The Square Root Of Two Is Not The Ratio Of Two Integers
The proof requires these facts about integers (which are the whole numbers, 0, 1, -1, 2, -2, 3, -3, etc.).
Suppose that x is the square root of 2 and that x is also the ratio of two integers. Let a and
be b relatively prime integers such that b is not 0 and x=a/b. Then
For integers a, if a2 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).
The 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).
So, 2 * (b2)= a2.
Therefore, a2 is even (because it is twice another integer). By (1) above, a is even.
Thus a=2*k for some integer k, and hence 2 * (b2)= (2k)2
Therefore, 2 * (b2)= 4*(k2).
Divide by two, to discover that b2= 2*(k2).
Therefore b2 is even (because it is twice another integer). By (1) above, b is even.
Therefore, b=2*j for some integer k.
Therefore, a and b are not relatively prime, because each has 2 as a divisor and 2 is bigger than 1
(their supposed greatest common divisor). This is a contradiction.
This style of proof can be modified to show that n(p/q) is irrational
when p and q are relatively prime integers (1 and 2 in the paragraph above) and n is not the
qth power of an integer.
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