While following my own lightly compulsive tracking of the blog’s viewer statistics and wondering why I don’t have more followers or even people getting e-mail notifications (at least I’ve broken 2,222 hits!) I ran across something curious. I can’t swear that it’s still true so I’m not going to link to it, and I don’t want to know if it’s not true. However.
Somehow, one of my tags has become Google’s top hit for the query “christiaan huygens logarithm”. Oh, the post linked to contains the words, don’t doubt that. But something must have got riotously wrong in Google’s page-ranking to put me on top, even above the Encyclopaedia Britannica‘s entry on the subject, and for that matter — rather shockingly to me — above the references for the MacTutor History of Mathematics biography of Huygens. That last is a real shocker, as their biographies, not just of Huygens but of many mathematicians, are rather good and deserving respect. The bunch of us leave Wikipedia in the dust.
I assume it to be some sort of fluke. Possibly it reflects how the link I actually find useful is never the first one in the list of what’s returned, so perhaps they’re padding the results with some technically correct but nonsense filler, and I had the luck of the draw this time. Perhaps not. (I’m only third for “drabble math comic”, and that would at least be plausible.) But I’m amused by it anyway. And I’d like to again say that the MacTutor biographies at the University of Saint Andrews are quite good overall and worth using as reference, and are also the source of my discovery that Wednesday, March 21, is the anniversary of the births of both Jean Baptiste Joseph Fourier (for whom the Fourier Series, Fourier Transform, and Fourier Analysis, all ways of turning complicated problems into easier ones, are named) and of George David Birkhoff (whose ergodic theorem is far too much to explain in a paragraph, but without which almost none of my original mathematics work would have what basis it has). I should give both subjects some discussion. I might yet make Wikipedia.
I’ve outlined now some of the numbers which grew important enough to earn their own names. Most of them are counting numbers; the stragglers are a handful of irrational numbers which proved themselves useful, such as π (pi), or attractive, such as φ (phi), or physically important, such as the fine structure constant. Unnamed except in the list of categories is the number whose explanation I hope to be the first movement of this blog: e.
It’s an important number physically, and a convenient and practical number mathematically. For all that, it defies a simple explanation like π enjoys. The simplest description of which I’m aware is that it is the base of the natural logarithm, which perfectly clarifies things to people who know what logarithms are, know which one is the natural logarithm, and know what the significance of the base is. This I will explain, but not today. For now it’s enough to think of the base as a size of the measurement tool, and to know that switching between one base and another is akin to switching between measuring in centimeters and measuring in inches. What the logarithm is will also wait for explanation; for now, let me hold off on that by saying it’s, in a way, a measure of how many digits it takes to write down a number, so that “81” has a logarithm twice that of “9”, and “49” twice that of “7”, and please don’t take this description so literally as to think the logarithm of “81” is equal to that of “49”.
I agree it’s not clear why we should be interested in the natural logarithm when there are an infinity of possible logarithms, and we can convert a logarithm base e into a logarithm base 10 just by multiplying by the correct number. That, too, will come.
Another common explanation is to say that e describes how fast savings will grow under the influence of compound interest. A dollar invested at one-percent interest, compounded daily, for a year, will grow to just about e dollars. Compounded hourly it grows even closer; compounded by the second it grows closer still; compounded annually, it stays pretty far away. The comparison is probably perfectly clear to those who can invest in anything with interest compounded daily. For my part I note when I finally opened an individual retirement account I put a thousand dollars into an almost thoughtfully selected mutual fund, and within mere weeks had lost $15. That about finishes off compound interest to me.
Continue reading “Some Names Which e Doesn’t Have”