Dynamics of Complex Numbers and Multiple Representations
LikeComplex numbers are amazing. Understatement. Allow me to go on. Who knew creating an abstraction for the square root of -1 would open up a whole field of mathematics? Let's dive into it.
Quick Introduction
Just to get an idea of what a complex number actually is:
where a, b are real numbers. The i here is core to the complex part of a complex number.
For a more formal description, they can be viewed as a 2-dimensional number, where a + ib can be denoted as (a, b) which follow the operations, addition and multiplication: (a, b) + (c, d) = (a + c, b + d) (a, b) * (c, d) = (ac - bd, ad + bc)
Motivation
What was the motivation for it? Seems to be a useless and non-practical thing which somehow just exists, for the sake of naming/abstraction. NO
x2 + 1 = 0, roots of this equation? You can catch on, just by observing a very basic equation.
Complex numbers were motivated by the fact that we could not figure out how to split 10, to get a product of 40. Well, not really, but you can get the point. It started out with solving polynomial equations, but evolved into something much more.
The concept of complex numbers have a subtlety to it which is not evident at first. You get the hang of it; the notation and the slight absurdity of it. But it slowly, but surely, unveils its magical powers.
Complex in Action
First of all, it allows you to express all the roots of an equation of any degree, whereas before, we just had to make do with saying it doesn't exist. See the equation above. Using the real plane of numbers, it can be a bit tedious to do some rotations or particular transformations mathematically, say of a picture, but the complex number system allows us to make this step many times easier.
These rotations and transformations are done through physics engines for FPS games, which is a monumental task, for realistic game-play and visual effects. Complex numbers have been used in physics, specifically, theoretical physics, to describe the nature of reality itself. String theory describes our reality as a 10-dimensional structure which has 4 dimensions describing our physical and perceivable reality and the remaining 6 dimensions are the 6-dimensional Calabi-Yau manifolds which are complex dimensions. It also shows up in the extremely well known Schrödinger's equation, which governs the wave-function of a quantum mechanical system. In the space of pure mathematics, we can assign meaning to something like logn(-1). Not only this, take it as far as you like. What would be the solution to sin(x) = π? cos(x) = -3? ln(i)?!?! This may seem absurd. Complex numbers can also be used during integrating some functions which may look cumbersome, mathematically. Say, you have to integrate the function 1/(x4 + 1). Now, obviously the denominator can't be factorized to ease the integration process, or can they? Looks like they can be, using complex numbers. x4 + 1 can be written/factorized as (x2 + i)(x2 - i). This can be done to factorized components again, till you get all of them to be linear equations, which can be much easier to handle as i is in fact a constant.
As we haven't seen the different representation of complex numbers, these equations would seem to be impossible to solve. But that is exactly the point. There is no answer on the real plane. We have to look to the complex plane for the solutions to some of the seemingly absurd equations pointed out in the last point.
Representations
As discussed above, complex numbers can be looked at as a 2D number. It has a real part which can be plotted on the X-axis and the imaginary part on the Y-axis. This is the simple, visual representation. Also as discussed, it has a rectangular representation, which is the simple, a + ib. Other than these two, we have a more interesting representation to observe. Any complex number, z, can be represented through a polar representation which is:
To describe some of the variables used in this, r = magnitude of the complex number(distance from origin to (a, b) = r), i is basically the same as defined above and θ = tan-1(b/a), also called the argument, which basically tells you the angle between the X-axis and the line connecting the point (a, b) and origin(complex number can also be looked at as a vector). Seems like an extremely weird representation. How did e come about in this? Not to go into too much detail, but it comes about through interpreting the rectangular form of the complex numbers in a more parametric way. A few steps are shown to give you an idea of what goes on, might write up a different article to show how the equality holds, but this can give you a decent intuition:
Digesting that maths, you can get an intuition of why any complex number can be written as z = reiθ. r is just a scaling factor for the unit vector, if you look at it through the lens of vector representation. The unit vector specifically, because, if r = 1, you are left with a complex number which has a magnitude of 1. For example, ei(0) = 1 + 0i, ei(π/2) = 0 + i, which both have a magnitude of 1, but have different angles from the X-axis. In the case of 1, as 1 lies purely on the X-axis, the argument is 0, but as i will be purely on the positive Y-axis, it has an argument equal to π/2.
Dynamism
Rotation
Most of the time, complex numbers are represented through the polar form. It gives us amazing capabilities to play around with objects, in this form. As you can see it has some angle in the representation, and it is the angle between the X-axis and the line from the origin to the point (a, b). Calculating rotation of vectors can be a bit tedious, more formally, to calculate a rotation of some vector by some angle, α. But if we look at the vector as a complex number, we just have to multiply it with a complex number of r = 1 and an argument θ = α. Why? Because a complex number, reiθ multiplied with eiα will equal to rei(θ + α), which basically means the original complex number has been rotated by α. Simple as that. You can then convert this back to rectangular form to get the rotated vector. A quick tip, if you want to rotate a vector by 90 degrees, just multiply the complex number for the vector by i. I think you can figure out why.
Logarithms
As complex numbers use the number e in their representation, you can see how logarithms can be used and produce weird, but spectacular results. You may know logarithm isn't defined for negative numbers, in the real plane. But in the complex plane, it sure is. Taking the natural log, ln, for a negative number say ln(-1), is defined and has an answer in the complex plane. Not just that, ln will be defined for even complex numbers. Why? The polar form representation.
As any number can be represented through the polar form, any number's natural log can also be calculated. In this case, ln(-1) = iπ.
sin(x) = π?!?!
How can sin(x) be more than 1? Just imagine complex angles. Again, we come back to the representation of the complex number, which has a lot of richness to it. sin(x) can be written as the difference of eix and e-ix divided by 2i.
If we take a = sin(x), X = eix which implies 1/X = e-iθ, then we get,
We get a quadratic equation which can be solved for X and subsequently for x, which gives us the solution for sin(x) = π or for any value. The variable a can be assigned any value, and you can generate an angle which will satisfy sin(x) = a. Taking it a little further, we can do the same for cos(x), tan(x) and obviously any other trigonometric function.
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Studying complex numbers, I have realized that many people prefer complex analysis to real analysis, just because of its amazing properties. As mentioned earlier, complex numbers can look pretty uninteresting, inconsequential and highly specific at first, but have amazing richness and use which can be taken apart and played with by just learning some of its different representations. If you feel this was an interesting introduction to the dynamics of complex numbers, you can continue the experiment.
