Cool Things I Learned From Studying Math

In the last three and a half years of pursuing an HBSc in Pure Mathematics, I have often been asked, “What do you do with a degree in Pure Math?” While I would be delighted to talk at length about all the jobs I could take up, the truth is I don’t really know. But what I do know is a bunch of random facts about numbers, matrices, integrals, graphs and other mathematical jargon. So here are some cool things I have learned from my time studying math.

Vacuously True Statements

In my very first semester of university, I took an Introduction to Proofs course. In the four months that followed, I quickly learned that math has a strict “innocent until proven guilty” rule. In simpler words, everything in math is assumed to be true, unless proven otherwise. On the contrary, even one piece of evidence (i.e. a counterexample) is enough to disprove a statement. During this course, I learned that mathematicians do not like being in a gray area: A statement can never be somewhat true or somewhat false.

This leads to one of my favourite concepts in math and logic — Vacuously True Statements. These are statements that are always true because the antecedent cannot be satisfied. For example, here is a vacuously true statement:

Every flying polka-dotted elephant in this room is smoking a cigar

On face value, this statement is obviously ridiculous and false. But a mathematician would argue that it is, in fact, always true.

This statement has two parts.

A: There exist flying polka-dotted elephants in this room

B: Every such elephant in this room smokes a cigar.

The premise of the statement (flying polka-dotted elephants) can never be true. Since flying polka-dotted elephants do not exist, you cannot find any such elephant that does not smoke a cigar. In other words, you cannot disprove B since A is false.

The Best Way To Hold A Pizza

From my many years of devouring pizzas, I learned that the best way to eat a large, flimsy pizza slice is to fold it into a U-shape. In fact, this practice is backed by differential geometry.

To understand why a U-shape is the best way to hold a pizza slice, we must first understand Gauss’s Theorema Egregium, which literally translates to “Remarkable Theorem”.

Imagine you are holding a sheet of paper. If you roll this up, it would form a cylinder. The surface of this cylinder is a curved surface, while the flat sheet of paper isn’t a curved surface. While this is obvious, Gauss wanted to define this transformation from a flat to a curved surface. This idea led to what he called a Gaussian Curvature.

To define Gaussian Curvature, picture an ant walking on the cylinder. The ant can go one of two ways (broadly speaking). It can either walk along the curve (represented by pink) or walk up the flat side of the cylinder (represented by blue). In this case, these are the two most extreme possibilities. In other words, the flat side has a minimum curvature of zero, and the curved side has maximum curvature (greater than zero). On multiplying the two curvatures, you get the value of the Gaussian Curvature. For the ant walking on our cylinder, the Gaussian curvature is zero. In a more practical sense, this means that the cylinder is flat. This explains why a sheet of paper can be folded into a cylinder.

The Gaussian curvature of a surface can be non-zero as well. Consider the ant is traversing a basketball. There are no flat surfaces. For the entirety of the curved surface, the maximum and minimum curvatures are always greater than zero. This means that the product of the maximum and minimum curvature, the Gaussian Curvature, is always greater than zero. This means that a sheet of paper can never be transformed into a ball.

How does this help us with pizzas, though? One consequence of Gauss’ theorem is that any flat surface will always have a Gaussian Curvature of zero, regardless of its shape. An unfolded pizza slice is flat. When it is folded, there are two extreme curvatures — the curved part (represented in blue) and the flat part (represented by green). When you multiply the two, you get a Gaussian curvature of zero. This preserves the pizza slice’s flatness, making the U-shaped slice the most convenient shape for munching on pizza.

Prime Numbers

As the extensive Wikipedia page on prime numbers will tell you, there is a lot to love and learn about the primes. Prime numbers are numbers that cannot be written as the product of two smaller numbers.

These numbers are the foundational unit of the number system. If you have learned any math at all, you have most likely heard about the fundamental theorem of arithmetic or some form of it. It states that “every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.” In other words, any natural number after 1 is a prime or can be broken down into a product of two smaller prime numbers.

But what fascinates me about the primes is that even though they are the building blocks of math, there is so much we still don’t understand fully. For instance, the set of all primes shows no clear pattern, which makes their infinity even more intriguing. Despite many attempts, no one has found a pattern, yet Euclid proved there are infinitely many primes, which is truly mind-blowing.

Euclid’s idea was simple, yet indisputable. If there are infinitely many numbers (which there are), there will always be a new way to construct a new prime number. This is one of the fundamental concepts of number theory and leads to many other fascinating concepts, such as divisibility, congruences, and residues. In applied mathematics, the primes provide essential results for cryptography, computer science and engineering.

While these are just some interesting concepts, mathematical logic and reasoning can explain most phenomena around us. By combining concepts with other fields such as physics, chemistry, and economics, mathematics attempts to form a complete system to understand the world around us.

If you want to learn more about any of these concepts, here are some useful resources to start with:

Vacuously true statements

• Vacuous truth: https://en.wikipedia.org/wiki/Vacuous_truth
• Trivialities and vacuousness: https://personal.math.ubc.ca/~PLP/book/section-10.html

Gaussian Curvature

• How a 19th Century Math Genius Taught Us the Best Way to Hold a Pizza Slice: https://www.wired.com/2014/09/curvature-and-strength-empzeal/
• Gaussian Curvature: https://en.wikipedia.org/wiki/Gaussian_curvature

Prime Numbers

• Prime Number: https://en.wikipedia.org/wiki/Prime_number#:~:text=The%20first%2025%20prime%20numbers,sequence%20A000040%20in%20the%20OEIS).&text=the%20product%20%E2%81%A0-,%E2%81%A0.,is%20called%20an%20odd%20prime..&text=the%20product%20%E2%81%A0-,%E2%81%A0.,is%20called%20an%20odd%20prime.)
• Euclid’s Theorem: https://en.wikipedia.org/wiki/Euclid%27s_theorem