Why is the square root of 7 not a rational number?

Answer 1

A rational number is a number that can be written in the form a/b with a and b relatively prime integers - a and b are whole numbers with no common factors (eg if a=3 then b can't be 3,6,9,12,etc). Rational numbers have decimal representations that either terminate (like 3/4=0.75) or are infinitely recurring (like 1/9=0.1111111111... or 5/7=0.714285|714285|714285...).

Irrational Numbers (numbers that aren't rational) have infinite decimals that never repeat (like pi=3.1415926535..., e=2.7182818284590...). It is possible to prove that unless n is a square number, the square root of n is irrational - if n can't be written as m^2 then n^0.5 is irrational.

Since you can't find a and b such that (a/b)^2=7 the square root of 7 is irrational. It should be noted that you can get as close as you like to 7^0.5 with rational numbers but you can never reach it exactly.