Mathematical Beauty, and Why it is a Hidden Gem

Have you ever wondered why mathematics is so hard? Personally, mathematics is my favorite subject, but I know this is not the case for most, if not all, of my friends. Happy upcoming Pi day, and today, I will be exploring where the beauty of math comes from, as well as the reason why you might be struggling with math.

For me, the beauty of math lies in how everything links together, and how there is usually a clear answer for every one of my questions. In terms of high-school mathematics, mathematics is one of the most “straightforward” subjects. It is not like English, where every person’s response will very likely be different. For subjects like English, there is a possibility of marking “inaccurately”, as the grade depends on “how clearly” you communicate. Whereas for mathematics, there is a definitive answer. And yet, interestingly enough, there are many different ways to get the same answer. One example I immediately thought of was finding the hypotenuse of a right-angled triangle, if you are provided with the other two lengths. People would immediately approach this with Pythagoras's theorem, but it is also possible to solve this with trigonometry, or even by drawing a scale diagram! 

However, as I explored university level mathematics and unsolved problems, I realized that for some questions, it is the lack of distinctive answers that makes mathematics beautiful. I will take Collatz Conjecture, which I have talked about before, as an example. The great thing about Collatz Conjecture is that it is easy to understand. Start by choosing any positive integer. Then multiply it by 3, then add 1 if it is odd, or divide by 2 if it is even. Continue to repeat this process with the new number that you got. Eventually, according to Collatz Conjecture, you will reach 1, no matter whatever numbers you chose at the start. This is one of the most famous unsolved problems. Even computers have been used to check the theory until huge numbers are reached, and yet, since mathematically, no counter-examples disprove the conjecture, this is near impossible to solve. In fact, no one in the world has been able to solve it.

But then, if mathematics is such an elegant puzzle, why does it torment so many students? One of the main reasons why mathematics is such a “difficult” subject is because math is a subject that builds on the previous topic. Everything you learn at the start, like numbers, fractions and decimals becomes a foundation for everything you learn. If your foundation in these initial topics is weak, then mathematics will become pure memorization. Mathematics is like a Lego building. If your base is strong and solid, building up is not too much of a struggle. The building is not shaky; it stands tall. But if your foundation is not solid, and the foundation is weak, the building is easier to break. Algebra is hard, some people say. But maybe the problem doesn’t lie in Algebra; maybe you are struggling to understand the basics that you learned before you did Algebra?

Overall, mathematics is a beautiful subject if you understand it. It is a nightmare if your previous ‘layer’ is weak. Both the beauty and nightmarish aspect of maths comes from the interconnectivity of the subject. Mathematics is not just the study of algebra, calculus and statistics; it is the study of numbers, but used in many different ways, one layer upon one another.