by Casia Siegel
When you think about math, what do you think? What feelings does it invoke? What kinds of interactions have you had with it? As someone who does math just about every single day, many people have expressed to me the negative connotations they have associated with the subject. Lots of the time, it comes from a bad experience in a math class, or a feeling that they aren’t good enough at it, or possibly just that they find the entire topic totally boring and uninteresting. I am not here to invalidate anyone’s feelings on the matter, but if you’d give me this chance, I’d love to share with you why I believe math to be an incredibly broad, beautiful, and enjoyable subject.
First, I want to address a massive misconception that people have about math. We all know some of the courses by name: pre-algebra, geometry, algebra I, algebra II, trigonometry, pre-calculus, calculus I, calculus II, calculus III, and then… what? Well, after those courses I mentioned and possibly a couple more, math starts to take on an entirely new shape. It stops looking like equations and numerical answers. It stops being just lists of instructions and rules to follow in order to get the correct answer. It completely morphs into something called proof writing. By your second or third year in undergrad as a math major, you would probably be writing entirely proofs for your homework. Proofs are justifications in paragraph form (still with mathematical notation mixed in) which concretely demonstrate a statement’s truth. For example, a homework problem might look like: “Show that the square-root of 2 is an irrational number,” meaning it cannot be written as a fraction of integers. I can tell you this is true, and maybe you believe me, but how would you show for certain that this statement is true? There are many methods of proof, such as direct proof, proof by contradiction, proof by contrapositive, proof by induction, and many more. To prove something demands thinking through something logically, step by step, and fully understanding the content you are working through. There can be many fully correct ways to prove the exact same statement. All this is to say that after a certain point, math becomes a lot more conceptual. After calculus, learning math switches from learning “how” to understanding “why.”
By your second or third year in undergrad as a math major, you would probably be writing entirely proofs for your homework. Proofs are justifications in paragraph form (still with mathematical notation mixed in) which concretely demonstrate a statement’s truth.
One of the things I used to find most intimidating about math is that it is an absolutely ginormous topic. Often, friends or family ask me what kind of math I am interested in, and unfortunately all too often the words I use to describe math normally don’t mean much to them—but this is not just because I am a math person and they are not. Math is such a rich and dense language, that even as a PhD student, when fellow math PhD students tell me what they are working on, if it is not close to my work then at best I have vague notions what the words that they use mean. There are so many different subfields and sub-subfields, that I still have no clue about. I was talking to some other PhD students in different departments, and by the time they get to grad-school they have mostly moved on from coursework to research. It is totally fine to not understand math words, or to remember the math you did years ago. Many people think that if these apply to them, then they are bad at math, so they don’t even both trying to learn any more. But I bet, if pressed, most math majors don’t remember so much trigonometry or geometry either, and neither me nor any of my peers even know what exactly we want to study yet. In math, you might be taking courses into your fifth year in your graduate program because there is just so much material to learn. While the breadth and depth of the topic can make it seem unapproachable, it also makes it less intimidating as time goes on. If you really give math a shot, you come to terms with the fact that it is so much bigger than any of us can every really know, and that feeling is actually super exciting.
Lastly, I just want you to see that math can be really cool and fun, by giving you three of my favorite math facts:
- If you drew a number line, and put a pin in it randomly, you would have a 100% chance of hitting a number that nobody in all of history has every talked about or used. (I don’t mean 99.9%, or 99.99999%, I genuinely mean 100%.)
- If 23 people are sitting in a room together, there are better than 50% odds that two of them share a birthday.
- We can put the numbers e=2.718…, π=3.141…, (which are called transcendental numbers!) and i= √(-1) (which is an imaginary number!) all together into the very neat equation eπi=-1.
I hope maybe you can walk away from this post with warmer feelings for math and a couple of fun math facts to show off at parties! (I promise, the birthday one can actually be fun to demonstrate).
