Vowels, and the Alphabet

One of the things I marvel at is that Spanish and Japanese get away with five vowels. Maybe a few dipthongs for Spanish. But English has at least 12 semantically distinct vowels:

bait, bat; beet, bet; bite, bit; boat, bought; boot, but; plus bout. Boit is not a word but seems like it could be, for a total of 12. There's also butte, but arguably that has an extra consonant /byoot/.

The other thing is how many syllables English has; it's a good thing we have an alphabet, and not the more commonly invented syllabary. The syllabaries I know of are in the 40-60 range. English: 12 vowels, plus 18 useful consonantal letters, plus th (thin), th (then), ch, and sh. 22*12 = 264 CV syllables. And that's not counting all of the consonantal combinations which don't deserve their own alphabetic letter: sk, br, gl, gr, -nt, -ng, kl, kr, etc., etc. It'd be a nightmare!

Blinding Flash of the Logarithmically Obvious

Feynman, in his Lectures on Physics, points out that the logarithm, while introduced late in the modern curriculum, is just as mathematically fundamental as roots, if not division. Subtraction is the inverse of addition, division of multiplication. Taking powers isn't symmetric -- 2^3 <> 3^2 -- so we get two inverses, root and logarithm. Given a^b = c, you can ask for a = c^(1/b) or for b = log_a c.

And, I realized while lying in bed this orning, for natural numbers the logarithm is actually much easier to calculate. Subtraction (a-b) can be thought of as subtracting 1 b times. Division (a/b) can be thought of as how many times you can subtract b from a until hitting 0 -- no guesswork needed (that's for long division.) And for log_a c (log, base a, of c) just count how many times you can divide c by a until reaching 1. (If you don't reach 0 or 1, of course, the division or log doesn't exist in the natural numbers.) Finding the roots of natural numbers takes guesswork or search.

In the real numbers it's reversed; there's convenient algorithms for finding roots, which you can find and verify without needing calculus (though calculus gives a proof), while finding logs needs tables of pre-calculated values.

UPDATE: Well, findings logs is a generalization of the natnum procedure; instead of dividing by your base, you divide by powers of your base. E.g in finding log_10 (2), you divide 2 by the fourth root of 10 (the square root is too big), then by the 32nd root of 10, and so on, and the log is the sum of the root powers (1/4 + 1/32 +...) You need the table of the various roots of 10.

Or, of course, you could just do trial and error. "Third root of 10 is bigger than 2, fourth root is smaller, so the log must be a rational in between..." It'll help to use the root algorithms, rather than needing trial and error on those, but it'd still be a fair bit of work.