A trip through middle school maths
At what point in your mathematical journey did you start to struggle? Was it in the death valley that is calculus? Maybe it was in the funky backstreets of logarithms, or functions, or perhaps it was even multiplication. I’m not sure I ever entirely understood addition (screw you, Collatz conjecture).
One of the topics that really stressed me out was functions, particularly the surprisingly shady world of polynomials and exponentials. And I don’t think I’m alone in that. Which is a real shame, because understanding these mathematical objects opens our minds to many of the more beautiful and useful areas of mathematics (read till the end, if you dare).
So in this article I want to pick apart functions, in as friendly a way as I can. I’ll talk about them in the context of understanding the messy reality we live in, avoiding scary symbols and equations where I can. And in the end, we’ll build to a really useful and applicable result which anyone can exploit.
On our journey, I think it’s worth first revisiting the notion of a function in mathematics. We can think of a function like a box with a handle on it. In the top there’s a slot, where you can put things — the input. You crank the handle on the side, and it spits out an output.
Maybe we can pop a slice of bread in as the input. A slice of sourdough, or a slice of ciabatta, or so on. And when we crank the handle, the machine performs a toasting action on the slice, and it spits out a toasted sourdough, a toasted ciabatta and so on 😋.
Yep, that’s a toaster. No surprises there. It is also a function. And no, I’m not being facetious. If you want to use the jargon of mathematics, we’ve defined:
f(x) from the domain, X = {sourdough, ciabatta, …} to the range Y = {toasted sourdough, toasted ciabatta, …}
Yes this is silly. But there is a purpose. I’m trying to help us to just forget about symbols and equations for a moment, and really digest the notion of a function. A function is just this: input → output, usually with some kind of action happening in between, so I like to write it as: input → action → output.
Now, putting toasters aside, think about all the situations in life with this kind of structure:
Input → Action → Output
Today’s COVID cases → … → Tomorrow’s COVID cases
Hours of practice → … → Skill at playing the piano
# of Doughnuts → … → Happiness
Radius of a circle → πr² → Area of the circle
Education → … → Salary
Some of those examples have clear-cut and well-known equations (actions), while some don’t, but all can be modeled by functions.
By building our understanding of functions, we can better understand the actions of the complex and messy world around us. From gravity, to the growth in your stock portfolio, to the happiness you gain from eating a doughnut, to the productive capacity of a nation.
Let’s look at the most basic of the ‘interesting’ functions that you learned at school, being polynomials: e.g. f(x) = x². The little two next to the number is just notation. It’s nothing special. It literally just means multiply x by itself.
f(x) = x² = x * x
It’s just repeated multiplication, and remember that multiplication is really just repeated addition. 3² is really just telling you to add 3, 3 times.
And, x² means this: starting at 0, add groups of x, exactly x times. Let’s visualise:
You’ll notice that the rate of growth is increasing: 3 → 5 → 7. Why is this?
We can also notice, that the change in the rate of growth, stays constant, at 2. We can think of this like acceleration. Say we hop into a car, and start accelerating at a constant rate — maybe increasing by 10km/h every 10 seconds or so. Our speed will increase, from 10km/h to 20km/h to 30km/h and so on. Our distance traveled over time will grow, increasingly fast.
Distance traveled changes somewhat quickly at 10km/h, but it’s a totally different story at 100km/h. While your speed increases linearly, your distance increases quadratically.
So that’s x². Pretty cool. If we look a bit further, and move from x² to x³ to x⁴, we are basically kicking this acceleration one step further down the chain. When we have x³, there is growth, in the rate of growth of the rate of growth (yep). This is delightfully known as jerk in physics, while x² is acceleration, and x¹ is speed.
And indeed cars demonstrate this.. jerk. You might notice that your car accelerates pretty quickly to begin with, but once you get faster and faster, the acceleration will tail off. While your speed is still increasing, your acceleration is decreasing! It’s negative jerk! You’ll notice this especially and frustratingly so with an older car, which struggles to make the last leap up to highway speeds.
Now let’s develop this a bit more to finish off. I did hint that there’d be something useful hidden in here somewhere. So let’s do something funky, like say:
Despite the similarity in notation, this exponential function (of base 2) is a considerably different beast. Let’s have a look at a picture:
You’ll notice that this function seems to grow in a more unrestrained way. While x² does keep pace for a little bit, the exponential function is a runaway train by comparison.
Under the exponential function, each unit makes itself a friend. We have doubling. So 2 becomes 4 becomes 8 and so on.
In fact, this is precisely the compounding effect (😲), and its why people recommend to start growing your wealth early in life. Each dollar in your bank account over the course of the year might make itself 10% of a friend (if the interest rate is 10%), giving you 1.10 dollars. And then next year, each of those 1.10 dollars makes themselves another 10%-friend, giving you $1.21. This is just the exponential function,
Coolios.
But what really interests me is the following. Perhaps there is some goal you desire to achieve: becoming a pro at basketball, or portrait painting, or public speaking. Imagine you improved just 1% a day. Seriously, imagine it. How much would you improve in a year? Try and figure it out if you can.
You might think it would be 365%, but it’s much, much more.
… Figured it out 😊?
In fact, a 1% improvement a day is a 3778% improvement per year!?!* In other words, a near 38-times improvement. Good god I was surprised when I heard this. We all should have listened more to our high-school maths teachers :))
-Marc 😊
*1.01³⁶⁵=37.78…
**please let me know if I’ve made any maths mistakes, it would not surprise me at all
