New improved number system!

Numbers have been represented using many different notations over the years. We use what is called the "Hindu-Arabic" system, but there have been many others in the past. There are systems developed by the Romans, Babylonians, Mayans, Egyptians, and many others.

It's a part of the standard elementary school syllabus here in New South Wales to teach about different number systems. This is a fun topic for including a bit of history in a class; and thinking outside the box, and helping students distinguish between the abstract concept of a number and the symbols we use to denote it.

One common feature of such lessons is to show the advantages of "our" Hindu-Arabic system over all those others, especially for manual calculations. But wouldn't it be fun if we could extend the lesson with a number system that was better than what we are using now?

Wait no longer. I now reveal for the world a new, improved number system!

The idea is simple. It is a minor modification to the conventional decimal system. It is still in base 10, so conversion between old fashioned notation and the new improved notation is easy.

Instead of using digits from 0 to 9, the new system uses digits from -4 to 5.

The symbols adopted for each digit are not all that important. One approach would be to add a sign designation of some kind to the digit, such as a strike through line. In this post I'll go with something easy to type, and to remember. Digits 0 to 5 are represented as normal. The four negative digits are represented using the first four letters of the Greek alphabet.

Negative digits	α	β	γ	δ
Value	-1	-2	-3	-4

A sequence of digits denotes a number just as in the Hindu-Arabic system. There is the ones place, the tens place, and so on. For example, 15 is unchanged in this new system; but 16 becomes 2δ in the new. This is 2 tens, and -4 ones, and so denotes the same number.

The speed of light is 299792468 m/s in old notation. In the new notation, it is 300βα25γβ m/s.

Why would anyone do this? It takes time to get used to a new number system, but if you grew up with it, you'd see a number of significant advantages that make it arguably superior to the system we are currently using.

Adding up large columns is easier

When you sum a column of n digits, randomly selected, the old system has a distribution of results with mean 4.5n and standard deviation around about 3*sqrt(n). In the new system, the standard deviation is unchanged, but the sum has mean 0.5n; about 9 times smaller. For a column of ten digits, the sum is, on average, around 5, with a standard deviation of around 10. You often don't carry anything at all; when you do carry any overflow to the next column, the amount carried is small.

Memorizing the times tables is easier

You now only need to remember up to your 5 times table. Multiplying digits of opposite sign just means reversing the sign of all digits in the answer. That's not quite true; but it will be true if we add the digit ε for -5. With ε, representations are no longer unique, but removing such a digit is easy. Just replace with 5, and carry α (that is, -1) to the next digit. Eg: 2ε becomes 15, βε becomes γ5.

Here is a complete multiplication table.

	ε	δ	γ	β	α	0	1	2	3	4	5
ε	25	20	15	10	5	0	α5	α0	β5	β0	γ5
δ	20	2δ	12	1β	4	0	δ	α2	αβ	β4	β0
γ	15	12	1α	1δ	3	0	γ	α4	α1	αβ	β5
β	10	1β	1δ	4	2	0	β	δ	α4	α2	α0
α	5	4	3	2	1	0	α	β	γ	δ	ε
0	0	0	0	0	0	0	0	0	0	0	0
1	α5	δ	γ	β	α	0	1	2	3	4	5
2	α0	α2	α4	δ	β	0	2	4	1δ	1β	10
3	β5	αβ	α1	α4	γ	0	3	1δ	1α	12	15
4	β0	β4	αβ	α2	δ	0	4	1β	12	2δ	20
5	γ5	γ0	β5	β0	α5	0	5	10	15	20	25

As a persuasive demonstration of the power of this system, we can calculate 257*473. In new notation, this is 3δγ*5γ3. Here are the multiplication tableaux, side by side:

   3δγ        257
   5γ3        473
   ---        ---
  1βγ1        771
 α23α       1799
13β5       1028
------     ------   
122δδ1  =  121561

In the first tableau, there is only one instance in which the sum requires a digit to be carried; in the second tableau you must carry three times; and one of those requires a two to be carried. This is typical of such calculations.

Long division is more robust

As with multiplication, the simplified times tables makes long division easier to carry through. If we are willing to be a bit flexible about digits, there are further advantages.

Have you ever been working through a long division problem, and made the wrong guess for the next digit? For example, consider division of 2930872 by 7364. In the current cumbersome system, you know that the answer is close to 400, so you have to guess the first digit as either 3 or 4. If you pick 4, then you'll need to back up the calculation, as the answer is actually 398.

In this new notation, getting the wrong digit by one is not a big problem, as long as you are willing to extend your repertoire to include some additional digits, like 6 and 7, or ε and ζ. This also means that you can make much cruder guesses for each digit in the division tableau, confident that recovery will be easy.

For example, here is a long division tableau for 121561 divided by 473. The answer, by the way, is 257, which in new notation makes finding the first digit a bit harder.

         3δγ
    --------
5γ3 ) 122δδ1
      142α
      ----
       β0γδ
       β11β
       ----
        αδβ1
        αδβ1
        ----
           0

But suppose that you got that first digit wrong, and proposed a 2 instead. It is a perfectly good strategy in this notation to ignore all but the most significant digit, and in this case we can guess a 2 because 5 goes into 12 about twice.

         2
    --------
5γ3 ) 122δδ1
      1α5δ
      ----
       3γ0δ

At this point, you can guess a 5 or a 6 for the next digit, because 3γ is 27, and 5 goes into 27 a bit over 5 times. We'll go with 6, because the next digit in the divisor 5γ3 is negative, so overestimates are sensible. Note that we are using a "non-standard" digit here.

         26
    --------
5γ3 ) 122δδ1
      1α5δ
      ----
       3γ0δ
       3β4β
       ----
        αδα1

Finally, 5 goes into αδ about γ times. (5 goes into -13 about -3 times).

         26γ
    --------
5γ3 ) 122δδ1
      1α5δ
      ----
       3γ0δ
       3β4β
       ----
        αδβ1
        αδβ1
        ----
           0

Of course, we need to "normalize" the result, by converting all non-standard digits, but this is a simple operation. The 6 is replaced by 1δ, and the 1 is carried over the next place, so the final result is, as before, 3δγ. This is 257, in positive digits.

This example gives the game away somewhat. We can regard the "new" system as allowing for alternative ways to write numbers in our existing system, by introducing some non-standard digits.

Negative numbers are easy

Negative numbers are nothing special in this system. They just use a negative valued digit in the most significant place.

Rounding is easier

If you take a fixed number of digits from the start of a number, you are effectively rounding. Actually, this is not quite true, and a deeper consideration of this issue suggests that the new system is actually more difficult for rounding to a nearest digit. So I'll gloss over the point and skip quickly to the conclusion.

Conclusion

I don't actually expect this new system to sweep the world. With some practice, it does help with certain kinds of manual calculations. But is it really worth the pain of learning a new system?

The real utility of the system is that it may help students think outside the box a bit with respect to number systems. It might be useful to have students imagine being aliens from a planet where negative digits are used from earliest childhood, and then try to think through what kinds of consequences might follow in manipulating numbers. Unfortunately, long division seems to be a lost art, so the force of this demonstration is somewhat diluted. The multiplication tableaux remain an intriguing exercise.

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Update. This post is part of the carnival of mathematics. Check out what other math bloggers are doing at Carnival of Mathematics Edition #7, hosted by nonoscience.

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Update, June 3, 18:05. This system is probably one that has occurred to many people. The earliest published reference I have found is Lehmer, D.N. (1903) in Note on Negative Digits, Science, Vol. 17, No. 430. (Mar. 27, 1903), p. 514. (Online at JSTOR). It would not surprise me to find older references still.