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.
In any event, e is a number just a little larger than two and seven-tenths. Even though its digits start with the appealing pattern 2.718281828, it is irrational, never settling into an endlessly-repeating sequence; and somewhere out there may be someone who knows what the first couple digits after that second 1828 are.
Florian Cajori’s A History Of Mathematical Notations — a name and a reference I’m surprised I managed to go four essays without citing, as his mathematical history work is indispensable even eighty years after first publication, comprehensive as well as entertaining in-between all the quotes from manuscript — notes that Gottfried Wilhelm Leibniz, otherwise famous for his work in getting the Duke of Brunswick-Lüneburg elevated to the rank of Elector for the Holy Roman Empire, used the letter b to represent this number in letters written to Christiaan Huygens, himself noteworthy for contributing to the 17th century Netherlands tradition of 31-tone music, on the 3rd and 13th of October, 1690, and again in January 1691. The connection between the base of the natural logarithm and b seems clear enough; why would it have shifted to another letter?
Well, so what if Leibniz liked b? Mathematicians are always inventing and reinventing notation, to fit convenience and the problems on which they are working. In the early days of a concept, when it is not clear just what it will be useful for, the choice of a symbol can be changed in ways which look capricious to the later reader. (One of the minor chores in preparing the second edition for a textbook I worked on is finding where we had, in good conscience, used the letter Omega Ω for a deserving quantity, as we’ve now found it conflicts with a quantity getting more space in the new edition and having a better claim to the letter.) Mathematics is a very human activity, and humans are always trying to redo in slightly better form what they had done before.
Other people developed other notations, some of them rather clever, even after the selection of e would seem to be fixed. Cajori notes an 1859 example by one B Peirce to introduce symbols which look like slightly opened paperclips, with the long leg on the right to replace e and the long leg on the left to represent π, which has some appeal. But no different letters have caught on for this, and none seem likely to in the future. It might be convenient to add a brand-new symbol, something outside the overloaded Roman and Greek alphabets, but it is hard to imagine a wholly new symbol getting into common use today. Just getting the major computer typefaces to support it would be overwhelming; getting people to type it too, when e already sits there so close to the home row of keys, might be impossible.
There was something else behind e beating out b and all other comers to represent this number, of course.