Square root of two: Difference between revisions

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Since <math>k</math> is an integer, <math>y</math> must ''also'' be even. However, if <math>x</math> and <math>y</math> are both even, they share a common [[factor]] of 2, making them ''not'' mutually prime. And that is a contradiction.
Since <math>k</math> is an integer, <math>y</math> must ''also'' be even. However, if <math>x</math> and <math>y</math> are both even, they share a common [[factor]] of 2, making them ''not'' mutually prime. And that is a contradiction.


[[Category:Mathematics Workgroup|CZ live]]
[[Category:Mathematics Workgroup]]
[[Category:CZ live]]

Revision as of 02:27, 29 March 2007

The square root of two (), approximately 1.4142135623730950488016887242097, is 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:

Assume that there exists two numbers, , such that and and represent the smallest such integers (i.e., they are mutually prime).

Therefore, and ,

Thus, represents an even number

If we take the integer, , such that , and insert it back into our previous equation, we find that

Through simplification, we find that , and then that, ,

Since is an integer, 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.