What are complex numbers and their practical applications?
In: Mathematics
A complex number is one that can be written as a+bi, where i^(2) = -1. i is referred to as an imaginary number, as you can’t multiply anything by itself and get a negative.
Complex numbers are common in cyclic physical phenomena, like damped spring systems, AC electrical circuits, and higher-level physics. In certain situations, if you try to solve the equations that describe the motion of these systems, the square root of negative numbers will become involved, hence introducing complex numbers. Euler’s formula states that e^(ix)= cos(x) + isin(x). It is here that we get from imaginary numbers in our equations, to a real component that describes physical systems.
Complex numbers are the way of solving something like sqr(-1). “Natural” numbers can be ilustrated as a “line”, for ilustrating “complex” numbers you would need a plane. Te lo explicaría mejor pero ingles es medio malo.
Think of them as 2-dimensional numbers. “Normal” numbers (ie, real numbers) can be represented on a line, complex ones on a plane. They have a real part (any normal number is complex number, you just set the imaginary part to be 0) and the imaginary part, which is the real new thing here.
The basis of said imaginary part is *i*, or the square root of -1. And it’s really as simple as not having an answer for what this square root was, so we just invented a symbol for it and went with it.
They have a number of interesting applications, especially when periodic behavior such as waves are involved.
Complex numbers are an expansion on the “regular” numbers, also called real numbers.
Remember when you, as a kid, were introduced to negative numbers, or decimals? They might have seemed weird at first, but once you got used to them they turned out to be really helpful. Now, you could answet queations like “What number is five less than two?”, “What if the temperature outside is 10 degrees and it drops by ten?” or “How much will everyone get if five people share 8 cookies equally?”
Complex numbers, which are very unfortunately named because they aren’t that complex and just as real as the real numbers, is sort of the same thing. You can think of it as adding another dimension to the number, perpendicular to the forwards-backwards way we usually count. Instead of just going back and forth on our regular old number line, we can also go back and forth on what is called an imaginary number line. That one also has a really bad name, because it is just as real as the real number line…
Complex numbers are immensely helpful for calculations including a wide range of topics, including but definitely not limited to quantum mechanics, electrodynamics, or just regular old newtonian mechanics.
By expanding the math we already have with complex numbers, initially making it slightly more complicated, we simplify a lot of later calculations!
Just to add a quip,
Yes, “complex” and “imaginary” are *extremely unfortunate* names as people get caught up in the words and what they mean, but the names come from a sort of historic precedent and, frankly, a bit of mathematician humor.
An example of an imaginary number is “3 + 2i”. Don’t think “algebra”, don’t think “arithmetic”, just because you see the “i” there, and the addition – think “notation” or “convention” rather than terms to be solved. What you have here is a 2 on the normal number line you’re used to, and a 2 along the second axis, what we call the “imaginary” axis. As others have said, this is effectively a singular 2D number, you can plot this on an xy graph and the construct represents a single number, just like you can plot “7” on a number line which is just a 1D graph.
I mean look, someone had to invent the decimal dot at some point, and I’m sure the notation we take for granted today blew the minds of everyone who saw it the first time.
And there are types of numbers that extend in multiple dimensions. Another one is the quaternion, which is in the form of “a + b*i* + c*j* + d*k*”. Beyond the complex number, these extend into 4 dimensions.
Where do we use them? A lot of rotations in math. I’m sure there are other uses, but in video games, this is what I principally use them for. Imagine a 2D graph, xy, like we’ve all done in grade school. Imagine an arrow from the origin, [0,0], to [1,0]. Basically, the arrow is 1 unit long and is pointing to the 1 on the normal number line. Often in video games we want to rotate a thing, and our arrow here represents our thing. The important thing is if I give some degrees of rotation, the arrow has to remain 1 unit long, because it’s easy to skew the length as you compute the new direction the arrow points. Because in video games and other things, while the line indicates direction and that should be all that matters, that arrow length can cause our thing to stretch, which is typically not what you want. We call this the unit circle. Quaternions are used to represent rotations in 3D space, because you can rotate along all 3 axis, xyz, at the same time.
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