Graphical representation of the complex number z

Graphical representation of the complex number z = x + i y

Interactive graphical representation of a complex number and the conjugate of the complex number. The complex number is shown red in the diagram and the conjugate blue. By moving the vector endpoint the number can be changed.

z = x + i y
Conjugate complex
+ i 

Polar co-ordinates

Axes ranges


Complex numbers


Gauss plane:

The complex numbers are two-dimensional and can be used as vectors in the Gaussian plane of numbers represent. On the horizontal axis (Re) of the real part and on the vertical axis is applied (Im) of the imaginary part of the complex number. Similar to vectors complex numbers can be expressed in Cartesian coordinates (x, y) or polar coordinates (r, φ).

Definitions and notations for complex numbers

A complex number z consists of a real part x and an imaginary part y. The imaginary part is characterized by the imaginary unit i .

z = x + i y

The complex conjugate to z consists of a real part x and the negative imaginary part y. This corresponds to a reflection in the real axis in the Gaussian plane.

z = x - i y

The amount of a complex number corresponds in the Gaussian plane with the length of the vector.

|z|2 = x2 + y2

The complex number can also presented in polar coordinates.

z = r cos(φ) + i sin(φ)

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