Learning Journal #1: Seeing Structure, pt. 1
Over dozens of asks by students, math educators eventually learn to weather the question “When will I ever use this?” Some professional mathematicians and popularizers like Jordan Ellenberg and Douglas Corey have put out their own answers to this, so I’ll throw in my own hat with mine: mathematics lets you see structure.
Let me clarify what I mean by this with an example. What was your first encounter with numbers? If you’re anything like me, you probably started out finger counting, getting down the numbers from one to ten before your teacher broke out the tens sticks and told you that hey, you can put ten of these small things together and treat it like one big thing! And if you’re anything like me, this caused you some grief at first. For many, this step from units (ones blocks) to groups (tens sticks) is their first big run-in with abstraction, the act of mentally putting a bunch of parts into a box and then using that box without worrying about the parts.
More than that, the transition from ones to tens is a stepping stone for the abstraction from tens to hundreds, hundreds to thousands, from thousands to millions. Going up and up and up and up through the numbers, you might have noticed some commonality between those special enough to get their own place value: there are ten ones in ten, ten tens in a hundred, ten hundreds in a thousand — the number 10 pops up again and again. Why?
For the reasoning behind this, an explanation scarcely mentioned by most grade school teachers, you need only look down and count your fingers.
Yes, that’s really it. The reason 10 has almost holy significance in our number system is because we have 10 fingers on our hands. This is the structure I’ve been talking about. Math teases structure out of patterns — in this case, a pattern of recurring tens tipped us off to the structure of our number system (the decimal system). Perhaps most liberating aspect of this realization is the fact that the choice of 10 was entirely arbitrary. A man-made construct.
“Math teases structure out of patterns.”
This fact wasn’t lost on the thousands of scientists and mathematicians who laid the foundation for computers like the one you’re reading this on now; computer scientists chose the number 2 for the number system (the binary system) used by electronic computers. Where you or I might learn to count like this:
1, 2, 3, 4, 5
A computer would learn it like this:
1, 10, 11, 100, 101
You know how we add a place value for every ten of the place before it in decimal? In binary, this happens for every two. Two ones makes a two (10 in binary), two twos makes a four (100 in binary), two fours makes an eight (1000 in binary).
Why two? Well, computers don’t have fingers, but they do have tons and tons of little on/off switches called transistors, and that quality of being either on or off can be encoded with the numbers 0 (for OFF) and 1 (for ON).
One of the powerful and beautiful things about being able to see structure is being able to build these bridges between seemingly disparate topics in surprising ways. Being able to question why the number 10 was so important in our number system leads to the discovery that the number we pick is really up to us, which leads to messing around with other choices of number — like 2 — which leads to the incredible insight that we can write numbers with electric switches.
Coming soon: What does this have to with Git?
