[jacobolus]

Personally I wish we had a good notation and separate single-syllable pronunciation for negative digits –1 through –9, and were willing as a society to accept numbers written with a mix of positive and negative digits.

Decimal arithmetic becomes quite a bit easier if you normalize your numbers to the digits from –5 to 5, and it would help prepare students for algebra.

[moefh]

An ugly thing about using negative digits with base ten (or any even base) is that you either have ambiguity on how to write numbers, or you have a different number of positive and negative digits.

For example, if you do base ten with digits EDCBA012345 (E=-5, D=-4, etc.) -- that's 11 digits -- then five can be written as 5 or 1E; fifteen is 15 or 2E, etc.

So you either:

- Accept that some numbers can be written in more than one way, which seems a huge disadvantage over what we have today.

- Rule out certain sequences of digits (E can only follow a negative digit, 5 can only follow a positive digit, if 5 follows a 0, then it depends on the digit before the 0, etc.). Not only is this inelegant, but it adds seemingly arbitrary rules for people to memorize when they're first learning how to write numbers.

- Remove one of the digits (e.g., remove E). Then things aren't symmetric around zero -- e.g. five is 5, but minus five is A5.

[jacobolus]

That numbers have interesting relationships to each-other is not an “ugly thing”, but rather a beautiful thing. The relationships exist whether we pay attention to them or not, so giving people a way to write and pronounce those relationships is a nice way to improve numeracy.

Writing numbers in multiple possible ways is really not that big a problem for most uses (for record-keeping in business transactions maybe). Personally I recommend becoming familiar with two different normalizations: all same-signed digits (the form we use now) or [–5, 5] with digits before a terminal 5 rounded away from zero. But in general for personal scratch work numbers don’t need to be normalized unless you feel like it. As long as you understand that 20–3 is the same as 10+7, it doesn’t really matter which one you think of as primary.

We use multiple number representations all the time, with e.g. the vulgar fraction 11/8 alternately representable as the “mixed number” 1 + 3/8, or as the decimal fraction 1.375, the simple continued fraction [1; 2, 1, 2], the percentage 137.5%, or the common logarithm ~10^(0.1383). Figuring out that 1.375, 1.385̅, 1.42̅5̅, 1.43̅5, and 2.6̅2̅5̅ are the same number isn’t that hard, with some practice.