrect assumption that 5+ rrot 2 is rational.
Hence, 5+root 2 is irrational.