One of the gifts I received for Christmas came from an aunt who likes giving me time-related items. (It's not a bad theme for me anyway.) This year she sent an ``irrational watch'', one whose face dispenses with such ordinary things like ``2'' in favor of labels like the golden ratio or the common logarithm of 9876. I was pleasantly surprised by it; I would have been completely surprised if she hadn't called Christmas day while we were opening presents --- and before I'd got to hers --- to ask if I'd gotten the irrational watch. But the spirit and, er, timing were good.
I'm a sloppy writer of thank-you notes, what with being sloppy about so many of these e-motions the hew-mons talk about. But I realized there was an obvious thing to do when I thought about her mentioning how she didn't understand any of the symbols on it: I could write a letter that explained each of them. It took me longer to get started than I wanted, and longer to complete since I felt I had to write it by hand, but I had a lot of fun doing it. It didn't hurt that the lead symbol was the Euler-Mascheroni constant, which is actually not proven to be irrational (although that's what everybody expects it to be; if it is rational, then the lowest denominator for its rational representation has to be greater than 10242080). And I realized that for e I could skip the whole dull compounding interest thing for other interesting constructions of it.
My aunt loved the letter, and took time to mention to my mother receiving this wonderful ten-page letter in dense tiny script (it's actually only six pages; my mother pointed out, mathematics wasn't my aunt's strong suit). And I really liked trying to think of ways to explain, for example, the Euler-Mascheroni constant to a person who didn't know any calculus (if she had known it, that was decades ago, and learned against her will), and I think I acquitted myself honorably.
It's left me thinking, it might be worth opening a Professional Blog, the sort of thing that doubles as a resume to prospective employers, using the gimmick I put into that letter as a starting point. I could probably go on for several thousand words just getting around to saying why e is interesting, particularly as I realized I was throwing around casually some pretty heady ideas that were worth careful explanation on their own.
Might be worth it at that.
Trivia: Gottfried Leibniz used the symbol
b as the base of the natural logarithm in letters written to Christiaan Huygens in October 1690 and January 1691. Source: A History Of Mathematical Notations, Florian Cajori.
Currently Reading: Florence, City Of Art, Loretta Santini.