The Maths History feed on Twitter mentioned that the 21st of August was the birthday of Augustin-Louis Cauchy, who lived from 1789 to 1857. His is one of those names you get to know very well when you’re a mathematics major, since he published 789 papers in his life, and did very well at publishing important papers, ones that established concepts people would actually use.
He’s got an intriguing biography, as he lived (mostly) in France during the time of the Revolution, the Directorate, Napoleon, the Bourbon Restoration, the July Monarchy, the Revolutions of 1848, the Second Republic, and the Second Empire, and had a career which got inextricably tangled with the political upheavals of the era. I note that, according to the MacTutor biography linked to earlier this paragraph, he followed the deposed King Charles X to Prague in order to tutor his grandson, but might not have had the right temperament for it: at least once he got annoyed at the grandson’s confusion and screamed and yelled, with the Queen, Marie Thérèse, sometimes telling him, “too loud, not so loud”. But we’ve all had students that frustrate us.
Cauchy’s name appears on many theorems and principles and definitions of interesting things — I just checked Mathworld and his name returned 124 different items — though I’ll admit I’m stumped how to describe what the Cauchy-Frobenius Lemma is without scaring readers off. So let me talk about something simpler.
Suppose you have a sequence, which is just an ordered list of numbers: the first number is, say, 1; the second number, say, 1/2, the third number, say, 1/4, and so on. A sequence might converge, which means roughly what you’d think it means from the ordinary meaning of the word, that the sequence eventually settles so close to some fixed number that you can’t see the difference after some point. Or it might not converge; for example, the sequence 1, 2, 4, 8, 16, et cetera clearly isn’t settling to anything. So too does the sequence 1, -1, 1, -1, 1, -1, and so on, not converge, even though it sticks very close to the numbers +1 and -1.
A sequence is called a Cauchy Sequence if, as you look at the M-th and the N-th terms as M and N get incredibly large, the difference between the M-th and the N-th terms gets arbitrarily close to zero. For example, consider the sequence 1, 1/2, 1/3, 1/4, and so on. The M-th term is 1/M, and the N-th term is 1/N. The difference between them is 1/M – 1/N, which is going to be incredibly tiny if M and N are very large at all. (Try M of 1,000, and N of 10,000; then try M of 1,000,000 and N of 10,000,000, if you want to convince yourself, although not your thesis committee. This tryout is just a sanity check.)
From that start — that this is a sequence where, eventually, the difference between any pair of entries gets arbitrarily close to zero — it would seem like a Cauchy sequence has to converge. And here’s a neat, counter-intuitive point: if the terms in the sequence are real numbers, then, yes, the Cauchy sequence has to converge to some real number. But if the terms in the sequence are rational numbers, then you’re not guaranteed that the sequence will converge to a rational number.
They will converge to some irrational number instead, but if you want to keep all your work in the rationals — and you might want to — then you’re left, sometimes, with a sequence that’s clearly converging to something but it’s just outside what you’re working with. If you spend a little time you can probably think of one, even if you haven’t got any mathematical training at all. Just think of the conditions: every term in the sequence has to be a rational number (if you write it out in decimals, it either ends or starts repeating itself), and eventually all the terms have to get so close together the differences between any pair are infinitesimally small, but the terms themselves get arbitrarily close to something that isn’t a rational number.