In light of those distinctions, I'm going to employ a certain formatting scheme in this post that will hopefully make it easier to compare base-ten with other systems. If I write out the name of a number with letters then I'm referring to the value. So by "ten" I mean a group of this many things: (o o o o o o o o o o). On the other hand, if I were to write "10" then I'm referring to the symbol formed by the digits themselves, and not a specific value or count of something. It gets a little less clear when we get to the name, since I'm already using the spelled-out word to indicate the value. But generally, if I put the word for the number in quotation marks I'm probably talking about it specifically as a name rather than as a representation of the value. A little confusing, but hopefully it'll make sense in context.
Now, if we were planning to switch to any system between base-one and base-nine we'd be all set for symbols: we'd have a few extra hanging around that we wouldn't need anymore. But since we're talking about switching to a system higher than base-ten, we're gonna need a couple more single-digit numerals to fill in the new gap before we reach "10" (which, remember, is twelve things in base-twelve). For now, we can just swipe a couple symbols from the Greek alphabet*. Doing so could give us a series of number symbols that looks like this:
1, 2, 3, 4, 5, 6, 7, 8, 9, γ, ψ, 10
Two values that were formerly represented by double-digit symbols, ten and eleven, are now represented by single-digit symbols, and the symbol "10" has been reassigned to represent a dozen of something.
Obviously, reciting that as "one, two, three, four, five, six, seven, eight, nine, ten, eleven, ten" isn't going to work. We need some new names for ten (γ, or this many things: o o o o o o o o o o) and eleven (ψ, or γ plus one) and to reduce confusion we should probably also rename twelve, which is now of course represented by the symbol "10." We could make them sound similar to their old names to ease the transition: γ (the old ten) can be "ben" and ψ (the old eleven) can be "elv." And 10 (the old twelve) can be "doz." (I'm pronouncing the "oz" in "doz" like the "aws" in "jaws.")
So, "1, 2, 3, 4, 5, 6, 7, 8, 9, γ, ψ, 10" would be read as "one, two, three, four, five, six, seven, eight, nine, ben, elv, doz."
Weird, definitely, but hopefully you can see what I'm getting at.
Our two new single-digit numerals would work just like the others, and so, for instance, the sequence would continue: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1γ, 1ψ, 20. This is where things start to get really messy, though, because now none of those symbols (or indeed any symbols from here on out) correspond with their values in base-ten. If we really wanted to separate base-ten and base-twelve we could potentially go further than γ and ψ and come up with a new symbol for every numeral for one through nine (zero could stay 0, I suppose) so that you wouldn't have to worry about any symbols having different values between the two systems (because the two systems simply wouldn't share any). Maybe that could be the better way to do it, but for this post I'm going to stick to just adding γ and ψ. It means you don't have to memorize a whole new list of numerals, but it also means having to realize that, say, "14" in base-twelve does not represent the same value as it does in base-ten. What is consistent, though, is that it still represents the notion of "one set + four;" we've just changed the definition of a "set."
Since we already assigned the values ([oooooooooo], [oooooooooo]o, [oooooooooo]oo) new names (ben, elv, doz), you might argue that we could therefore let the symbols that used to represent them ("11" and "12") keep the old names ("eleven" and "twelve"). But that's just asking for miscommunication, I think, so the values now represented by "11" and "12" need new names. Deciding what they should be could be interesting/complicated. The names "eleven" and "twelve" are actually kind of weird to start with, because they don't fit the pattern of the rest of the nearby numbers. Based on the others, you'd expect them to be "oneteen" and "twoteen" or something like that ("thirteen" and "fifteen" are almost what you'd expect, which would be "threeteen" and "fiveteen"). So maybe we could just call them that? But then there's the fact that all the names in English for the values between ten and twenty (in base-ten) don't match the pattern that's subsequently followed. Really, you'd expect them to be "ten, ten-one, ten-two, ten-three, ten-four..." Or, if "ten" was named like the rest of the two-digits: "tenty, tenty-one, tenty-two..." Or perhaps something like "unty" would be even more in-line with the pattern, so: "unty, unty-one, unty-two..."
Fuck it. I went and wrote that fairly sensible paragraph, but I've changed my mind. We're still not gonna redo the numerals or give them new names, but we are totally going to rename the resulting double-digit symbols, because that'll make talking about them a lot less ambiguous.
In our base-twelve system, two-digit numbers after doz (10) are named like this: [first numeral]duz-[second numeral]. Therefore "11" is oneduz-one, "25" is twoduz-five, and so on. The number names in English for three-digits and above also have a slew of problems, but I've gotta draw the line somewhere, so we'll just adopt them into our system more-or-less how they are. For three- and four-digit numbers, replace "hundred" with "grondred" and "thousand" with "zousand." Yes, it sounds dumb; I don't care, it makes (some sort of) sense. Sadly we can't just have million, billion, trillion, etc. all just go by "zillion," so they'll be mizion, bizion, and so on. Again, dumb; again, I don't care. And I think that should cover everything. Hooray! The old "teens" get a little clunkier, but the rest of the two-digits (plus the three- and above) all have the same number of syllables as before. Saying the names obviously won't be nearly as natural, but the extra bit of thought you have to put into it is probably a good thing.
|Shut up, it's not like that at all...|
|Let the beauty of Paul Rudd assuage all your doubts...|
Now, with the business of names and symbols out of the way, it's time to address the issue that's been sitting menacingly in the corner: Changing our number system automatically changes math. In a base-twelve system, 10 ÷ 4 = 3, and 6 × 3 = 16. Half of something is 60%. It's weird.
The realization that our system of math isn't absolute may lead you to wonder whether it means anything at all. All these equations and formulas that we use to describe our world through physics and chemistry... is it all pointless? Are we just taking abstract concepts that have no bearing on reality and pretending they combine in meaningful ways? Is everything I know a lie?!
No. Now get a grip. The fact of the matter is that regardless of which number you base your system on, this many (o o) of something plus this many (o o) of something will always give you this many (o o o o). And this many things (o o o o o o o o o o o o) divided into this many (o o o) groups will always look like this (oooo oooo oooo). The values stay the same; all that's changing are the names and symbols you attribute to the values. Your number system and its inherent math are just a language to describe relationship of quantities. And just like the sun's gonna rise no matter what you've named it, the values are going to combine the same way no matter what you've decided to call a full 'set.'
Admitting that the values are constant but that the symbols for them are arbitrary may very well break a few hearts, though. The much-loved ratio of a circle's circumference to its diameter, π, is no longer 3.1415926535... Those symbols themselves have no objective significance, only the underlying the ratio of values that they express does. In base-ten that ratio is roughly 22/7, but in our nomenclature for base-twelve it becomes 1γ/7, which comes out as 3.184809493ψ... To be clear: those two different-looking strings of numbers are actually describing the same value; they're just in different languages.
So, this was a long post, and no one will blame you if it hurt your head a little bit (hopefully all the pain was from the concepts themselves rather than me choosing suboptimal ways to relate them). Now that we sort of see what a base-twelve system would look like, you might be asking yourself, Would it really be worth it to switch? Or maybe there's no question for you, and you've decided, No, it would not be worth it to switch. And if that's where you're at, I've gotta say I think I agree with you, despite all the effort I've put into describing it. I fear there's probably no plausible way to switch our number system at this point; we've come too far with base-ten. It's a shame, but it's the truth. That doesn't mean I think it's worthless to understand what it would look like, though. As many people can surely attest, learning a foreign language can be very helpful in understanding the nuances of your own.
I have a few more miscellaneous thoughts on this subject, but I'll save those for another post.
* I should note that my choice of letters from the Greek alphabet was largely arbitrary; I basically just chose two that weren't too similar to the existing numerals and which I thought looked nice with the font I'm using. There have of course been may other proposals for the symbols which should represent the values ten and eleven in base-twelve. A and B are common ones, as are T and E. I went off on my own and selected Greek letters because I wanted symbols that had very little meaning for most people outside of how they're being used here. Obviously all the Greek letters are used heavily in math and science so the solution is hardly ideal, but I think it's decently illustrative of the points I'm trying to make.
Also, the names I've listed—for those two numerals, "10," and the double-digits—are entirely my own. I think they're sensible, but I'm sure there are better-known (and maybe better overall) alternatives out there. And I could have just called "100" in base-twelve a "gross" and "1,000" a "great gross" since those are existing names for those quantities, but I wanted their names to emphasize how they're related to the values with the same symbols in base-ten, hence "grondred" and "zousand."