This post includes my thought process as I look at a particular piece of math. That part of the post contains a little more technical stuff, with less explanation, than many of the posts on this blog. If there is something you don’t understand, feel free to ask about it below. Or skim it – you won’t miss out on the main idea.
I think most people learn math backwards. For most of my life, I had the impression that math is a neat package of rules. The teacher would say something like “The square root of 2 is irrational,” “,” or “Dividing by zero is undefined.” Then I would memorize the rule and apply it to a set of problems. This, unfortunately, is what math class means to most people: you are told a rule, and then use it to get the right answer to a question.
Following rules of math is a bit like running laps around a basketball court. It is related to what you might do in a basketball game. It can even make you a better player. But it is certainly not the same as a game of basketball. Following mathematical rules is a helpful tool in doing real math, but it is not the same thing.
There is certainly value in knowing how to use a rule. But how do we know about the rules in the first place?
I’ll let you in on a little secret. Mathematicians make up their own questions. They ask about what they’re interested in or what they think will be useful. They find their own answers.
That’s how math gets developed.
That’s not to say that there is no structure to math – to come up with a valid mathematical idea, you need to stick to the principles of logic. There are true mathematical statements, and false ones. Some are applicable to physical situations, and some are not. Some are interesting, some are beautiful, some are useful, and some are not. There is a whole scene of math research and development that is thriving nowadays – and it has been around since at least the Ancient Greeks.
In school, we usually are usually given a rule, given a problem, and asked to find an answer. When a person studies math on their own, they ask their own questions. They look at an idea from their own perspective. They make their own mini-discoveries about it.
That’s what I’d call doing math forwards. To fully appreciate math, you need to mess around with some ideas on your own.
Take, for instance, the irrationality of the square root of 2. You may remember it as an “obvious” fact, or something that you accepted because your teacher said so. While this is a true fact, it is not at all an obvious one. At the time of its discovery, it was scandalous – perhaps even cause for murder. There’s a reason is irrational. But people had to discover that fact. They discovered it by looking at the number and asking their own questions about it.
A bad math class lacks the drama, the debate, and the flashes of insight that make the subject interesting. I know many people who love math. But I would be hard-pressed to find someone who loves following a rule that he doesn’t understand. The way people approach math on their own is very different from the way it is presented on a seventh-grader’s homework.
In this post and the next, I will take you through my thought process as I look at a particular piece of math. It is not going to represent the way all mathematicians think, or even all of the ways I look at math. It is just a sample.
This thought process is a bit messy in places. There are unanswered questions. And that’s fine, because I am not trying to solve anything.
I’m just playing with an idea.
Let’s look at this mathematical object.
Here’s the first thing I would ask: What kind of object is it?
Math deals with all types of objects, like numbers, shapes, vectors, functions, and sets. The list is practically endless. Each type of object has its own special properties.
We make this kind of categorization all the time in everyday life, without even thinking about it. It makes no sense fry a baseball or play catch with pancake batter. In the same way, it makes no sense to multiply two circles or rotate the number 78. Before I can do anything interesting with a mathematical object, I need to know what kind of object it is.
In this case, I’d say the object we’re dealing with is a number. Now I know which steps are possible, and which steps are not. For instance, I might add 5 to it, but I won’t draw a line through it.
It may have been easy for you to see that this is a number. Truthfully, in most situations, I do not spell out this kind of thought at all. It usually happens unconsciously. But I find that this can be an important first step to articulate. When I am faced with more complicated objects, this question can be extremely helpful.
An important thing to notice about this number: This is a fraction, which has a fraction inside it, which in turn has a fraction inside it, and so on. It starts with . The “…” is replaced by another , which all together makes the fraction. Next it becomes , and so on. No matter how many layers are explicitly written, it represents the same number. The “…” tells the reader that the pattern repeats forever.
As another example, 2+2+2+2+2+… means that you start with 2, then 2+2, then 2+2+2, and so on. In this case, the “…” means that the pattern of adding 2 is repeated forever.
Next thought: Is there any obvious idea I might overlook?
Have you ever struggled with the volume controls on your TV, only to realize that it was on mute? That is the kind of thing I want to avoid by asking this question. In this case, I think it makes sense to check whether our mystery number,, is equal either to 0 or to infinity. I want to rule out those possibilities, because neither are particularly interesting.
I know to check this because of experience. I have seen a lot of sums with a “…” that equal infinity or equal 0. For example, … gets bigger than any particular number: it is infinite. And gets smaller and smaller. It is equal to 0.
These examples really deserve more explanation in the context of infinite series. But that’s a separate discussion. Back to our number.
Well, is 1, plus a positive fraction (I don’t know what the fraction is equal to yet). So can’t be equal to 0.
I just ruled out something — that’s good news in my book!
Infinity seems more difficult to deal with, but it turns out that I can use the same trick. We just found out that is greater than 1, which tells me that is less than 1.* That tells me that we are adding 1 to a number less than 1. So that sum must be less than 2. If our mystery number is less than 2, it certainly can’t be infinity.
*If that is confusing, think of it this way. 2/3 < 1 because 2 < 3: if you split 2 cakes among 3 people, everyone gets less than a whole cake. A positive fraction is
Now we know something about our mystery number. Not only is it not 0 and not infinite, it is somewhere between 1 and 2. That’s way more than I knew just a few minutes ago.
So far, I have been asking questions without a particular goal in mind. No one gave me a question and told me to answer it. My purpose was simply to get information, because the mystery number looked kind of weird and interesting.
But now I have a question I really want to answer. I’m curious about this.
What “regular,” “non-…” number is our mystery number equal to? From the discussion so far, we know it is equal to something between 1 and 2. But what? 1.5? 1.000007? ?
I worked on this for a while, and got an answer to this question in two completely different ways. The mystery number, it turns out, has something to do with this flower.
I’ll talk about both techniques in the next post.