Complex Analysis: This area of mathematics exploring the domain outside of real numbers used to be my favorite subject in mathematics. Yes, used to be because I am now aware I only learned a surface-level of what the subject is really about. Nevertheless, ever since I took Complex Analysis course during my undergraduate, I’ve been in love with all the ideas that the subject paints. I felt that the subject was, beautiful.
Recently, I took a course about Complex Analysis for my doctoral program, and this course will remain as one of the most interesting mathematics courses that I have ever taken. Well, firstly, the professor was mostly unprepared (sorry professor, but that’s how I felt!!) which made each class quite chaotic, so everyone had different ideas on what was going on in the course. This made class discussions, to put it mildly, intriguing: there are so many different misunderstandings that one simple concept can provoke!
Second (and better) reason why the course was memorable was because of this one midterm question: “What is the most beautiful concept that you learned in class, and how is this beauty of mathematics different from other types of beauty that human beings typically encounter?”
Thanks to this question, I spent 6-7hours to complete that untimed (thankfully!) exam.
Before introducing my answer to this beautiful question, let me briefly introduce the subject. The subject of Analysis typically examines ways to understand behaviors of extremely complicated functions or mappings by using functions and mapping that we do know (e.g., polynomials). Thus, Complex Analysis is about doing exactly that by incorporating complex numbers (i.e., numbers that are “beyond real numbers”). Any complex number can be considered as a composition of its real and imaginary parts (i.e., given any complex number z, it can be written as x+iy, where x and y are real numbers; x is called the Real Part of z (= “Re(z)”), and y is called the Imaginary Part of z (= “Im(z)”).
While this idea seems simple and straightforward, it has a profound impact on how we perceive numbers and points on the Cartesian plane. When we examine the regular xy-plane, we learned that x and y are completely independent and that they never can affect each other. No operations are possible.
(Well, I guess, that is unless you consider the points as vectors on the plane. But even if you do so, there are only limited operations you can do, and no basic arithmetic operations other than addition and subtraction that produces another vector on the same plane)
However, considering a complex number in terms of its real and imaginary parts, one could consider any points on a Cartesian plane and consider them as numbers, now able to utilize many arithmetic/algebraic facts from real numbers that they are familiar with. Now, you can multiply, divide, and exponentiate any two points on the Cartesian plane. (Well, the Cartesian plane would need to be defined as a Complex plane, but you get the gist!)
And this idea of marrying the two ideas of arithmetic and xy-coordinates was how I began my answer.
As many students study mathematics in school, one big theme that stands throughout their journey is how to define numbers. The first definition of numbers that students encounter is quite limited, not even being able to subtract a bigger number from a smaller number. Gradually, this confined idea of natural numbers expands to Integers, to rational numbers, irrational numbers, real numbers, and so on.
To me, this idea of complex numbers, in a way, sets the “destination” for numbers before delving into the more abstract structures of mathematics. And observe how this destination connects to other areas of mathematics, in this case, the Cartesian plane.
Studying mathematics, one would see that this connection, as much as it is beautiful itself, also enables so many other different ideas, marking the departure to the ideas of Complex Analysis.
I believe this is the beauty of mathematics: there is something – some kind of abstract structure that exist in this world (or, as Plato would say, outside of this physical world) that the semiotic language that is mathematics allows us to investigate. We do not know where this abstract structure would take us, but our language of mathematics renders us a tool to explore. Sometimes, we may face the dead-end, or, sometimes, we may discover something that would lead us to whole new domains to study. I know Karl Weierstrass called it poetic, or others would call it explorative; doesn’t matter—to me, this poetic, explorative characteristic of mathematics is beautiful.
Aligning with this idea, I suggested that this kind of beauty of mathematics is very different from other kinds of beauty that human beings typically encounter. In daily life, the definition of beauty often is confined to senses: that is, pleasing in visual, auditory, or other sensory manners.
But mathematics don't always have this kind of beauty.
Though, one cannot expect to always find beauty in such sense in the field of mathematics. Sure, we can design cool visuals or establish geometric intuitions by manipulating ideas in mathematics such as Julia sets, visualization of Mobius transformations with Riemann Sphere, or all sorts of cool functions.
However, such visual/geometric representations are often insufficient to depict the beautiful ideas that mathematics renders. (Not saying Julia sets and other stuff are bad. But we cannot only rely on visuals or geometrical intuitions when learning mathematics is what I’m saying.)
For example, before the establishment of epsilon-delta definition of limits, mathematicians used to think of limits as “approaching” a certain value. While such loosely constructed intuition can be useful, it may be misleading, as it allows, for example, one to say that 1.99999… is a number that is approaching 2 (when it is exactly 2). With the usual kind of beauty, the idea of limit approaching a certain value may be more beautiful. However, the concept that allows all kinds of other beautiful mathematics to happen by bringing the abstract idea of limit into a rigorous, precise concept is the epsilon-delta definition. Blinded by geometric, intuitive beauty, one may overlook the other kinds of beauty that mathematics allow us to cherish.∎