## complex numbers

*last updated 28 December 2021*

Complex numbers are a mathematical concept that permits the solution of equations where real numbers aren’t sufficient. They have many uses in mathematics and physics.

The key point is defining the value “*i* ” as equal to the square root of -1. Then complex numbers are simply the combination of a real number and an “imaginary”, part, which is a real number multiplied by *i*.

Most common mathematical operations can be performed on complex numbers, such as adding or multiplying them, using rules that are based on the fact that *i* x *i* = -1.

In electrical engineering, we are already using the letter *i* for current, so we use “*j*” instead for imaginary values.

Complex numbers are perhaps easiest to think of as a 2-dimensional graph, where real numbers form one axis and imaginary ones on the other. Any point on the graph can be specified by a real value and an imaginary value, but the two are not interchangeable.

This like specifying a point on the surface of the Earth using longitude and latitude: they are basically at right angles to each other, and both are required to identify a specific location.

So when we talk about an impedance, it often is in the form : 40 + j30 ohms. The (real) resistance is +40 ohms, and the reactance (the imaginary part) is +30 ohms. A positive reactance is inductive, and a negative one is capacitive. But simply adding the two numbers together (40 + 30) gives a totally different value.

We can also express a point on the complex plane as a distance from the origin * along with an angle*: this corresponds to the “magnitude” of the impedance and the “phase angle”. These forms are interchangeable, as any point can be identified either way. But the magnitude by itself doesn’t tell us the location of the point, because it could be in any direction.

You can find more information on complex numbers with an online search. This isn’t a complete mathematical treatise, just a quick introduction to those who haven’t used them before.