History and Other Stuff

Now that I've discussed some of the other words I thought about, I'd like to go back and do the same for written representations.

The idea of coloring the digits red may seem to be obvious, and to be the obvious best choice, but in fact it was one of the last things I thought of, and it has some flaws that almost kept me from picking it. If I hadn't wanted to talk about fractions and decimal expansions, I probably would have gone with underlining! One large flaw is that color information is easily lost in copy and paste operations. (But, most other choices have the same problem.) One tiny flaw is that I already used red digits for contrast in the figures in Powers of N. (I also used purple digits in a couple of other places.) The main flaw, though, is that I think adding colored text to the site is a step in the wrong direction … a small step, maybe, but a step nonetheless. Now that I've started down this garish path, where it will end? Blinking text for imaginary digits? That might be convenient for power-series coefficients, you know. Animated figures? Videos? Banner ads?

One last flaw is that although color is the best choice in this situation (on a web site), it's a pretty poor choice in other situations. In print it's expensive, and in handwritten notes it requires a second pen and a lot of switching back and forth. As a result, it would probably be a good idea to pick an alternate representation to use in those cases, but I'm really not sure what's best. Instead, I'll just tell you all the other options I thought of and let you pick for yourself. They're all based on the idea of modifying the existing digits.

• Inverted text. The idea here is to take a rectangular area around each digit and invert it, as in a photo negative. The symbolism is great, but inverted text is impractical to draw by hand, and the fact that it appears to be selected is surprisingly distracting. (Does anyone else like to select pieces of text while reading?)
• Different font, size, or style. I couldn't find any combinations that weren't ugly, but perhaps someone more talented could make it work.
• Strikethrough. If you want to combine a minus sign with a digit, strikethrough is the natural choice, so this is another option with good symbolism. Too bad it's ugly! Slash, backslash, and vertical bar are no better. Also note that Europeans use strikethrough on 7 to distinguish it from 1, and that computer folks sometimes put a slash on 0 to distinguish it from the letter O.
• Overline or underline. If you move the minus sign up or down you get an overline or underline, so the symbolism here is pretty good too. However, overline is already the standard notation for repeating digits in decimal expansions, and underline is the notation I've been using in place of that (see Fractions). So, if you're me, they're both taken, and if you're you, you still have to remember which line is which.
• Overdot or underdot. I thought about these, too, but they're small and easy to miss, and not especially compelling.
• Hat. For me, the hat symbol is pretty strongly bound to the Fourier transform, but I've seen it used for other transforms, especially ones that are more or less self-inverse, and negation of digits certainly fits in that category. If you use wide hats to cover adjacent digits, then hat is a lot like overline, only better because it's clearer how many digits are covered; a horizontal square bracket would be even better.

In typed notes I sometimes do things like write 101 as 10n1, but that option doesn't belong in the list because it doesn't modify the existing digits. It also doesn't handle numbers like 1334 very well.

Of course, there's also the hash mark option that I mentioned right at the start. It's not part of a complete system of digits, but it's still useful since 1 is by far the most common negative digit in theoretical work. How can I be so sure? Because up until now, that's what I've used! Here's a quick history that can also serve as a review of what I know about the theoretical value of negative digits.

• At the end of Fractions in Base 2, I talked about numbers as polynomials over the base, and joked that there ought to be a digit with value −1.
• Almost immediately I started taking the joke seriously. A few months later, in an essay about divisibility tests in other bases, I officially started using the hash mark as a negative digit. Here are the main facts translated to the new notation.

  101	= 11 × 11
  1001	= 11 × 111
  1001	= 11 × 111

• A year and a half after that, in Repunits, I put the hash mark to good use once more, and the letter X had a cameo as Digit with Value −2 in this crazy number.

  1110011211001111110010101010100111111001121100111

• I used the hash mark one more time in The Multiplication Table even though I was already planning to write this essay.

Here's another interesting bit of history. Just over a year ago, a friend of mine (Dave Greene) brought to my attention the article The Reverse Notation, about how to remove the digits 7–B from base 12 and replace them with negative digits. The ideas of working in base 12 and of removing digits didn't do much for me, but I did like the idea of obtaining a full set of names and symbols for negative digits by modifying the names and symbols for positive digits. What about the specific choices the author made? The symbols were OK—overline, plus some interesting handwritten forms—but the names were unbearable. Of course, there's no way the author, writing in 1950, could have known that changing the initial consonant(s) to “r” would one day be polluted by association with Scooby-Doo!

Shortly after that, I drove down to Texas and had some time to think in the car, and I came up with the table and the notes from the previous page. I figured I could make a nice little essay out of that, but I wasn't in any rush since the whole thing seemed kind of frivolous. It wasn't until I finally sat down to write it that I really understood the connection to complements and mental arithmetic. The last piece of the puzzle was the idea of coloring the digits red. Once I had that, I just had to do a bunch of writing, and voilà.

Just for the record, I don't have anything against base 12, I'm just not interested in moving away from base 10. Base 12 is actually quite nice, as I explain at some length in Duodecimal.

Now I have to backtrack a little. The point of The Reverse Notation is that almost everything is easier if you use the digits 6–6 instead of 0–B: addition, subtraction, multiplication, long division (though I'm not convinced on that point), rounding, etc. I liked the part about multiplication enough that I worked it into my earlier comments about mental arithmetic.

… but when you multiply, in general the best you can do is rewrite both numbers so that there are no digits outside the range 5–5.

So, we can write numbers in the usual way, with all positive digits, and we can rewrite them using the digits 5–5. Are there any other ways we can rewrite them? How about with all negative digits? At first sight that seems like a stupid idea that doesn't even work, but if we allow an initial 1, it works just fine, and in fact is the standard second complement. To repeat an earlier example, 674 = 1326.

That gives us three different canonical forms for integers. I don't know any other useful forms, but I do know an interesting related question: given some particular number, in how many ways can we rewrite it? To get a simple answer, we need to make two assumptions.

• The digit 0 is allowed, and counts as different than 0. This too isn't as stupid as it seems. In fact, I've already used 0 several times: when I was talking about fractions (e.g. 0.077077), when I made the table on the previous page, and when I was talking about nen (10). When necessary, I'd read 0 as “noh” or “nero”.
• The number of digits in the number is allowed to increase, but only by 1. So, rewriting 97 as 103 counts, but rewriting 674 as 199326 doesn't, even though it's a valid rewrite.

After that it's easy—all we have to do is walk from right to left through the n digits of the number and decide whether each one should be red or black. Thus, the number can be written in 2ⁿ ways, and in fact the ways correspond naturally to n-digit binary numbers.

Speaking of 199326 and canonical forms, I ought to point out that the negative form (second complement) isn't well defined unless we specify that the length is minimal (n+1), and that the digits aren't really all negative unless we allow 0 (or count 0 as negative). Also, in what we might call the centered form, we have to decide whether we want to allow both 5 and 5. If we do, the form isn't well defined, but if we don't, it's not symmetrical.

Finally, here are some examples of numbers as polynomials over the base that I hope can also serve as examples of the aesthetic value of negative digits. From this familiar identity …

(x ± y)² = x² ± 2xy + y²

… we can discover some nice connections between the values of the squares.

	132 = 169
72 =	132 = 169 = 49
	312 = 961
292 =	312 = 961 = 841

And, from this familiar identity …

(x + y)(x − y) = x² − y²

… we can discover a new way of looking at some old facts.

9 × 11 = 11 × 11 = 101 = 99
8 × 12 = 12 × 12 = 104 = 96
7 × 13 = 13 × 13 = 109 = 91

And, last and probably least, you know how small multiples of 9 have digits that add up to 9? Well, small multiples of 11 are even easier to write down!

7 × 9 = 7 × 11 = 77 = 63
8 × 9 = 8 × 11 = 88 = 72
9 × 9 = 9 × 11 = 99 = 81

What's more, larger multiples of 9 sometimes have digits that add up to 9, but larger multiples of 11 always have digits that add up to 0.