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MATH
[1251]

Methods from Complex arithmetic

All methods from Complex arithmetic.

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Add, multiply and divide complex numbers

For z1=x1+iy1z_1=x_1+iy_1 and z2=x2+iy2z_2=x_2+iy_2,

z1+z2=(x1+x2)+i(y1+y2),z_1+z_2=(x_1+x_2)+i(y_1+y_2),z1z2=(x1x2−y1y2)+i(x1y2+y1x2).z_1z_2=(x_1x_2-y_1y_2)+i(x_1y_2+y_1x_2).

The additive inverse is −z1-z_1. For a non-zero z=x+iyz=x+iy,

z−1=xx2+y2−iyx2+y2,zz−1=1.z^{-1}=\frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}, \qquad zz^{-1}=1.

Division is defined using this inverse. In practice, when z2≠0z_2\ne0,

z1z2=z1z2‾z2z2‾.\frac{z_1}{z_2} =\frac{z_1\overline{z_2}}{z_2\overline{z_2}}.

The multiplication formula follows by expanding and replacing i2i^2 by −1-1. The conjugate in the division formula makes the denominator real:

z2z2‾=∣z2∣2.z_2\overline{z_2}=|z_2|^2.
  1. [1]:Complex number
  2. [2]:Calculate with Cartesian forms

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Add, multiply and divide complex numbers