Perfect Moments: My Mathematical Breakthrough (Age 16)

[Image: blurry version of the Borromean-ring logo of the International Mathematics Union (IMU). For the non-blurry version, just click the blurry one. And no, I had nothing at all to do with the creation of this logo, or with the sciences of either mathematics or physics, really. My “breakthrough” was much smaller and more personal.]

I sort of straddled peer groups at my fairly small high school: sort of an academic/bookish nerd (hello, Latin honors student!), sort of an arty writer/photographer-wannabe (remind me to tell you about my James Thurber ripoffs), and sort of a pilotfish attached to the flank of the social center (it helps to develop crushes on school yearbook and newspaper editors). I even dabbled very superficially in athletics, and acquired a varsity letter in tennis to prove it. All of which pretty much set me on course for the life which followed: not quite one thing, not quite another, and really not very much of something else, but a mishmash of them all.

My personal Venn diagram might look something like the Borromean rings shown above: indistinct and formless enough that at any one time, I’m not quite sure what domain I’m occupying… assuming it is indeed just one domain at the moment. The so-called “credential” which I use most often when answering questions on Quora is: Not an expert in much; probably interested in it anyhow.

I took three year-long classes with one teacher, Mr. Hanlon: trigonometry, physics, and calculus. (Aside: all nominally in the nerd category. But because Mr. Hanlon also introduced me to photography, via the Camera Club, his influence also fed the wannabe artiste in me.)

Like many students in those three classes, I often had to grapple with what we (and maybe everyone) called “word problems.” These dressed up pure-math theory in the garb of applied math, so you might be asked something like, “If you’re in a train heading east at 30 miles per hour, and there’s a train 15 miles away, heading west at 45 miles per hour, how long do you have until the head-on collision?” Solving such problems, for me, often involved two steps: (a) untangling the language so I understood more or less exactly what the premise was, and (b) sketching out the specifics, often on graph paper, if (as here) the specifics lent themselves to being sketched.

But there was a specific subset of word problems quite different, in that while it involved words, and of course numbers, “solving” a given problem required no particular knowledge or manipulation of either. Instead, the solution — for me — simply consisted of… well, for lack of a better way of expressing it, simply consisted of flipping a switch. These were problems — and/or solutions — of unit conversion.

Here’s a more or less simple (?) example of one such problem: Given (a) the distance from some Star X to the Sun is 8.7 light years, and (b) the speed of light is 1,080,000,000 kilometers per hour, how many hours does it take a photon to travel from X to the Sun?

Now, a couple of conversions need to be performed to work through the whole thing. Until now, I’ve never tried to explain how I do this to anyone but Mr. Hanlon, and don’t know if it’ll make sense to you, stranger. But I’ll try. Start by looking at the general shape of the problem: we’ll need to juggle times (T), distances (D), and velocity (V). We’re being asked to calculate a time, given (a) a distance and (b) a velocity. Here’s how velocity is expressed, generally:

V = D / T

or specifically, in the specific units provided above:

V = km / hr

What had always interested me (if “interested” is even the right word… “noticed,” maybe?) was that in this case, km / hr looks like a fraction, using the slash (virgule) as a separator between numerator and denominator. And I knew the peculiar property of fractions — whatever it’s really called, I don’t know — that when you put them together in calculations, it’s easier if you “make them the same” and thereby get them to “cancel each other out.”

What’s an example, hmmm…? Well, consider this quite simple one: what is one-third of three-sixteenths, i.e.:

1 / 3 * 3 / 16 = ?

The way I’d learned to solve this kind of basic fractional arithmetic is to note the 3 in the first fraction’s denominator, and the 3 in the second fraction’s numerator. Since they’re both the same, you can ignore them, then grab the first fraction’s numerator and stick it over the second’s denominator:

1/3 * 3/16 = 1/16

See? It’s like the 3s aren’t even there. Magic!

So now back to the word problem. Ignore the numbers, look at the units. We want to come up with time, expressed in terms of hours (hr), given information about distance (km) and velocity (km/hr). We need to express all terms as if they were fractions: if they’re pure units, like distance alone, then we express the not-a-fraction distance as a distance divided by 1. To make the units — represented as “fractions” — “cancel out” properly in a calculation we need to make the whole thing work something like this:

hrs = km / 1 * hr / km

See what’s going to happen there? The two kms are about to “cancel each other out,” leaving you with:

hr / 1

(Note by the way that the velocity in the problem is km/hr. So we’ve got to remember to first invert the 1,080,000,000 km/hr figure given to its hr/km equivalent, i.e., 1 hr/1,080,000,000 km — that is, 0.000000000925925925 hr/km — in order to have a workable “fraction” with the proper units on the proper sides of the slash.)

There’s one more unit conversion to make in order to solve the overall problem — we’d need to convert the 8.7 light years’ distance to km (we can’t have an ly in a calculation involving km everywhere else — but that’s pretty straightforward multiplication. The real trick I’d stumbled on, apparently, was the analogy between word problems which juggle different units (time and distance, say) and their expression in the form of “fractions.”

I don’t remember the specific problem we’d been asked to solve at the time. But I remember Mr. Hanlon writing it out on the board. And I remember the actual numbers involved weren’t as complex as in this example — the multiplications and divisions were all simple to do mentally. When he asked us to solve it, my hand went up almost immediately.

How did you do that so fast?!? he asked, genuinely surprised that I hadn’t even bothered to write anything down. I explained it, kinda, as I did above: well, they’re like fractions, and I just canceled out the matching numerator and denominator. Mr. Hanlon said something like, You figured that OUT on your own?!?

But really, it wasn’t anything like a mathematical feat. I’d just manipulated words and slashes, and the answer presented itself at once.

To this day, I have no idea if this is a “thing” — something others do or not. I don’t know if Mr. Hanlon himself knew about it, or ever did it in his own mind. I still do it, though.

You?