# Complex Numbers

In electrical engineering, complex numbers are used to simplify difficult computations. In Cartesian form, it is expressed in the following way:

a+jb

Where:

a = real component

b = imaginary component

j = \sqrt{-1}

In polar form, it is expressed in the following way:

r e^{j\theta} = r \angle \theta

Where:

\begin{aligned} r &= \sqrt{a^2 + b^2} \\ \theta &= \arctan(\frac{b}{a}) \end{aligned}

## Operations

Complex addition and subtraction is easily done in Cartesian form:

(a + jb) + (c + jd) = a + c + j(b + d) \\ (a + jb) – (c + jd) = a – c + j(b – d)

When dealing with polar form, it’s easiest to first convert to Cartesian form and add or subtract from there.

### Multiplication and Division

Conversely, complex multiplication and division is easily done in polar form:

r_{1} \angle \theta_{1} \times r_{2} \angle \theta_{2} = r_{1}r_{2} \angle(\theta_{1} + \theta_{2}) \\ r_{1} \angle \theta_{1} \div r_{2} \angle \theta_{2} = \frac{r_{1}}{r_{2}} \angle(\theta_{1} – \theta_{2})

Therefore, when dealing with Cartesian form, it’s easiest to first convert to polar form and multiply or divide from there.