Question

prove that square root of 5 is an irational number

Answer 1

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 natural numbers m and n. Then √5 can be expressed in lower terms as 5n − 2mm − 2n, which is a contradiction.
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Answer 2

take √5=a/b that is and you can prove