Complex plane

The graph of the complex number a+bi, where a is the real number, and bi is the imaginary number

In mathematics, the complex plane is used to help visualize complex numbers. The vertical line is called the imaginary axis, and the horizontal line is called the real axis. Any complex number can be written as a+bi, where a and b are real numbers, and i is the principal square root of -1. In the complex plane, you go a spaces on the real axis, and b spaces on the imaginary axis. The complex plane can also be expressed in the form of polar coordinates, where the size of the line is equal to the square root of a squared plus b squared, and the direction is an angle.

The complex plane can be used to draw the graphs of some functions, called complex functions, that can be used to study all kinds of things, like cars, airplanes, atoms and electricity.

Complex conjugates

The graph of the complex number a+bi and it's conjugate a-bi.

Every complex number z=a+bi has a complex conjugate. It is written as [math]\displaystyle{ \overline{z} }[/math] and is equal to a-bi. The complex number and its conjugate can be represented on the complex plane—as mirror opposites about the real axis.

Complex Plane Media

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  Geometric representation of the complex-valued point z = x + yi in the complex plane. The distance along the line from the origin to the point z = x + yi is the modulus or absolute value of z. The angle θ is the argument of z.

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  Riemann sphere which maps all points on a sphere except one to all points on the complex plane

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  Mandelbrot fractal, imaged on a complex plane

Related pages

• Riemann sphere