Square root of two: Difference between revisions
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The square root of two | The [[square root]] of two, denoted <math>\sqrt{2}</math>, 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 == | == In Right Triangles == | ||
The square root of two plays an important role in [[right triangle|right triangles]] in that a unit right triangle (where both legs are equal to 1), has a [[hypotenuse]] of <math>\sqrt{2}</math>. Thus, <math>sin(\frac{\pi}{4}) = \frac{ | The square root of two plays an important role in [[right triangle|right triangles]] in that a unit right triangle (where both legs are equal to 1), has a [[hypotenuse]] of <math>\sqrt{2}</math>. Thus, <math>\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}</math>. | ||
== Proof of Irrationality == | == Proof of Irrationality == | ||
There exists a simple proof by contradiction showing that <math>\sqrt{2}</math> is irrational: | There exists a simple proof by contradiction showing that <math>\sqrt{2}</math> is irrational: | ||
Assume that there exists two numbers, <math>x, y \in \mathbb{ | Assume that there exists two numbers, <math>x, y \in \mathbb{N}</math>, such that <math>\frac{x}{y} = \sqrt{2}</math> and <math>x</math> and <math>y</math> represent the smallest such [[integer|integers]] (i.e., they are [[mutually prime]]). | ||
Therefore, <math>\frac{x^2}{y^2} = 2</math> and <math>x^2 = 2 \times y^2</math>, | Therefore, <math>\frac{x^2}{y^2} = 2</math> and <math>x^2 = 2 \times y^2</math>, | ||
Revision as of 13:53, 4 April 2007
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:
Assume that there exists two numbers, , such that and and represent the smallest such integers (i.e., they are mutually prime).
Therefore, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 = 2 \times y^2} ,
Thus, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2} represents an even number
If we take the integer, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} , such that , and insert it back into our previous equation, we find that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (2 \times k)^2 = 2 \times y^2}
Through simplification, we find that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \times k = 2 \times y^2} , and then that, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \times k = y^2} ,
Since is an integer, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} must also be even. However, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} are both even, they share a common factor of 2, making them not mutually prime. And that is a contradiction.
