Square root of two
The square root of two, denoted , is the positive number whose square equals 2. It is approximately 1.4142135623730950488016887242097. It provides a typical example of an irrational number.
In Right Triangles
The square root of two plays an important role in right triangles in that a unit right triangle (where both legs are equal to 1), has a hypotenuse of . Thus, .
Proof of Irrationality
There exists a simple proof by contradiction showing that is irrational. This proof is often attributed to Pythagoras. It is an example of a reductio ad absurdum type of proof:
Suppose is rational. Then there must exist two numbers, , such that and and represent the smallest such integers (i.e., they are mutually prime).
Therefore, and ,
Thus, represents an even number; therefore must also be even. This means that there is an integer such that . Inserting it back into our previous equation, we find that
Through simplification, we find that , and then that, ,
Since is an integer, and therefore also must also be even. However, if and are both even, they share a common factor of 2, making them not mutually prime. And that is a contradiction, so the assumption must be false, and must not be rational.