The Irrationality of

Problem:
Prove that is an irrational number.

Solution:
The number, , is irrational, ie., it cannot be expressed as a ratio of integers a and b. To prove that this statement is true, let us assume that is rational so that we may write

= a/b

1.

for a and b = any two integers. We must then show that no two such integers can be found. We begin by squaring both sides of eq. 1:

If b is odd, then b² is odd; in this case, a² and a are also odd. Similarly, if b is even, then b², a², and a are even. Since any choice of even values of a and b leads to a ratio a/b that can be reduced by canceling a common factor of 2, we must assume that a and b are odd, and that the ratio a/b is already reduced to smallest possible terms. With a and b both odd, we may write

where we require m and n to be integers (to ensure integer values of a and b). When these expressions are substituted into eq. 2a, we obtain

Upon performing some algebra, we acquire the further expression

The Left Hand Side of eq. 6 is an odd integer. The Right Hand Side, on the other hand, is an even integer. There are no solutions for eq. 6. Therefore, integer values of a and b which satisfy the relationship = a/b cannot be found. We are forced to conclude that is irrational.