# Complex Numbers

June 2, 2021

Back to: Fundamentals of Signal Processing

A complex number *z* is an ordered pair of two real numbers *a* and *b*. There are three standard ways to represent z. Put another way, there are three numerals that represent the identical number z:

- Ordered pair (a, b)

Example: z = (√3, 1)

- Rectangular coordinates a + in

Example: z = √3 + i1

- Polar coordinates |z|∠z

Example: z = |z|∠z = 2e^{iπ/6}

In the three representations:

- a is the real part of z
- b is the imaginary part of z
- |z| is the magnitude of z
- ϕ=∠z is the phase of z

While the compound quantities (a,b) and ze^{iϕ} are complex numbers, all the components (a, b, |z|, and ϕ) are real numbers.

### Graphical Representation of Complex Numbers

A complex number is an ordered pair z=(a, b), so it is convenient to represent it on a two-dimensional graph where the real part a is the abscissa (x-axis) and the imaginary part b is the ordinate (y-axis).

### Conversion Between Representations

Trigonometric definitions and two common theorems make for easy conversion between the three representations of z:

- Tangent function: tan(ϕ) = b/a
- Arctangent function: ϕ = atan(b/a)
- Pythagoras’ theorem: r
^{2}= a^{2}+ b^{2} - Euler’s identity: e
^{iϕ}= cos(ϕ) +*i*sin(ϕ)

#### Examples

**If**: z = (a,b) = (√3, 1) = √3 + i

**Then**: r = sqrt[(√3)² + (1)²] = √[10] ≈ 3.16

**And**: ϕ = atan(√3/1) = π/6 (30°)

**If**: z = (a,b) = (4, 3) = 4 + 3i

**Then**: r = sqrt[4² + 3²] = sqrt[25] = 5

**And**: ϕ = atan(3/4) ≈ 0.644 (36.9°)

**If**: z = (a,b) = (-4, 3) = -4 + 3i

**Then**: r = sqrt[4² + 3²] = sqrt[25] = 5

**And**: ϕ = atan(-3/4) ≈ π-0.644 (180°-36.9°=143.1°)

### Star-algebra Structure

Complex numbers with the conjugate operator * is a special case of a star-algebra. The conjugate z* of a complex number z=a+ib is defined as z* = a – ib, where z = a + ib.

Using the conjugate operator, it is easy to extract the real and imaginary components from a complex number z = a + ib:

- Real part of z= [1/2](x + x*) because x+x* = (a+ib)+(a-ib)=2a
- Imaginary part of z= [1/2](x – x*)because x-x* = (a+ib)-(a-ib)=2b