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.
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).
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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