EECS20N: Signals and Systems

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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.