A rational number is any number that can be expressed as a/b, where a and b are integers. Setting a/b equal to the square root of three and squaring, we have:
a^2/b^2=3
a^2=3b^2
However, because 3 is not a square number, three times a square cannot be another square. So, there are no such a and b, and thus we have proven that the square root of three is not a rational number.