What is a complex number 1

Complex numbers basics

What are complex numbers actually? This is exactly what we'll take a closer look at in the next few sections. We start from the very beginning, with the basics of the complex numbers. What do you need this for? How do you look? How do you represent them? This article is part of our Mathematics section.

Most of them have already learned about some types of numbers at school. For example there were the natural numbers like 1, 2, 3, 4, 5, there were negative numbers like -1, -2, -3 and also irrational numbers like the square root of 2 (i.e. about 1.4142 .. .). In the course of study - sometimes also in the upper level - there is another type of number. These are called "complex numbers". And these are really unusual.


What are complex numbers?

With the whole "normal" numbers that you usually get to know in school, you cannot get a number whose square is negative. Let's take as a simple example:

In school you would often simply say that this is not solvable, because what should you substitute for x so that after the square you get -1?

So we need something new that we now have to build for ourselves. For this purpose we now use the letter i - whereby in electrical engineering or physics a j is sometimes used instead of the i - and say the following:

From now on we will refer to the i as an imaginary unit. A complex number is made up of two components: a real part a and an imaginary part b, which we multiply by the i. A complex number looks like this:

  • Complex number: z = a + bi

Inset: Why do you need complex numbers?

What is this rubbish supposed to be? We just saw briefly how a complex number is structured. But why do you need something like that? Well, the complex numbers help in calculating problems in various natural sciences. In electrical engineering, for example, the complex numbers can be used to calculate alternating currents. It may be hard to believe at first, but without the complex numbers, the calculations would be even more difficult.


Complex numbers representation

How can one imagine the complex numbers? To do this, let's look at something that is reminiscent of the x-y plane in mathematics. If you have memory problems here, please take a look at the article Coordinate System Introduction. There is something similar for the representation of complex numbers. One speaks here, however, not of an x-y coordinate system but of the Gaussian plane of numbers (or Gaussian plane for short). Then one no longer writes x and y on the axes, but real part (Re) and imaginary part (Im).

Fig. 1: The complex numbers 2 + 3i and 4 - i are shown.

Let's just take the first complex number. That was the 2 + 3i. And then the picture is expanded a little. I entered a and b in red and the hypotenuse in green. I also entered an angle that is created in green.

Image 2: We enter a, b, the hypotenuse and an angle.

And as one can almost assume with the term "hypotenuse", a connection can be found here with the Pythagorean theorem (if you don't know it anymore, see article Pythagorean theorem). This would then be as follows:

In addition, these two connections can be found:

Working with Complex Numbers

We just discussed some of the basics about complex numbers. But not any more. In subsequent articles we will now look at how to deal with complex numbers, i.e. addition, subtraction, multiplication and division. The terms complex conjugate extension and polar coordinates are also discussed.

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