Beauty of mathematics

Written on 2019-06-08 in 574 words. Part of math studying

So I study mathematics. I study, reason and argue about structures emerging from models describing our world. Often enough, I consider mathematical results as boring and auxiliary side results bore the heck out of me. But there are times when these results merge again and you begin to see the inherent beauty of it.

I will give it a shot today. Specifically, I just want to show you a formula which seems boring to me. But if you begin to reason about it, we discover links to other fields of mathematics and recognize how many details are hidden in simple formulas.

Formula:

[ prod_{i=0}^n (x - x_i) ] here $i$ is an integer. $x_i$ is the known $i$-th root of a polynomial. $x$ is an unknown, real-valued variable. What is it good for? This formula describes polynomials made up of linear factors. Boring, right?

  1. We subtract and multiply real numbers in the formula. Thus, we manipulate elements with the laws of ℝ. But we also have a variable $x$. This extends our elements from ℝ to ℝ[x]. Field extensions are algebraic objects yielding beautiful patterns.
  2. Polynomials are pleasant objects. They are continuous and thus their graph does not jump around unpredictably. Even though it might look as crazy as Weierstrass' function. Additionally, their derivatives can be computed easily (as you might remember from high school).
  3. Derivatives? Why do we need that? It gives us an easy process to determine the notion of a “slope“ for a given function. Cool, right? It gives us an idea how much the function values change next to some given point. And these derivatives are polynomials themselves!
  4. Polynomials might have roots (hence some $x$ for polynomial $P$ such that $P(x) = 0$). They necessarily have roots (if they are not just a constant) in (Fundamental theorem of Algebra). So we can simply take the same real-valued polynomial, use the same formula and exchange real values with complex numbers. We get the same formula (by invariance of notation), but suddenly the polynomial has roots. Is it perplexing that polynomials like x² + 1 = 0 don't have roots in ℝ, but ℂ?
  5. Roots are interesting. Any minima and maxima of polynomials are necessarily roots of the derivative. What a surprising relation! This makes it easy to find them. But finding roots of generic functions is so interesting that it drives an entire industry.
  6. If $P(x) = 0$ we have $prod_{i=0}^n (x - x_i) = 0$. So if $x$ is any root (let's say $x_j$), we have $x - x_i = x_i - x_i = 0$ for some factor. Some factor is zero and only one factor needs to be zero to render the entire expression zero. And because ℝ is an algebraic domain, we know that $P(x) = 0$ implies that one of the factors is zero. So $x$ must have been a root.
  7. One factor from the formula above is the generic notation of a linear factor. Any polynomial can be written as product of linear and quadratic factors. Why not cubic or quartic? Fascinating!

I hope my wording made my excitement apparent. I can totally see that people don't find it as fascinating as me. But sometimes you just need to dig far enough to see the inherent structures and discover its beauty. Discovering mathematical properties takes quite some effort, but sometimes it is worth its beauty. Even if math is not your main field of interest.