Complex arithmetic
Sums
In order to add two complex numbers, we separately add their real and imaginary parts,The complex conjugate of x + iy is defined to be x - iy. The complex conjugate of a complex number z is written z^{*}. Notice that
z - z^{*} = 2iIm{z}.
Hence, the real and imaginary parts can be obtained using the complex conjugate,
Im{z} = (z - z^{*}) / 2i
Products
The product of two complex numbers works as expected if you remember that i^{ 2} = -1. So, for example,(1 + 2i)(2 + 3i) | = | 2 + 3i + 4i + 6i^{ 2} |
= | 2 + 7i - 6 | |
= | -4 + 7i, |
In general,
If we multiply z = x + iy by its complex congugate z^{*} we get
zz^{*} | = | (x + iy)(x - iy) |
= | x^{ 2} + y^{ 2}, |
which is a positive real number. Its positive square root is called the modulus or magnitude of z, and is written | z |,
| z | | = | √zz^{*} |
= | √(x^{ 2} + y^{ 2}). |
Ratios
How to calculate the ratio of two complex numbers is less obvious, but equally mechanical. We convert the denominator into a real number by multiplying both numerator and denominator by the complex conjugate of the denominator,(2 + 3i) / (1 + 2i) | = | (2 + 3i) / (1 + 2i) ´ (1 - 2i) / (1 - 2i) |
= | [(2 + 6) + (-4 + 3)i] / (1 + 4) | |
= | 8 / 5 - (1 / 5) i. |
The general formula is
In practice it is easier to calculate the ratio as in the example, rather than memorizing the formula.