38 lines
1.7 KiB
Markdown
38 lines
1.7 KiB
Markdown
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---
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title: Complex Number Plane
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---
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## Complex Numbers
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The complex number plane expresses a complex number in graphical form. The complex number is an extension of the real number
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line that adds an imaginary axis. This creates a two dimensional space with real and imaginary coordinates.
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Complex numbers take the form (a + bi) with the real part being "a" expressed on the x-axis and "b" expressed on the y-axis. See the
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graph below.
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<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/a/af/Complex_number_illustration.svg/183px-Complex_number_illustration.svg.png" />
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A complex number that is on the x-axis is called purely real; while a complex number that is on the y-axis only is
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called purely imaginary. The x-axis or real number line includes all real numbers. Therefore, the set of all real numbers is actually
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a subset of the complex numbers. All real numbers, then are complex numbers who imaginary component is zero.
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## Complex Polar Coordinates
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In polar form the cooordinates are the radius to the point in the complex plane and the angle from the x-axis.
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The radius[r] is found from the pythagorean formula applied to the real and imaginary componenets.
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r = sqrt(a^2 + b^2)
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The angle for the polar coordinate is found from taking the inverse tangent of the real and imaginary coordinates.
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@ = arctan(b/a) where x > 0
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@ = arctan(b/a) + pi where x < 0
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@ is undefined when x = 0
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<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/69/Complex_conjugate_picture.svg/300px-Complex_conjugate_picture.svg.png" />
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#### More Information
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-[Wikipedia:Complex number](https://en.wikipedia.org/wiki/Complex_number)
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-[Wolfram](http://mathworld.wolfram.com/ComplexNumber.html)
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