The Set of Complex Numbers

The set of complex numbers is two-dimensional, and a coordinate plane is required to illustrate them graphically. This is different to the real numbers, which are one-dimensional, and can be shown by a simple number line. The rectangular complex number plane is constructed by arranging the real numbers along the horizontal axis, and the imaginary numbers along the vertical axis. Each point in this plane can be assigned to a unique complex number, and each complex number can be assigned to a unique point in the plane.

A complex number really does keep track of two things at the same time. One of those things is the real part while the other is the imaginary part. For example, z = 3 + 2i is a complex number. The real part of z is 3 and the imaginary part of z is 2. The everyday meaning of ''imaginary'' is something which doesn't exist. The meaning in math is quite different. Identifying the imaginary part of a complex number is easy because it has a label. The imaginary part is the number multiplying the label i'. The imaginary part of 3 + 2i is the 2. Be careful because the imaginary part is not 2i. The imaginary does not include the label.

At first solving problems such as “√-39” and “x2+1=0” were impossible.

However, mathematicians soon came up with the idea that such a number to solve these equations could be created. The number is √-1, known as i. It’s a good thing that scientists, mathematicians who didn’t want a new number created, and other non-believers finally allowed i (and complex numbers) in the number system. “i” is very useful to the world today. “i” is an amazing number. It is the only imaginary number until you square it then it becomes real. It wasn’t instantly created though, it took several centuries to convince certain mathematicians to accept this new number. Eventually, though, a section of numbers called “imaginary” was created (which also includes complex numbers, which are numbers that have both a real and imaginary part), and people now used i in everyday math.

In about 50 c.e. an impossible problem emerged when Heron of Alexandria, a Greek mathematician, was trying to evaluate the square root of a negative number. He later found out that the square root of a negative number cannot be evaluated using only real numbers. To find a solution, another new number system needed to be invented. In the mid-sixteenth century, an Italian mathematician named Girolamo Cardano began to do just that. He is recognized as the discoverer of the imaginary numbers that play an essential role in understanding the complex number system. Carl Friedrich Gauss, a nineteenth-century German mathematician, is credited with inventing and naming the complex number system. Complex numbers are generally expressed in the form a + bi, where a and b are real numbers and i is the imaginary number. The “a” part is considered the real part of the complex number and the “bi” part is the imaginary part of the complex number. Upon further inspection, we can see that the set of complex numbers includes all the pure real numbers, together with all the pure imaginary numbers, together with many more numbers that are sums of these.

In other words, whenever a complex number has b = 0, it is actually a pure real number too because it is equal to a + 0i, which is just a real number. Whenever a complex number has a = 0, it is actually a pure imaginary number because it is equal to 0 + bi, which is just bi, an imaginary number.

Electrical engineers often use complex numbers in their calculations every day. For example, when electricity flows through an electrical circuit component like a light bulb that resists the flow of electricity by using some of it to do work, the current flowing through the bulb is purely real and can be measured by a current meter accurately. When electricity flows through other devices where no real current can go, the current flowing through that device is purely imaginary. If it were measured the current flow through the device would show zero because a meter can only measure the real part of the current. Think of it like this: a coil is just a wire, so if you run electricity through it there is no real voltage drop because there is no real resistance. A capacitor is just two pieces of metal that do not touch so if you put a voltage through it no real current can flow.

If I needed to describe the distance between two cities, I could provide an answer consisting of a single number in miles, kilometers, or some other unit of linear measurement. However, if I were to describe how to travel from one city to another, I would have to provide more information than just the distance between those two cities; I would also have to provide information about the direction to travel, as well.

The kind of information that shows a single dimension, such as linear distance, is called a scalar quantity in mathematics. Scalar numbers are the kind of numbers you’ve used in most all of your mathematical applications so far. The voltage produced by a battery, for example, is a scalar quantity. So is the resistance of a piece of wire, or the current through it.

However, when we begin to analyze alternating current circuits, we find that quantities of voltage, current and resistance are not the familiar one-dimensional quantities we’re used to measuring in DC circuits. Rather, these quantities, because they’re dynamic, possess other dimensions that must be taken into account. Frequency and phase shift are two of these dimensions that come into play. Even with relatively simple AC circuits, where we’re only dealing with a single frequency, we still have the dimension of phase shift to contend with in addition to the amplitude.

In order to successfully analyze AC circuits, we need to work with mathematical objects and techniques capable of representing these multi-dimensional quantities. Here is where we need to abandon scalar numbers for something better suited: complex numbers. Just like the example of giving directions from one city to another, AC quantities in a single-frequency circuit have both amplitude and phase shift. A complex number is a single mathematical quantity able to express these two dimensions of amplitude and phase shift at the same time.