One of my favorite quotes about mathematics is from John von Neumann:
“In mathematics you don’t understand things. You just get used to them.”
On one level, this runs against everything I believe as a teacher. Mathematics should not be an intimidating collection of inscrutable methods! It should be a timidating collection of scrutable methods! We should accept nothing on authority. Everything in mathematics is there to be understood.
And I do believe that.
But I also know that mathematics is full of startles and shocks. I know that even the simplest objects can bury deep secrets.
Take the number line.
Boring, right? Nothing could be more prosaic.
Well, that’s only because you’ve gotten used to it. To my mind, the number line merits only one possible reaction:
It’s not the integers that spook me, although those are plenty weird.
To wit: although we distinguish numbers like “4986” vs. “4987,” not all human languages are so fussy. After all, does the difference between these numbers really matter? Are you good enough at counting to reliably tell one from another? The fact that we assign these distinct identities is a bit of a mind-popper.
No, what keeps me up at night is the stuff between the integers: the numbers a 12-year-old student of mine recently dubbed “the disintegers.”
When we draw a line connecting 0 to 1, what exactly are we drawing?
You might think we’re drawing a line of fractions. After all, fractions are like bedbugs: tiny and everywhere, living in every nook and crevice you can imagine.
And yet there are lots of numbers that aren’t fractions, perhaps the most famous being the square root of 2.
The square root of 2 can’t be written as a fraction (by which I mean, one whole number divided by another). Sure, you can get close – say, 665,857 / 470,832, which is the square root of roughly 2.0000000000045. But you can’t get the square root of 2 precisely.
Okay, so the disintegers aren’t all ratios of whole numbers. But that’s okay. Maybe, like the square root of 2 (which is the solution to the equation x^2 = 2), they’re just the solutions of equations. That’s not so bad, right?
We call such numbers – the solutions to simple polynomial equations – algebraic.
Unfortunately, some numbers escape even this widened circle.
There’s no nice algebraic equation that gives you pi or e. These numbers are called “transcendental.” For them, the best you can do is to describe a process by which a computer could, in theory, compute the number.
(For example, if you want to compute e, you can take a big number n, find 1 + 1/n, and then raise this sum to the nth power. It won’t give you e precisely, but the bigger the n you choose, the closer you’ll get to e.)
At this point, we’re forced to widen our conception of “number” again. Numbers, we shall say, are “things you can compute.” Maybe they’re not fractions. Maybe they’re not even the solutions to equations. But they are things that you could tell a computer to calculate.
Unfortunately… that’s not quite right.
There are non-computable numbers.
In fact, almost everything on the number line is non-computable.
What does this mean?
If you pick a random number between 0 and 1, you will arrive at a number that literally cannot be communicated or expressed. Sure, you can list the decimal digits, but if you ever stop, even after a trillion years, then you will have failed to specify your number – and there is no process by which you can tell a computer to carry on the job.
These are numbers that cannot be touched by the human mind.
And this is most numbers.
Well, you know what they say: In mathematics, you don’t understand things. You just get used to them.
Fawn Nguyen taught a badass lesson where she posed kids the philosophical and mathematical challenge of finding the square root of 7 on a number line.
Simon Gregg calls noncomputable numbers “the dark matter of the number world” and points out that biology has its own version of the “most of what’s out there is beyond our imagination” problem.
Also worth reading is Mr. Sock Monkey’s comment below, which speaks for us all:
monkey still not sure of definition of word number or if number have existence independent of human mind. when monkey read definition of number by mr russell it make simple brain of monkey spin.