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The complex plane

The plane whose points are complex numbers, x+iy plotted as (x,y).

The complex plane is the ordinary Cartesian plane relabeled so that every point corresponds to exactly one complex number: the point (x,y) stands for z = x+iy, with the horizontal axis carrying the real part and the vertical axis carrying the imaginary part. Arithmetic on complex numbers then becomes geometry on this plane -- addition is vector addition, and multiplication combines lengths and rotates angles -- which is why complex analysis leans on pictures as much as on algebra.

Examples

  • The number 3-4i sits at the point (3,-4), three units right of the origin and four units down.

  • The equation z=2i names a single point on the vertical axis, two units above the origin.

See also

Region Distance between complex numbers
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