Online graphical multiplication of complex numbers

Multiplication of the complex numbers z₁ and z₂

The operation with the complex numbers is graphically presented. By moving the vector endpoints the complex numbers can be changed. The red arrow shows the result of the multiplication z₁ ⋅ z₂.

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z₁ = x₁ + i y₁ = + i

z₂ = x₂ + i y₂ = + i

Product

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).

Multiplication of complex numbers

The multiplication is done by multiplying out the brackets considering the relation i²= -1.

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

$z_{1} \cdot z_{2} = (x_{1} + i y_{1}) \cdot (x_{2} + i y_{2})$ $= x_{1} \cdot x_{2} - y_{1} \cdot y_{2} + i (x_{1} \cdot y_{2} + y_{1} \cdot x_{2})$

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

$With z_{1} = r_{1} (\cos φ + i \sin φ) = r_{1} e^{i φ}$

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

$z_{1} \cdot z_{2}$ $= r_{1} r_{2} (\cos (φ + ψ) + i \sin (φ + ψ))$ $= r_{1} r_{2} e^{i (φ + ψ)}$

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