I studied applied mathematics, in which I did both a bachelor's and a master's degree.
Everyone knows the numbers 1, 2, 3 and so on. Also everyone knows that it is possible for numbers to become negative. Furthermore, we can have fractions, such as 1/2 or 27/36. Not all numbers can be represented as a fraction though. The most common example of a number that is not a fraction is pi. It starts as 3.1415 and continues on forever with no clear pattern in it. These numbers are called irrational numbers. This gives us a couple of sets of numbers.
- Natural Numbers: Natural numbers are all positive numbers larger than 0. So 1, 2, 3 and so on. Whether zero also belongs to this set is a discussion between mathematicians, but is not of real importance.
- Integers: The set of integer numbers is the set of all natural numbers and all their negative counterparts. So this set consists of 0, 1, -1, 2, -2 and so on. So as you can see the natural numbers are a subset of the integers.
- Fractions: These are numbers that can be written as a division between two integer numbers, so 1/2 or -7/324. Clearly, all integer numbers are also part of the fractions since any integer number x can be written as x divided by 1. Therefore the integers are a subset of the fractions, and since the natural numbers are a subset of the integers, they are also a subset of the fractions
- Real Numbers: These are all numbers that appear on a number line. So if you point at one specific location on the number line you will point at some number, which may or may not be a fraction. For example, it might happen that you exactly point out pi, which is not a fraction. All these numbers form the real numbers. Clearly the real numbers include the fractions and hence they also include the integers and the natural numbers.
You might think that the set of real numbers contains all numbers, but this is not the case. We still have the complex numbers. These numbers are not necessarily on the number line, but instead they lie in the complex plane.
In the sixteenth century two Italian mathematicians tried to find a general formula to calculate the roots for third degree polynomials, i.e. solutions of equations of the form ax^3 + bx^2 + cx + d = 0. They succeeded in finding such a formula but they had one problem. For some third degree polynomials it might happen that you had to take the square root of a negative number to find one or more of the roots. This was thought to be impossible. However, the formula seemed right, since all solutions it gave for which no negative square root had to be taken were correct. If you would assume that you could take the square root of a negative number, it might give other solutions which are correct too.
This is how the imaginary number i originated. i is defined to be the square root of -1. Therefore, if we have to take the square root of -7, which is the square root of -1 times the square root of -7, it is equal to i times the square root of 7.
In the eighteenth century Gauss and Euler did a lot of work on this topic and they founded the fundamentals of the complex numbers as we know them nowadays.
Characterization of a Complex Number
A complex number can be written down as a+b*i. Here a and b are real numbers and i is the imaginary number that is the square root of -1.
To make notation a little bit easier, we call a complex number z. Then a is the real part of z, and b is the imaginary part of z.
As you can see, all real numbers are also complex numbers since they can be represented as a + b*i, where b = 0.
The Complex Plane
A complex number can be drawn in the complex plane. In the complex plane the horizontal axis is the real axis and the vertical axis is the imaginary axis. A number a +b*i corresponds to a point (a,b) in the complex plane. Then the absolute value of a complex number is equal to the length of the vector that goes from (0,0) to (a,b) in the complex plane. This means the absolute value of a complex number is the square root of (a^2 + b^2).
The complex plane gives us the option to represent a complex number in a different way. In the picture we see the angle theta, which is the angle between the real axis and the vector that corresponds to the complex number. This angle is called the argument of z. Now a is equal to the cosine of the argument times the absolute value of z and b is equal to the sine of theta times the absolute value of z. Therefore we have:
z = r(cos(theta) + i*sin(theta))
Here r is the absolute value of z and theta the argument of z.
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The famous mathematician Leonhard Euler found that the following statement holds for any number x:
e^(i*x) = sin(x) + i*cos(x)
Here e is the natural logarithm. In particular, when we fill in x = pi we get what is often called the most beautiful mathematical formula since it contains e, pi, i,1 and 0 and the three most common operations in math:
e^(pi*i) + 1 = 0
This formula implies that any complex number can be represented by a power of e.
z = r*e^(-i*theta)
Here r is again the absolute value of the complex number z and theta is the argument of z, which is the angle between the real axis and the vector that goes from the point (0,0) to the point (a,b) in the complex plane.
Euler's formula also gives the opportunity to represent the sine and cosine in a different way using powers of e. Namely:
sin(z) = (e^(iz) - e^(-iz))/(2i)
cos(z) = (e^(iz) + e^(-iz))/2
Applications of Complex Numbers
Complex numbers are not only just a tool to find the non-real roots of a polynomial or to find the square root of a negative number. They have numerous applications. A lot of them are in physics or electrical engineering. For example calculation regarding waves are made much easier when using complex numbers, because it allows to use powers of e instead of sines and cosines.
In general, working with a power of e is easier than working with sines and cosines. Therefore using complex numbers in settings where a lot of sines and cosines appear might be a good idea.
Also, some integrals become a lot easier to compute when we can look at it in the complex setting. This might seem very vague, and the explanation goes beyond the scope of this article, but it is an example in which complex numbers, or more general, functions of complex numbers, are used to simplify computations.
Complex numbers are an extension of the real numbers. A complex number can be expressed in multiple ways. The easiest is a + b*i where i is the imaginary number which equals the square root of -1. They can also be expressed using powers of e or sines and cosines. Both use the fact that a complex number can be represented as a point (a,b) in the complex plane.
Complex numbers are useful in practice since they allow you to take the square root of negative numbers. Often this makes computations easier.
This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.