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. To show that is irrational, we must show that no two such integers can be found. We begin by squaring both sides of eq. 1:

From eq. 2a, we must conclude that a² (and, therefore, a) is even; b² (and, therefore, b) may be even or odd. If b is even, the ratio a²/b² may be immediately reduced by canceling a common factor of 2. If b is odd, it is possible that the ratio a²/b² is already reduced to smallest possible terms. We assume that b² (and, therefore, b) is odd.

Now, we set a = 2m, and b = 2n + 1, and require that m and n be integers (to ensure integer values of a and b). Then

Substituting these expressions into eq. 2a, we obtain

The L.H.S. of eq. 6 is an odd integer. The R.H.S., 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.