Square root of two: Difference between revisions
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imported>Sébastien Moulin m ([[Category:|CZ live]]) |
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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]] | [[Category:Mathematics Workgroup|CZ live]] |
Revision as of 01:27, 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.