| Property | Formula |
|---|---|
| Double conjugate | (z-bar)-bar = z |
| Sum | (z1 + z2)-bar = z1-bar + z2-bar |
| Product | (z1 * z2)-bar = z1-bar * z2-bar |
| Quotient | -bar = z1-bar / z2-bar |
| Power | ()-bar = (z-bar)^n |
| Modulus relation | z * z-bar = |z|^2 |
| Real/Imaginary | z + z-bar = 2Re(z), z - z-bar = 2i*Im(z) |
| z is real iff | z = z-bar |
| z is purely imaginary iff | z + z-bar = 0 (and z != 0) |
Division technique: To compute z1/z2, multiply numerator and denominator by z2-bar: z1/z2 = (z1 * z2-bar) / (z2 * z2-bar) = (z1 * z2-bar) / |z2|^2
This converts the denominator to a real number.