While this is fun speculation for written forms, it is actually a serious issue for algebraic software. The answer seems to come down to which operations are more common: comparisons v. additions. It would be interesting to take a poll of symbolic algebraic manipulation (SAM) systems to see how they represent polynomials & integers. At 06:37 PM 3/1/2005, David Gale wrote:
How do you write your polynomials, ax^2+bx+c or c+bx+ax^2?
Traditionally it seems the first way is most common. Why? The second would seem more natural. Don't we generally like to have things (like exponents) increasing from left to right? By contrast, for power series we have
a_0+a_1x+a_2x^2+ . . ., (never. . .+a_2x^2+a_1x+a_0).
For that matter, why do we write our numerals so that ten is written 10 rather than 01? As with polynomials, the digits of a numeral represent coefficients of powers of ten in decreasing order from left to right. Aha, I get it! Of course. They're Arabic numerals and therefore were read by their creators from right to left. That doesn't explain the polynomials though, or did we get them from the Arabs too? Some further random related observations.
1. Reversing the digit order convention would make addition and multiplication feel more natural since in applying the algorithms one first learns the units digit, then the tens and so on. When we "carry" we would move digits "forward", when we borrow we'd move them backwards.
2. How would you feel about pi ~95414.3? It would certainly take getting used to.
3. Lets look at the "names" for the numerals.
English: seven-teen, eight-teen,. ., so far so good, increasing exponent order, but then we do twenty one, twenty two, Arabic order.
French: At least it's consistently Arabic. dix-sept, dix-huit, and vingt six, vingt sept. Spanish, Italian the same, but,
4.German: consistently anti-Arabic, achtzehn, neunzehn, zwanzig, ein und zwanzig, but after 100 we get, for example, 123 is ein hundert drei und zwazig so 123 gets permuted to 132, the middle digit comes last!
5. The above suggests that the different digit orderings is a Romance vs. Germanic language thing. However, going back to the Latin is no help since they didn't have the Arabic place system. For example 18 I just learned is "duodevigniti". How would a Roman say "You owe me XXXIV denarius" I wonder.