Glimpsing Connections

Learning a new subject in mathematics is always interesting. You learn new techniques to analyze problems and get to investigate the relationships between objects. Whether you’re learning about probability, algebra, geometry, or any other field, you will do these two things. The idea is to expand your toolbox to apply to various mathematical problems.

That’s fine, but it’s not the reason that I enjoy mathematics. For me, it comes from a slightly different place.

The reason I enjoy mathematics is because one can often analyze a problem or an object in different ways. In school, classes outside of mathematics and science often operate along the lines of, “as long as you can argue persuasively for your position, that’s what matters.” In mathematics, it’s more like, “as long as you get to the right answer at the end using logical steps, you’re good to go.” My point here isn’t to get into a debate about the “right” way to do mathematics education. Rather, I want to show that the method doesn’t matter so much as where it leads you. This opens up the possibility for many approaches to analyzing an object or solving a problem.

I came across this idea while working through a proof in my complex analysis class. I needed to show that when you add the real and imaginary parts of a complex number, the result is less than or equal to the magnitude of that complex number multiplied by the square root of two. My first instinct was to use the triangle inequality (since this result was an inequality). However, while I was able to get the result, I wasn’t very satisfied with the proof. It didn’t provide a sense of why it was true. I started thinking that there must be another way to do this.

After some thinking, I stumbled upon two more proofs that highlight this fact in a much more straightforward manner. One involves combining the real and imaginary parts (after taking the absolute value of each component) using trigonometric identities in order to get one trigonometric function at the end. Since this function only oscillates between 1 and -1, the maximum of the function is the factor multiplying the function in front, namely the magnitude of the complex number multiplied by the square root of two. Therefore, the inequality is proved.

The second way is similar, and involves the fact that we want to maximize the scalar product of two vectors: (cosθ,sinθ) and (1,1) (this is for the first quadrant, but can be extended to the others). Doing so means the vectors are parallel, which implies cos&theta = sinθ, giving us our desired angle and hence the inequality.

The first approach is more algebraic in nature, but the advantage is that it becomes clear why the maximum occurs where it does. When you combine both trigonometric functions, there’s no question about where the maximum is. The second approach is more geometric in nature, but I think it provides a good sense as to how we require both components to be equal in magnitude. In my mind, these two approaches make the result a lot more obvious than through pure manipulation. Is it more brief? Not really. But I think it makes up for that in its ease of understanding.

Without going through all the details, you might not share my enthusiasm about these proofs. But the reason I enjoy them is two-fold. First, they don’t follow the “standard” approach. I’m sure the usual solution would present the argument using the fact that you can square the sum of the real and imaginary parts and obtain some inequalities. However, what I like about these solutions is that they are (at least, to me) more creative. When you look at complex numbers, your mind might not jump to combining sines and cosines, or using the scalar product to find the bound. This is one reason I like these proofs. Second, these proofs connect different parts of mathematics together. Although complex numbers can be thought of as a purely algebraic system, one can also translate this into geometry using the language of two-dimensional vectors. Why does this work? Because it turns out that the complex numbers and the two-dimensional real plane are isomorphic to one another. Essentially, this means that I can translate a problem from one setting to the other without “losing” any crucial information.

This is what I love about mathematics. I get most excited when I see a problem get picked up from its usual setting and transported to another. The strange thing is, this often doesn’t help one solve the problem. I just find it to be a cool thing you can do, and it illustrates how mathematics is a lot more interconnected than you might realize from school.

I like to look out for this now. When I see a problem, I try to recast it into a different (but equivalent) setting. I want to see if I can connect a problem to another area, no matter how far or seemingly irrelevant.

Of course, if you look and try hard enough, you can always find connections between ideas. That’s just the way our mind works. I realize that some connections might be useless, but I’m not trying to make problems easier. I just want to see how many problems or ideas I can explain in different ways. It’s a challenge in translation.

The most important thing is that you’re able to get to a result. As long as you can do that using logical steps, you’re fine. However, the fact that most problems tend to have multiple approaches makes mathematics such a lovely subject. With enough time spent thinking about a problem, you can find many more connections than you would expect at first glance. Furthermore, having different ways to work through a problem increases the chance that a result seems “obvious” to you. Remember, just because you understand each step of a proof doesn’t mean the result will seem obvious or inevitable. By understanding how to tackle a problem in various ways, you increase that chance that the solution will be clear.

In my eyes, mathematics is about the art of glimpsing connections between different areas of study. Often, these come in the form of delightful proofs that use novel techniques or methods which buck the standard approach. These help highlight the deeper connections between subjects in mathematics.

Remember, there’s a whole rich web of connections right between your feet. You can spend your whole life above this mathematical web, but it’s worth taking the time to dive deep into it and glimpse the many connections that lay there.