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I'm quite confident that the idea of negative digits has value (practical value in mental arithmetic, etc.) and that the convention of writing ordinary digits in red is sound. The next part I'm not so sure about. It addresses a need, and is the best way to address that need (in my humble opinion), but at the same time it's kind of silly, and I haven't even fully adopted it yet myself. So, read and enjoy, but feel free to detach and ignore if you want—not that you need my permission for that.

Probably you've already noticed what the need is. If not, please read the following equation aloud: 97 = 103. Yes, the trouble is that we have concepts and symbols for negative digits, but no words, and it's hard to think well about things without words (see Words Are Ideas).

That's not entirely true, so let me elaborate a little.

  • Do we really need words to do mental arithmetic? That's a tough question! Maybe there are people who do mental arithmetic visually, as if they were manipulating numbers on paper, or symbolically, as if they were running Mathematica. All I can say for sure is that I'm not one of them. A lot of what I do is symbolic—keeping track of what I'm doing, performing individual operations—but the way I remember intermediate results is totally verbal. It feels slow, and it's probably the main reason I'm not a savant, but I guess I'm stuck with it.

    In any case, we definitely need words if we want to talk about mental arithmetic with other people.

  • Do we really have no words for negative digits? That can't be right. I've been using negative digits for years, and my memory is verbal, so I must have some words, and in fact I do. If I want to think of 97 as 103, I say “one hundred less three”, and if I want to think of 147 as 267, I say “two hundred less sixty less seven”, or maybe “two hundred minus sixty plus seven” if the double negative gets on my nerves. (Double negatives like that are closely related to the nested rewrites I mentioned earlier.)

    On the other hand, I find those words clumsy and unsatisfying. They're also incomplete: they don't provide any way to read numbers as strings of digits, as we do when reading phone numbers and credit card numbers and the like. Of course, such numbers really are just strings of digits with no arithmetical meaning, so there's no reason they'd ever contain negative digits, but I sometimes like to think of numbers in calculations in the same way.

So, the problem isn't simply that we need words for negative digits, it's that we need better words so that we can think and communicate better.

Anyway, after all those preliminary remarks, here's what I propose. Just as we did with symbols, let's construct new words by modifying and reusing the existing ones, in this case by changing the initial consonant(s) to “n” for “negative”. There are some cases where that doesn't work, and some where the spelling needs to be adjusted afterward, but that's the basic idea. And, here are the results in a table, with exceptions in bold.


As you can see, the table includes words not just for negative digits but also for negative multiples of ten. That's as far as we need to go, though! The words “hundred”, “thousand”, “million”, and so on don't change, so for example we can read 123456 as “one hundred nenty-three thousand nor hundred and nifty-six”. We also don't always need to be so formal. We can read 103 as “one hundred and nee” or just “one oh nee”, and 267 as “two hundred and nixty-seven”, “two nixty-seven”, or “two nix seven”, depending on the situation.

Here are notes on all of the exceptions and some of the non-exceptions.

  • un. The written word “one” doesn't have an initial consonant, but the spoken word does, a “w” as in “won”. If we change that to “n” we get “none”, but that's a terrible result because “none” is a fairly common word that already has a mathematical meaning (that isn't the one we want). However, if we drop the consonant we get “un”, and that's pretty good! As a prefix it indicates negation, and if we bring to mind the facts that “digit” can mean a finger (or toe) and that fingers can be used for counting, then the digit “un” is in fact an un-digit, a single anti-finger. By the way, the component “un” in systematic element names like unnilhexium means positive one and is pronounced “oon”.
  • noo. The written word “two” doesn't transform well, but “too” gives us “noo”, which is fine. I thought about “nu”, but no … it would be a huge distraction since I'm already familiar with the Yiddish meaning.
  • nee. Here I thought about “ni”, but again, no … the association with Monty Python is just too strong. Although, I admit it would be amusing to make saying “nu” for “ni” into a numerical mistake like saying “five” for “three”! Another important factor here is that “ni” doesn't sound the same as “nee”.
  • nor. The unadjusted form “nour” looks like it ought to rhyme with “sour”, but “for” gives us “nor”, which is fine. It's already a real word, but it's not too common and doesn't have a mathematical meaning (unless you count NOR gates).
  • nix. This is a real word that has a mathematical meaning, but it's quite rare. Even as a verb it's not very common.
  • nate. The word “eight” doesn't have an initial consonant, but we can always add one. Unfortunately, the result is “neight”, which looks way too much like something a horse would say. What can we do? Time for another homophone: “ate” gives us “nate”, which is fine.
  • ine. The word “nine” already starts with “n”, so we can't change the consonant, but we can drop it to get “ine”. That looks a little weird at first, but you get used to it, and the pronunciation is fairly intuitive. (If you look at words ending in “ine”, there are some two-syllable and longer ones that have the wrong sound, but all the one-syllable ones I can think of are right.) I thought about the alternate spelling “ayn”, which I've seen as the name of that one letter in Arabic and as Ayn Rand's first name, but I find the pronunciation totally unintuitive. When I see it by itself I want to read it as “ane”, and in Ayn Rand's name, well, for years I thought it was just a trendy way of spelling “Ann”.

    Here's an amusing coincidence: “ine” is pronounced much like the German word “ein”, meaning “one”, and ine is closely related to one; for example thirty-ine (39) equals twenty-one.

  • teeny. I didn't want to make negative forms of the irregular words “eleven” through “nineteen”, especially since each one would need two points of variation (for example, 14 can turn into 1414, or 14), but I also didn't want to break the symmetry between positive and negative. So, the only choice was to make regular positive forms first! That's easy enough, though. The common element is obviously “teen”, and to make regular forms we need a root word that ends in “y”, so, “teeny”. Then, instead of “fourteen” we can say “teeny-four”, instead of “eleven”, “teeny-one”, and so on. The fact that “teeny” means “tiny” is unfortunate, but I can live with it.

    I don't expect the regular positive forms to replace the normal irregular words; they're just alternatives that are convenient here. Also, I might be the first to write a paragraph about them, but I'm definitely not the first to think of them. I found some isolated instances with just a quick search online.

  • neeny. Once we have “teeny”, we get “neeny” for free; it's not even an exception. Then, here's how the regular forms handle the two points of variation.


    However, I also want to propose one irregular word. Calling 20 “twenty” is fine, but calling 10 “teeny” is just too weird. We should continue to call it “ten”, and then by extension we should call 10 “nen”. It's not great that the sound is so close to “nenty”, but I don't think it'll cause too much trouble.

  • norty. This too is not an exception!
  • nifty. This is a real word, but it's not too common and doesn't have a mathematical meaning.
  • natey. As before, from “eighty” we get “neighty”, and as before it looks terrible. This time there aren't any homophones we can use, but we can fake it and obtain “natey”, which isn't too bad. I thought about “naty”, but it looks too much like a typo for “natty”.
  • inety. The same logic that gave us “ine” gives us “inety”. That looks even weirder, but what's the alternative? The word “aynty” is completely awful—it looks sort of like “auntie” and sort of like a bunch of random letters. Maybe “einty” has some appeal if you're familiar with German, but it has the wrong sound if you're not.

When you consider how short all these words are, it's really remarkable that there are no significant collisions with normal English words!



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@ February (2012)