EECS20N: Signals and Systems

# Complex arithmetic

### Sums

In order to add two complex numbers, we separately add their real and imaginary parts,

(x1 + iy1) + (x2 + iy2) = (x1 + x2) + i(y1 + y2)

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* = 2Re{z},

z - z* = 2iIm{z}.

Hence, the real and imaginary parts can be obtained using the complex conjugate,

Re{z} = (z + z*) / 2,

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,

(x1 + iy1 )(x2 + iy2 ) = (x1 x2 - y1 y2 ) + i(x1 y2 + x2 y1 ).

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

(x1 + iy1 ) / (x2 + iy2 ) = (x1 x2 + y1 y2 ) / (x22 + y22) + i [(-x1 y2 + x2 y1 ) / (x22 + y22)].

In practice it is easier to calculate the ratio as in the example, rather than memorizing the formula.