## The Square Root Of Two Is Not The Ratio Of Two IntegersThe proof requires these facts about integers (which are the whole numbers, 0, 1, -1, 2, -2, 3, -3, etc.). -
For integers a, if a
^{2}is even, then a is even. The definition of even is that c is even if and only if there is some integer k such that c=2*k. -
For integers a, if a
^{2}is odd, then a is odd. The definition of odd is that c is not even; necessarily, this means that there is some integer k such that c=2*k+1. - If a number x is the ratio of two integers, then it is the ratio of two relatively prime integers. That is, if x is a ratio of two integers, then there are some integers a and b such that b is not 0, x=a/b, and the greatest common divisor of a and b is 1 (1 and -1 both divide a and b, but no other integers do that).
2=xThe contradiction means that something in our "Suppose that x is the square root of 2 and x is also the ratio of two integers" is false. So, if one part of this "and" sentence is true, the other part must be false. In particular, if x is indeed the square root of 2, then x is NOT the ratio of two integers. In mathematical terminology, x is irrational. "Ir" as
a prefix means "not". So x is NOT rational where rational means being the
ratio of two integers (with the denominator integer non-zero).
This style of proof can be modified to show that n Your comments and questions are welcome. Please email them to my email address. |