Division of complex numbers graphically

Division of the complex numbers z1 and z2

The ratio of the complex numbers is graphically presented. By moving the vector endpoints the numbers can be changed. The red arrow shows the result of the division z1 / z2.

z1 = x1 + i y1 = + i 
z2 = x2 + i y2 = + i 

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).

Division of complex numbers

The division is carried out by the fraction is expanded with the complex conjugate of the denominator.

With z1 = x1+iy1 and z2 = x2+iy2 is

z1z2 = x1+iy1 x2+iy2 = x1+iy1 x2+iy2 x2-iy2 x2-iy2 = x1x2+y1y2 x22+y22 +i x2y1-x1y2 x22+y22

The division of complex numbers can also be done in trigonometric or exponential form.

Withz1 = r1(cosφ1+isinφ1) = r1eiφ1 andz2 = r2(cosφ2+isinφ2) = r2eiφ2is

z1z2 = r1r2(cos(φ1-φ2)+isin(φ1-φ2)) = r1r2ei(φ1-φ2)

withr=|z|=x2+y2 andφ=atanyx

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