Prove that `sqrt(2)` is not a rational number.

Explanation

Let assume `sqrt2` be a rational number and simplest form be `a/b` then, a and b are integers having no common factor other than `1`
`sqrt2= a/b`
​⟹ `2=(a/b)^2` (On squaring both sides )
or, `2b^2=a^2` .......(i)
⟹ `2` divides `a^2` (∵`2` divides `2b^2`)
⟹ `2` divides `a`
Let `a=2c` for some integer c
Putting `a=2c` in (i)
or, `2b^2=4c^2`
⟹ `b^2=2c^2`
⟹ `2` divides `b^2` (∵`2` divides `2c^2`)
⟹ `2` divides `a`
Thus `2` is a common factor of `a` and `b`
`a` and `b` have no common factor other than `1`.
The contradiction arises by assuming `sqrt2` is a rational.
Hence, `sqrt2` is irrational.