Do you know what a complex number is? Or probably you’ve heard about the imaginary unit? How did mathematicians happen to create them and what for? If you ever take a course in complex analysis, you already know the answer. If you don’t, it’s not a problem either as getting the basics of complex numbers doesn’t require you to be a math master, a school course is nearly enough. Historically, the concept of complex numbers arose from the problem of solving quadratic equations. Let’s recall the general form of the quadratic equation:
ax^2+bx+c=0
where $a,b,c$ are constants. Roots of such an equation can be found with the well-known formula:
x_{1,2}=\frac{-b \pm \sqrt{D}}{2a}
where $D=b^2-4ac$ is the discriminant of this equation. As we know, if D>0, then the equation has two distinct real roots. In case D=0, there’s one real root. In the case D=0, elementary algebra shows that the equation has no real roots. Why so? Let’s consider an example. Suppose we have an equation:
x^2+4x+5=0
First, we need to find its discriminant:
D=4^2-4 \cdot 1 \cdot 5=16-20=-4
Therefore, the formula for roots goes like this:
x_{1,2}=\frac{-4 \pm \sqrt{-4}}{2}
As you can see, we have \sqrt{-4}. But square roots are defined only for non-negative numbers! And that makes sense, as notation of a square root \sqrt{-4} is simply a way to say that this number raised to the square will give -4. But we won’t find such a number among real numbers whose squares are negative. That is why elementary math teaches that there are no square roots of negative numbers.
Watch our video tutorial on complex numbers:
Imaginary unit
So, there’s no place for \sqrt{-4} among real numbers. Can we imagine some other type of numbers containing it? Actually, mathematicians have already done this a long time ago. They started with denoting \sqrt{-1}=i, the so-called imaginary unit. In fact, any other symbol could be used instead of $i$, it simply seemed the most natural to adopt. Let’s consider the square of the imaginary unit:
i^2=\sqrt{-1}\cdot \sqrt{-1}=(\sqrt{-1})^2=-1
The expression for “i” squared may also be given as follows:
i^2=\sqrt{-1}\cdot \sqrt{-1}=\sqrt{(-1)^2}=\sqrt{1}=1
Both ways are seemingly correct and yet provide different outcomes. This makes us think that the whole approach is controversial. The point is that the notation i=\sqrt{-1} is obsolete mainly since it causes such misunderstandings. The proper definition of the imaginary unit is the following:
i^2+1=0 \rightarrow i^2=-1
Roots of negative numbers
To fully understand this we should involve complex analysis and other tricky stuff. We won’t do that now, and for starters just remember one thing. If you come across a square root of a negative number then immediately replace it as follows:
\sqrt{-4}=i\sqrt{4}=i\cdot 2=2i
Or in general:
\sqrt{-b}=i\sqrt{b}, b>0
We now know how to deal with roots of negative numbers, therefore we can get back to our initial equation and finally solve it:
x_{1,2}=\frac{-4 \pm \sqrt{-4}}{2}=\frac{-4 \pm i\sqrt{4}}{2}=\frac{-4 \pm 2i}{2}=-2 \pm i
That’s it. $i$ here can be treated as a flag indicating that the number next to it is of a different, unusual nature.
Numbers of different nature
Consider the resulting number as follows:
-2 \pm i=-2 \pm 1\cdot i
to indicate that $1 \cdot i$ is also a number, only of a different kind. We could go even further adopting a sign for real numbers as well:
-2 \pm i=-2\cdot r \pm 1\cdot i
This makes little sense, though, as we’re already used to reals.
Despite being of a different nature, imaginary numbers can be added, subtracted, multiplied or whatever. Addition and subtraction are no challenge, let’s see:
i-15i+3i=i(1-15+3)=-11i
Here we treat “$i$” as a factor that can legally be pulled out in front. Along with that, we can consider “$i$” as “$1$” of a different nature, subtract “$15$” of a different nature from it, and then add “$3$” of a different nature. The result is obviously “$-11$”, of a different nature. Both ways yield the same “$-11i$”, an imaginary number. Thus, we can regard $i$ either as a flag to mark a number as being imaginary, or as a factor beside real numbers.
Complex numbers
Now let’s move on to another example.
i+4i-3+\frac{5}{4}i=i(1+4+\frac{5}{4})-3=\frac{25}{4}i-3
We’ve obtained a sum of an imaginary number and a real number. Such structures are called complex numbers because they are literally a complex of two numbers of different nature – real and imaginary. It’s just like you have money of two different currencies in your pocket and you can’t just add them up to obtain a single value, of course, without converting. Consider one more example:
\sqrt{-23}\cdot 2+i-2=i\sqrt{23}\cdot 2+i-2=i(2\sqrt{23}+1)-2
Again, this is a complex number as well. In fact, we could say this number has this much 2\sqrt{23}+1 of imaginary part and this much -2 of real.
A complex number is a compound of two numbers – real and imaginary. The latter are recognized by the i mark used to identify them.
The whole point of this imaginarity and different nature we’ve been talking about is the way multiplication of such numbers is performed. Suppose we want to multiply two imaginary numbers:
2i\cdot 3i=6i^2=6\cdot (-1)=6
We multiplied two imaginary numbers and obtained a real number.
2i\cdot 3=6i
Here by multiplying an imaginary number by real number we still obtain an imaginary. And that’s worth remembering.
General representation of complex numbers
Complex numbers are often denoted by letter z. Real part of a complex number z is denoted as {\rm Re}(z). Imaginary part of a complex number is denoted as {\rm Im}(z):
z=\frac{25}{4}i-3
{\rm Re}(z)=-3
{\rm Im}(z)=\frac{25}{4}
In general, a complex number can be represented as follows:
z={\rm Re}(z)+i{\rm Im}(z)
We’ve emphasized that a complex number contains a real part and an imaginary part. Therefore, it’s convenient to represent complex numbers as pairs:
z=({\rm Re}(z),{\rm Im}(z))
Thus, the first number in a pair stands for the real part and the second for the imaginary. So, if we have z=(2,-1), it’s just another way of notation z=2-i. Remember that the order of numbers in a pair is fixed, otherwise how could we tell which number is real and which is imaginary? The numbers in a pair cannot be switched without consequences. Let’s see what happens if we do switch them:
z=(2,-1) \Leftrightarrow z=2-i
z=(-1,2) \Leftrightarrow z=-1+2i
As you can see, this seemingly minor change resulted in a completely different number.
Comparison of complex numbers
Considering real numbers, we’re used to comparing them. For example, we can say that -4<1. But how do we compare two complex numbers? The answer to that is in no way. Complex numbers cannot be put in any sort of order, we cannot say which of the two complex numbers is less or greater than the other one. In fact, we can only compare their real parts and imaginary parts separately. However, two complex numbers can be equal. This happens if and only if their real and imaginary parts are correspondingly equal.
Summing up. The main idea is that the real numbers we’re all used to are not the only ones. In fact, we can create various types of numbers by defining rules for operations with them. The only thing stopping us from doing that is a lack of sense. However, complex numbers do make sense, as, for example, they allow to solve any quadratic equation and write down its solution, even if its determinant is negative. Of course, this approach leads much further than just solving quadratics! For example, look what beautiful stuff can be generated with the help of complex numbers:
This is called the Mandelbrot set. Surely, you’ve heard about fractals! This is one of them. To generate it, we need to take a complex number, multiply it by itself, and add it to the original number; then take that result, multiply it by itself, and add it to the original number; and so on. If the number generated during the iteration grows ever and ever larger, then the original complex number is not in the Mandelbrot set. If the sequence converges, drifts chaotically, or cycles periodically, then the number is in the set.
Hopefully, now you’re a bit acquainted with complex numbers. In the next sections we’ll consider how to perform operations with complex numbers and their geometrical representation, and a lot more. This section was an introduction to give you a mere idea of the concept of complex numbers. Next, we’re going to offer a bunch of examples with detailed explanations on how to deal with complex numbers. By the way, since complex numbers are also numbers, there are plenty of online calculators available, for example, this. If you have any questions about complex numbers, don’t hesitate to ask and get complex analysis answers for free. Do math!
