What complex numbers are and what they are used for.
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The words complex and imaginary that usually get assigned to this concept are really poor names unfortunately. They tend to make people think they aren’t real or complicated or esoteric.
Complex numbers can be thought of as 2dimensional numbers. Just like how on graph paper you can measure how far left-right you are AND up-down you are from point to another, you can do the same with complex numbers. That’s all they are. Instead of having a number line, you now have a number “grid” or plane to be more technical.
They are used everywhere amazingly enough. You’ll find that some of the most common applications is for things that oscillate or rotate. There are ways to describe this behavior as rotations in complex numbers. Turns out, a lot of stuff oscillates and rotates in one way or another. There’s even math you can do that allows you to describe ANYTHING (even if it’s not oscillating) as some combination of oscillations and rotations. So the applications are practically endless. To name some concrete examples and further reading terms: quantum mechanics, Fourier analysis/transformation, electrical engineering
Complex numbers come with trigonometry “built in”. So anytime you have some circular geometry to describe, you can probably do it most elegantly with complex numbers.
They are used to pack an extra dimension into algebra and trigonometry so you don’t have to use integral calculus to solve engineering problems. The real part of the number represents the x axis, the imaginary part of the number represents the y axis, and Euler’s formula of e^ix = cos x+ i sin x relates them to pi and the natural log because e^(i*pi)=-1.
Natural logarithms model natural processes perfectly with simple equations. For example, charging a capacitor with a fixed current results in a voltage that has an e^x shape. When we want to calculate the current that results from changing the voltage on a capacitor, we have to use the formula I=C*dV/dt, then integrate.
In electrical engineering we use complex numbers to describe the behavior of AC circuits by mapping the periodic cycle of a sine wave onto a unit circle. This allows us to use trigonometry to transform amplitude and phase angles into vectors. This allows us to calculate the impedance of a capacitor or inductor at a specific frequency using simple linear algebra instead of solving an integral, because Euler’s formula maps e^x to complex space. Thus, an electrical circuit that would take hours to solve with calculus takes minutes to solve with linear algebra, all because Euler’s formula relates imaginary numbers to logarithms to x-y coordinates.
When dealing with something oscillating, a number that records the characteristics of oscillation isn’t sufficient, because the underlying process that we observe as oscillating could be actually rotating. We need two perpendicular views of oscillation to fully describe a rotation. With only one view, there wouldn’t be enough data.
In other words, when you are not seeing something moving doesn’t mean it’s not moving. It could be it’s moving in another dimension.
Complex numbers are invented for this purpose. It separates the part of what we see from what is actually going on.
True ELI5: Maths wizzes figured even though there are infinite numbers, there weren’t enough of the normal numbers to do the calculations they needed in a nice way, so they invented more numbers. As many more new numbers as there are normal numbers. They called these new numbers ”complex”. Now they could do the calculations they needed more conveniently.
So, complex numbers are made up of 2 parts, a real part and an imaginary part. The real part are the numbers we know and love.
So imaginary numbers are used to find solutions to problems where the real number system doesn’t provide an answer.
So, take squares. One squared is one. So what number squared will give you -1?
Well it won’t be a real number, but let’s just call it ‘i’.
Now we don’t know what ‘i’, because it’s not part of the ‘universe’ or real numbers. However, it’s an incredibly useful concept that allows us to solve all kinds of equations. Now that doesn’t mean it’s just an academic thing. It turns out that it’s very useful in physics, especially electro-magnitism.
I see a lot of explanations here, and I don’t know if this would count as an ELI5, but if you want some history and visual aids to go with an explanation, Veritasium did a great video on topic that may help you understand the concept of complex numbers and what they’re used for with less confusion:
# Veritasium | [How Imaginary Numbers Were Invented](https://www.youtube.com/watch?v=cUzklzVXJwo)
To expand on some of the other info here, the sinus and cosinus functions are defined using a circle. While a normal equation, like x+3, would yield a straight line on a graph, the sin and cos functions draw a perfect circle. However, when you do math with these, you get some weird numbers that don’t make sense in reality, like a negative force when it should be positive.
Mathematicians then came up with “complex numbers” and when you use these numbers to calculate things with sin and cos, the results make sense and align with the real world
Here’s a slightly different take from other explanations. In school, we have been taught about how plus, minus, multiplication, reciprocals etc are computed between numbers.
But it turns out, mathematicians have taken that basic idea and generalized it to different types of ‘objects’ besides numbers.Objects include ‘real’ numbers (the numbers you are famliliar with), integers (whole numbers without a decimal point), vectors (a sequence of numbers), matrices (a two dimensional table of numbers), rational numbers (like 3/4) and so on.
For each of these object types, mathematicians have type-specific rules for how to add and multiply and subtract and take reciprocals. When you say a + b = c, if a and b are matrices, then the operation will yield the matrix c. Similarly, there are rules for multiplication, division, reciprocals etc. There’s a lot more that can be said on this topic, but it is not germane to your question.
Now to your question. A complex number is one such type of object; it is merely a pair of real numbers. Forget the terrible terms complex and imaginary.Suppose complex number p is a pair of the form (a,b) where a and b are real numbers, and q is (c,d), then the rule for addition is _defined_ as follows: p + q = (a,b) + (c,d) = (a+c, b+d). That is, the result is a pair where each is a pairwise summation of the first and second parts of p and q. Say, p = (10,20) and q = (30, 40).Verify that p + q = (40, 60).
Likewise, multipication p * q *is* *defined* *as* (a*c – b*d, a*d + b*c).
Verify that p * q = (-500, 1000).
This looks like an arbitrary rule, and there’s nothing intuitive about it on the face of it. But it turns out complex numbers map directly to 2D geometry. A point in 2D space can be represented by a complex number since it is a pair of numbers. Imagine a point p = (10,20), and imagine a line joining the origin (0,0) to p. Without any confusion, we can refer to the line also as p. Each complex number has a corresponding radial line leading to it.
Verify that p * (0, 1) = (-20, 10). Call this point (and corresponding radial line) q.
If you plot the lines p and q, you will find that the act of multiplying p by (0,1) has the effect of rotating p counter clockwise by 90 degrees, yielding the line q. Check it out. Similarly, p + (10, 0) has the effect of shifting p 10 units to the right on the x axis, and p + (0, 20) has the effect of shifting the point 20 units vertically.
Now, If you have a shape defined by a bunch of points, then multiplying each vertex by (0,1) would end up rotating the shape 90 degrees, since all points rotate counterclockwise the same way.
More generally, multiplying by arbitrary complex numbers ends up rotating and shearing shapes in different ways. I’ll stop here, except to add in passing that since we are speaking of rotations (and hence angles), you can imagine deep connections to trigonometry.
Complex numbers are often presented as “a real number & an imaginary number”. However, this is not IME the representation that truly capture what complex numbers means. There is a mathematically equivalent representation of complex numbers which is
**Complex number = Real number & Angle**, where every 180° in the angle means multiplying by -1.
As such, a complex number can be seen as “you were in the middle of changing a +3 into a -3, but you stopped at the middle of changing the sign so you’re in between positive and negative”. Said otherwise, the angle can be seen as representing a rotation or an oscillation of some sort.
It means that complex numbers find use everywhere there is some rotation or oscillation. In electricity, alternative current oscillate => complex numbers are used to represent the current. In quantum mechanics, you will often use both an amplitude (the real number) and a phase (the angle).
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