A Leap Day 2016 Mathematics A To Z: Fractions (Continued)

Another request! I was asked to write about continued fractions for the Leap Day 2016 A To Z. The request came from Keilah, of the Knot Theorist blog. But I’d already had a c-word request in (conjecture). So you see my elegant workaround to talk about continued fractions anyway.

Fractions (continued).

There are fashions in mathematics. There are fashions in all human endeavors. But mathematics almost begs people to forget that it is a human endeavor. Sometimes a field of mathematics will be popular a while and then fade. Some fade almost to oblivion. Continued fractions are one of them.

A continued fraction comes from a simple enough starting point. Start with a whole number. Add a fraction to it. 1 + \frac{2}{3}. Everyone knows what that is. But then look at the denominator. In this case, that’s the ‘3’. Why couldn’t that be a sum, instead? No reason. Imagine then the number 1 + \frac{2}{3 + 4}. Is there a reason that we couldn’t, instead of the ‘4’ there, have a fraction instead? No reason beyond our own timidity. Let’s be courageous. Does 1 + \frac{2}{3 + \frac{4}{5}} even mean anything?

Well, sure. It’s getting a little hard to read, but 3 + \frac{4}{5} is a fine enough number. It’s 3.8. \frac{2}{3.8} is a less friendly number, but it’s a number anyway. It’s a little over 0.526. (It takes a fair number of digits past the decimal before it ends, but trust me, it does.) And we can add 1 to that easily. So 1 + \frac{2}{3 + \frac{4}{5}} means a number a slight bit more than 1.526.

Dare we replace the “5” in that expression with a sum? Better, with the sum of a whole number and a fraction? If we don’t fear being audacious, yes. Could we replace the denominator of that with another sum? Yes. Can we keep doing this forever, creating this never-ending stack of whole numbers plus fractions? … If we want an irrational number, anyway. If we want a rational number, this stack will eventually end. But suppose we feel like creating an infinitely long stack of continued fractions. Can we do it? Why not? Who dares, wins!

OK. Wins what, exactly?

Well … um. Continued fractions certainly had a fashionable time. John Wallis, the 17th century mathematician famous for introducing the ∞ symbol, and for an interminable quarrel with Thomas Hobbes over Hobbes’s attempts to reform mathematics, did much to establish continuous fractions as a field of study. (He’s credited with inventing the field. But all claims to inventing something big are misleading. Real things are complicated and go back farther than people realize, and inventions are more ambiguous than people think.) The astronomer Christiaan Huygens showed how to use continued fractions to design better gear ratios. This may strike you as the dullest application of mathematics ever. Let it. It’s also important stuff. People who need to scale one movement to another need this.

In the 18th and 19th century continued fractions became interesting for higher mathematics. Continued fractions were the approach Leonhard Euler used to prove that e had to be irrational. That’s one of the superstar numbers of mathematics. Johan Heinrich Lambert used this to show that if θ is a rational number (other than zero) then the tangent of θ must be irrational. This is one path to showing that π must be irrational. Many of the astounding theorems of Srinivasa Ramanujan were about continued fractions, or ideas which built on continued fractions.

But since the early 20th century the field’s evaporated. I don’t have a good answer why. The best speculation I’ve heard is that the field seems to fit poorly into any particular topic. Continued fractions get interesting when you have an infinitely long stack of nesting denominators. You don’t want to work with infinitely long strings of things before you’ve studied calculus. You have to be comfortable with these things. But that means students don’t encounter it until college, at least. And at that point fractions seem beneath the grade level. There’s a handful of proofs best done by them. But those proofs can be shown as odd, novel approaches to these particular problems. Studying the whole field is hardly needed.

So, perhaps because it seems like an odd fit, the subject’s dried up and blown away. Even enthusiasts seem to be resigned to its oblivion. Professor Adam Van Tyul, then at Queens University in Kingston, Ontario, composed a nice set of introductory pages about continued fractions. But the page is defunct. Dr Ron Knott has a more thorough page, though, and one with calculators that work well.

Will continued fractions make a comeback? Maybe. It might take the discovery of some interesting new results, or some better visualization tools, to reignite interest. Chaos theory, the study of deterministic yet unpredictable systems, first grew (we now recognize) in the 1890s. But it fell into obscurity. When we got some new theoretical papers and the ability to do computer simulations, it flowered again. For a time it looked ready to take over all mathematics, although we’ve got things under better control now. Could continued fractions do the same? I’m skeptical, but won’t rule it out.

Postscript: something you notice quickly with continued fractions is they’re a pain to typeset. We’re all right with 1 + \frac{2}{3 + \frac{4}{5}} . But after that the LaTeX engine that WordPress uses to render mathematical symbols is doomed. A real LaTeX engine gets another couple nested denominators in before the situation is hopeless. If you’re writing this out on paper, the way people did in the 19th century, that’s all right. But there’s no typing it out that way.

But notation is made for us, not us for notation. If we want to write a continued fraction in which the numerators are all 1, we have a brackets shorthand available. In this we would write 2 + \frac{1}{3 + \frac{1}{4 + \cdots }} as [2; 3, 4, … ]. The numbers are the whole numbers added to the next level of fractions. Another option, and one that lends itself to having numerators which aren’t 1, is to write out a string of fractions. In this we’d write 2 + \frac{1}{3 +} \frac{1}{4 +} \frac{1}{\cdots + }. We have to trust people notice the + sign is in the denominator there. But if people know we’re doing continued fractions then they know to look for the peculiar notation.