Is square root of 5 a rational number?

Its irrational because it comes out to be a number with decimals (2.236067977). So, no.

This irrationality proof for the square root of 5 uses Fermat's method of infinite descent: Suppose that √ 5 is rational, and express it in lowest possible terms (i. e., as a fully reduced fraction) as mn for naturalnumbers m and n. Then √ 5 can be expressed in lower terms as 5n - 2mm - 2n, which is a contradiction.