Negative Digits

What does that even mean? How can a digit have a negative value? It's very simple, really. If, say, we assign the hash mark # the value −1, then just as the number 29 is shorthand for 2×10 + 9, the number 3# is shorthand for 3×10 + (−1). The two numbers happen to have the same value, but there's nothing wrong with that, just as there's nothing wrong with writing 1/3 as 2/6.

We could go on from there and pick eight other arbitrary symbols to use as negative digits, but that would only lead to hard-to-remember numbers that looked like cartoon curses. What I want to do instead is forget about the hash mark and instead modify the existing digits 1–9 so that we can reuse them. To be specific, I want to color them red! The use of red to indicate negative numbers is apparently traditional in accounting—that's why people say “red ink” and “in the red”—but what I want to do is a little different since it's per digit rather than per number. (Still, I expect that many accountants have had the same idea over the years.)

Some examples are definitely in order here. We already know that 31 = 29; we can think of that formally as 3×10 + (−1) or perhaps 3×10 − 1, or informally as 30−1. We don't always have to look at the whole expansion; for example we can replace 31 with 29 even when it's part of a larger number.

531 = 529
315 = 295

We can have more than one negative digit at a time.

671 = 529

Formally that's 6×100 − 7×10 − 1, but informally we can group the negative digits together and think of it as 600−71. We can also have multiple groups of negative digits.

123456 = 82556

We can even make believe we're accountants and write entire numbers in red, like 84 = −84. Formally that's (−8)×10 + (−4), but by good fortune it amounts to the same thing. Beyond that lies only madness like 31 = −29, perhaps better written as 31 = −31.

Now that we know what negative digits are, we can move on to the next question, which is why anyone would ever want to use them. There are several possible answers. Negative digits have some theoretical value that I'll talk about later, and some aesthetic value due to how they make certain patterns and symmetries more apparent, but the main answer is that they have practical value in mental arithmetic. Over the years I've developed something, some system of concepts or neurons, that's reasonably effective at manipulating numbers, and negative digits are an important part of that system. I hope that having a way to capture these vague entities in a definite, tangible form on paper will both sharpen my own thinking and make the same thinking available to others.

To give you an idea of the practical value, here's a fairly complete catalogue of how I use negative digits in mental arithmetic.

• If I have to add 29 to some other number, most of the time what I'll actually do is add 30 and then subtract 1. In other words, I'll rewrite 29 as 31! Consider 456 + 29. It's certainly not hard to compute by normal methods, but with the rewrite it's trivial. The rewrite method isn't restricted to numbers that end in 9 (e.g., 17 = 23) or to single-digit transformations (e.g., 289 = 311).
• Similarly, if I have to subtract 29 from some other number, most of the time what I'll do is subtract 30 and then add 1. I want to call that subtracting 31, but it doesn't feel like a double negative when I'm doing it, so I guess my internal representation is closer to adding 31. Although I discussed addition first, I associate the rewrite method much more strongly with subtraction, because for some reason borrows give me more trouble than carries.
• I don't always do this, but given any two numbers you want to add or subtract, you can always rewrite the second number so that there are no borrows and no carries. The “forward” proof where you examine the digits of the two numbers from right to left is left as a not-too-interesting exercise for the reader. The following “backward” proof is nice, though: take the first number and subtract the answer, digit by digit. The result is a series of digits in the range 9–9, so, a valid number. But, that number (or minus that number, if you're adding) has to be equal to the second number, and hence a rewrite of it!

  Note that the rewrite depends on the first number. For example, for 526 − 147 you want 253, but for 528 − 147 you want 267. Also note that rewriting really just packs the borrows and carries into the second number. In the last example, normally you'd borrow one unit from the 5 and move it right; all the rewrite does is move one unit left into the 1 instead. Nevertheless, for some reason the rewrite method still seems simpler to me.

• Here are a couple of real-world examples adapted from last time I balanced my checkbook. First, 520.36 − 31.84. I typically group the digits in pairs, so the first thing I'd do here is notice that 84 is greater than 36 and then rewrite 31.84 as 32.16. To be consistent, we also ought to notice that 32 is greater than 20 and then continue on to 168.16, but it's easy enough to just subtract directly. 520 less 30 is 490, 490 less 2 is 488, and 36 plus 16 is 52, so the answer is 488.52.

  Second, 52.77 − 35.81. Once again, the first step is to invert the cents. That yields 36.19, and then we can subtract. 52 less 36 is 16, and 77 plus 19 is—um, rewrite—77 plus 21 is 96, so the answer is 16.96. Nested rewrites like that are pretty common. They definitely exercise the part of my brain that does the inverting, but they're not too difficult. They wouldn't be necessary if I didn't group digits, but oh well, grouping has other advantages.

• Negative digits and rewrites also give us a more productive way to think about complements. Instead of saying an unhelpful bunch of words like “the second complement of 674 is 326”, we can write down solid mathematical facts like 674 = 1326 (or 674 = 1334, which is the point I was trying to make at the end of the explanation of second complements). The exact relationship between negative digits, rewrites, and complements is this: if you want to rewrite a block of digits—to invert them, to change them from positive to negative or vice versa—all you have to do is take the second complement, change the color, and increment or decrement the digit(s) in front. For example, 279 = 321, while 9979 = 10021.

  Complements also turn up once in a while when you're working near zero. Suppose you have a running total of 4.63 and you subtract 6.00 to get −1.37, and then later you add 10.00 to get 8.63. The 37 in the middle is the second complement of 63, but it's a waste of time to compute. In that situation you can use negative digits to avoid taking the complement—just think of the intermediate value as 2.63.

• Negative digits can help with multiplication too. It's trivial to figure out that 21 × 3 is 63, but what about 19 × 3? It should be just as effortless.

  19 × 3 = 21 × 3 = 63 = 57

  In the same way, multiplying by 97 is a chore, but multiplying by 103 is easy, and so on.

  The situation is a little different than with addition and subtraction, though. When you add or subtract, you can always rewrite the second number to make the operation trivial, 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. (Just work through the digits from right to left.) It's usually easier to multiply after you've done that, but it's not always trivial, especially since you have to keep track of positive and negative digits. So, again, I don't always do it. For example, if I needed to multiply something by 93, I'd most likely transform it to 107, not 113; or perhaps I'd multiply by 31 and then 3.

• One amusing special case is that it's relatively easy to multiply by prime numbers. If the prime is 2 or 5, of course that's easy. If not, the prime's last digit has to be 1, 3, 7, or 9; but 1 and 3 are easy, while 7 and 9 can be rewritten to 3 and 1, easy. So, multiplying by a n-digit prime is about as hard as multiplying by a generic (n−1)-digit number! That's great for two-digit primes, and not bad for small three-digit primes, and that's about as far as I've ever needed to go. For the case when both factors are prime, see the discussion of split factors in Multiplication.
• Negative digits can probably help with division as well, but I can't tell you how since I don't do much of that. However, I would like to say a few words about fractions! Just as we can rewrite the integer 857 as 1143, we can rewrite the decimal expansion of 1/7 as follows.

  1/7 = 0.142857 = 0.143143

  Isn't that better? What happens to 1/13 is even more remarkable.

  1/13 = 0.076923 = 0.077077

  If we divide that by 77, we get the fact underlying both equations.

  1/1001 = 0.000999 = 0.001001

  Other fractions with even repeat lengths often have complementary halves (first complement), but it's usually not as much fun to rewrite them. For example,

  1/17 = 0.0588235294117647 = 0.0588235305882353.

  The number 5882353 is prime, by the way.