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.

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z₁ = x₁ + i y₁ = + i

z₂ = x₂ + i y₂ = + i

Quotient

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.

$With z_{1} = x_{1} + i y_{1} and z_{2} = x_{2} + i y_{2} is$

$\frac{z_{1}}{z_{2}} = \frac{x_{1} + i y_{1}}{x_{2} + i y_{2}}$ $= \frac{x_{1} + i y_{1}}{x_{2} + i y_{2}} \frac{x_{2} - i y_{2}}{x_{2} - i y_{2}}$ $= \frac{x_{1} x_{2} + y_{1} y_{2}}{x_{2}^{2} + y_{2}^{2}} + i \frac{x_{2} y_{1} - x_{1} y_{2}}{x_{2}^{2} + y_{2}^{2}}$

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

$With z_{1} = r_{1} (\cos φ_{1} + i \sin φ_{1}) = r_{1} e^{i φ_{1}} and z_{2} = r_{2} (\cos φ_{2} + i \sin φ_{2}) = r_{2} e^{i φ_{2}} is$

$\frac{z_{1}}{z_{2}}$ $= \frac{r_{1}}{r_{2}} (\cos (φ_{1} - φ_{2}) + i \sin (φ_{1} - φ_{2}))$ $= \frac{r_{1}}{r_{2}} e^{i (φ_{1} - φ_{2})}$

$with r = | z | = \sqrt{x^{2} + y^{2}} and φ = atan \frac{y}{x}$

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