Let be rational,

Then, can be expressed as where p and q are co-prime integers and

q ≠ 0,

we have,

As p and q are integers, 5p and 3q are also integers

is a rational number.

But √3 is an irrational number, so the equality is not possible.

This contradicts our assumption, that is a rational number.

Therefore, is an irrational number.