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MATH
[1251]

Theorems from Complex arithmetic

All theorems from Complex arithmetic.

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Rules for conjugation

For z,z1,z2∈Cz,z_1,z_2\in\mathbb C with z2≠0z_2\ne0,

z‾‾=z,z1+z2‾=z1‾+z2‾,z1−z2‾=z1‾−z2‾,z1z2‾=z1‾ z2‾,\overline{\overline z}=z,\qquad \overline{z_1+z_2}=\overline{z_1}+\overline{z_2},\qquad \overline{z_1-z_2}=\overline{z_1}-\overline{z_2},\qquad \overline{z_1z_2}=\overline{z_1}\,\overline{z_2},z1z2‾=z1‾z2‾,Re⁡(z)=z+z‾2,Im⁡(z)=z−z‾2i.\overline{\frac{z_1}{z_2}} =\frac{\overline{z_1}}{\overline{z_2}}, \qquad \operatorname{Re}(z)=\frac{z+\overline z}{2}, \qquad \operatorname{Im}(z)=\frac{z-\overline z}{2i}.

Each identity can be checked by writing the numbers in Cartesian form. The last two recover the real and imaginary parts of zz from zz and z‾\overline z.

  1. [1]:Complex conjugate

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Rules for conjugation