NathanielJ wrote:The argument that e isn't a "compact" base because 4 has an infinite expansion in it doesn't hold much water in my mind though. If you think either base 2 or 3 is more efficient at representing information than base e, then I'd like to see you write e2 in one of those bases.
Try this argument on for size then. Say you want to write down some reduction formulae that describe the base in a way that's purely symbolic. What I mean is, you can define the real numbers as decimal expansions
but is it possible to do so for base e? Can you formulate the real numbers in terms of their e-expansions, without reference to anything but strings of symbols?
What you realise as you try to answer this question is that you'll need to determine a method for determining equivalence classes of strings of symbols. Now, perhaps you're much more cunning than I, but it seems quite difficult to write out some finite set of rules which tell you when and how a particular string can be rewritten, though in decimal that's pretty easy. A canonical form, which in decimal is merely "doesn't end in an infinite string of 9s" seems likewise quite intractable. Furthermore, doing practical additions and multiplications seems nigh impossible.
Now, this isn't to say that there aren't non integer bases that I like. I'm quite fond of base phi, but it has lots of positive points going for it that e doesn't have. Its quite easy to write down what the canonical form should be, and addition and multiplication are all about substituting strings of the form 011 with those of the form 100, and for convenience I also remember 0200 = 1001.
But there are no rules like that for e.