“Oh, You’re Studying Math?” A First Small Step into Mathematics

“That sounds hard. I used to like math, but then I had a bad teacher and realized I just wasn’t any good at it anymore. I think it’s really cool that you like it though, and that you can teach other people about it. That’s what you want to do, right? Become a math teacher? That sounds really great. I couldn’t be a teacher, especially for math, that sounds way too difficult.

“What always bothered me about my math classes was that we never did anything useful. I’m over twenty years old now, and I haven’t had to use the quadratic formula, or know how to find the area of a pentagon or multiply matrices. And guess what? They never even taught us how to do our taxes. That would be useful in a math class. Also understanding loans.

“It just seems so ridiculous that they put us through so many years of stuff we don’t need to know for our job. Engineers and scientists should have to take it, but I don’t want to do any of that. I just want to go into business and be able to do my taxes and make sure I’m not getting taken for a ride if I buy a house.”

And so goes many conversations I have about being a mathematics major. I don’t wish to encourage everyone to study math, nor am I writing to tell you of the wonderful contrived applications of math that you hear about in school and see infographics about when discussing the need for STEM majors in college. What I am aiming for in writing these Small Steps into Mathematics, is to help people develop an appreciation for math beyond its utility and difficulty. Most people are aware that we need math in order to be at the technological level that we are at, and that math, on the whole, is a rather difficult subject for many people. I hope to show that math transcends these characteristics and can be understood on a level of beauty and passion, similar to art.

I likely lost many of you in that last sentence. How can math be related to art, how is it beautiful and full of passion that is not hatred? Some of what will follow will be largely influenced by A Mathematician’s Lament by Paul Lockhart (do read it!). So let us begin our first small step into the wonderful world of mathematics.

When I tell people how math is like art, I am greeted by many puzzled glances but also a certain curiosity. To some extent, I believe there is a visual beauty to some particular formulas in math, such as Euler’s Identity:

$e^{i\pi}+1=0$

If you have done math through precalculus, you might be familiar with this formula, how it has five of the (arguably) most important numbers in mathematics, all cohesively together. If you have not gone through this level, or it has been quite a while since you have looked at any math, that is alright! I hope that you can find some excitement in the mess of symbols either way. However, this visual beauty is not the main similarity I think of between math and art. I think of creativity, the ability to create a world that is your own, explore new meanings and find new ways to view the world.

Consider the process of creating art. It is something most people begin to do as soon as they can get paint on their fingers or a crayon in their hand. They represent the world in different colors, textures and forms. As we grow older we often doodle, idly making creations to occupy our brains. We go through school and learn certain rules about aspects of art. We learn our colors, we develop fine motor skills to improve our ability create art, we may learn different techniques to create works of art on our own. Math can be like this as well.

Math is about exploring our world in new and different ways. Using shapes, symbols and numbers we find a way to explore our perception of the world. We create different realities where we view mathematical objects as something tangible. We expand our definitions of these objects, and begin refining where we can get with them. Math creates a brand new world to play with and enjoy with far less background than most people believe is necessary. It encourages creativity and exploration into realms that we can control. To see this, I’ll give a simple example, a small step, where hopefully you can see some of the beauty and creativity that is inherent to mathematics.

A good example is a conjecture based only on arithmetic. It is a rather simple problem to state, one that was likely an idle musing of a mathematician, yet has evaded all attempts at solving it. It is called the Collatz Conjecture. Here is what it states:

Let us begin with a positive whole number $n$. If it is even, then divide it by 2. If it is odd, then we multiply by three and add 1. Take your new number and repeat the process. Eventually you will obtain the number 1.

So it can be reduced to this formula (don’t worry if it is difficult to parse, the explanation I just gave translates the same):

$a_1 = n, \; a_{n+1} = \left\{ \begin{array}{lr} \frac{a_n}{2} & n\textnormal{ is even}\\ 3a_n+1 & n\textnormal{ is odd} \end{array} \right.$

Then no matter what positive whole number we begin with, if we repeat this cycle we will eventually reach 1.

For a concrete example, let us start with $n=10$. 10 is even, so we divide by 2 and get 5. Now 5 is odd, so we multiply by 3 and add 1. We have $5\cdot 3 = 15$, and adding 1 we get 16. Well 16 is even, so we divide by two and get 8. Then we get 4, then 2, finally 1. We have run computers that have shown if we start with any number up to around $2^{60}$, which is a monstrously big number, we eventually get to 1. Yet we haven’t proven it yet.

This is the sort of wonder and excitement and strangeness inherent in mathematics. It takes creativity to attack a problem like this, and it is fantastic that such a simple statement has gone unproven for a very long time. This is today’s small step into mathematics. While there is much more to be said about how math is a wonderful and beautiful subject, I will devote another post to this in a more direct way.

To those of you wondering why I chose to major in math, and why others do: The subject is rich and beautiful and forces creativity. At a certain level, computational prowess and memorization is not enough. It is challenging and fun and frustrating and rewarding in the best of ways, and no matter what job I end up with the time I spend doing math will always influence how I experience life.