[sanityUnbounded]

Curious, why do you have to remember that 2 + 2 = 4 & 3 + 2 = 5? Once you know the values the symbols represent, at that point isn't it similar in simplicity to roman numerals?

II + II = IIII

2 + 2 = 4

Don't see how the latter problem lends itself to any more memorization beyond symbols

[Spivak]

Because if you don't memorize your 'primitive' algebraic rules you'll end up just doing a pullback with roman numerals in the middle. Integers don't have any values, they're mathematical objects with certain properties. Asking about 'the value of 5' doesn't make sense unless you're trying to convert to another, already known, number system.

What is 2 + 5? Well 2 is II and 5 is V which is IIIII. So then we have IIIIIII which is (IIIII)II or VII which is 7.

[GenWintergreen]

I just want to point out that it's humorously appropriate for a user named "Spivak" to mention pullbacks.

[jdminhbg]

The equivalent to II + II = IIII isn't 2 + 2 = 4, it's 2 + 2 = 22. Simply jamming the symbols together gives you the right answer.

[dllthomas]

Right, much like I + V is IV.

(There is more help; it's not as simple as people have been portraying, though...)

[Spivak]

Sure, but that notation was likely born out of convenience by people who wanted a shorthand for situations where it's easier to represent a number as a subtraction but it does make the number system more complicated.

[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.

[wvenable]

You just need to add a sort to to get VI from that -- something explained in the article regarding multiplication.

[crowbahr]

V + I = VI though.

You just have to put the larger number first for the addition.

[dllthomas]

That might be enough to prevent you from getting any well-formed wrong answers.

It's not enough of the story, though - consider:

    VI + V = VIV
    V + IV = VIV