Balanced Duodecimal

This is a long-winded, but hopefully sufficient explanation for a Sporcle quiz on a constructed number system.

Bases

First, let us understand different bases of numbers. Our system is what is known as a base 10 positional number system. This means that we organize the system into powers of 10. E.g. the number 2743.81 can be written as

2•(1000) + 7•(100) + 4•(10) + 3•(1) + 8•(0.1) + 1•(0.01)

But since this is a series of numbers multiplied by powers of ten, it can be expressed as follows:

2•(10³) + 7•(10²) + 4•(10¹) + 3•(10⁰) + 8•(10⁻¹) + 1•(10⁻²)

In other words, we have a separate symbol for every number up to 9, and beyond that, we begin grouping them in tens. When we have ten or more clusters of tens, we start grouping them into hundreds, etc. The number system is called "base 10" or "decimal" because we group the numbers as multiples of powers of 10. It is called a "positional" system because the size of the grouping is indicated by the position of a digit (each space to the left means a higher power of 10).

But there are other sorts of number systems that are possible. The choice of base 10 is common because people often counted on their fingers, and most of us have ten of those. Another common base to use is 20, as is seen in Mayan, Eskimo-Aleut, and Celtic number systems, among others (perhaps these people counted on their toes as well). Naturally, when using a smaller base, you don't need as many symbols, and when using a larger one, you need to use more. Here is a sampling of different base number systems for comparison. For bases greater than 10, we will use letters of the alphabet as digits, starting at 10: A=10, B=11, C=12, etc.

Base 2 (binary)	Base 5 (heptal)	Base 10 (decimal)	Base 12 (duodecimal)	Base 16 (hexadecimal)
1	1	1	1	1
10	2	2	2	2
11	3	3	3	3
100	4	4	4	4
101	10	5	5	5
110	11	6	6	6
111	12	7	7	7
1000	13	8	8	8
1001	14	9	9	9
1010	20	10	A	A
1011	21	11	B	B
1100	22	12	10	C
1101	23	13	11	D
1110	24	14	12	E
1111	30	15	13	F
10000	31	16	14	10
10001	32	17	15	11
10010	33	18	16	12
10011	34	19	17	13
10100	40	20	18	14
10101	41	21	19	15
10110	42	22	1A	16
10111	43	23	1B	17
11000	44	24	20	18
11001	100	25	21	19
11010	101	26	22	1A
11011	102	27	23	1B
11100	103	28	24	1C
11101	104	29	25	1D
11110	110	30	26	1E
11111	111	31	27	1F
100000	112	32	28	20
100001	113	33	29	21

So, for example, in base 5, things are grouped in powers of 5. The base-5 number written as 231.4 would be equal to the following in base-10:

2•(5²) + 3•(5¹) + 1•(5⁰) + 4•(5^-1)

or: 2•(25) + 3•(5) + 1•(1) + 4•(1/5) = 66.8

A sensible question might be whether 10 is the best choice of base for a number system. One side-effect of this number system is that we are often inclined to group things in nice round numbers, such as groups of 10, 20, 100, etc. But if we group things in this way, we might run into problems trying to divide them into smaller groups. E.g. if you buy 10 items, it is easy to divide them among 2 or 5 people, but not 3 or 4. While it's common to want to divide things into groups of 2, it's not terribly common to want to divide them into groups of 5.

By comparison, a group of 12 can be easily divided into groups of 2, 3, 4, or 6. So perhaps base 12 would make a better system than base 10 for many purposes. The Babylonians took this even further, using a base 60 positional number system so that natural groupings could be divided by any integer up to 6. Remnants of this system can be seen in the 12 positions of a clock face or the 360° of a circle (which, by a lucky coincidence, is quite close to the 365 days of the year, and with a little correction lent itself nicely to calendar calculations).


"Balanced" numeral systems: (negative digits)

Thus far, this should explain the "duodecimal" or "base 12" part of the number system used in the quiz linked above. So what's all this business about negative digits? To understand the motivation, note that a side-effect of the positional systems above is that the first digit (from the left) is not always the best possible approximation of the number. American vendors often take advantage of this when setting prices. E.g. $19.99 is a lot closer to $20 than it is to $10, but the psychological tactic they employ is to make the leftmost digit as low as possible to make the customer feel like they are getting a lower price. Pretty much everyone knows that's what the vendors are doing, but they persist in doing it anyway, and it seems successful every time one hears someone estimate a price by truncating the lower digits instead of rounding them properly. One way around this is to use digits that represent negative numbers. The table below shows how counting in this "balanced" system works. Each digit of the latter system will be enclosed in parentheses for clarity. Keep in mind that the latter is grouped in clusters of 12:

Base 10 (Decimal)	Balanced Duodecimal:	New Numeral	I.e.:
-6	(-6)		-6•1
-5	(-5)		-5•1
-4	(-4)		-4•1
-3	(-3)		-3•1
-2	(-2)		-2•1
-1	(-1)		-1•1
0	(0)		0•1
1	(1)		1•1
2	(2)		2•1
3	(3)		3•1
4	(4)		4•1
5	(5)		5•1
6	(6)		6•1
7	(1)(-5)		1•12 + -5•1
8	(1)(-4)		1•12 + -4•1
9	(1)(-3)		1•12 + -3•1
10	(1)(-2)		1•12 + -2•1
11	(1)(-1)		1•12 + -1•1
12	(1)(0)		1•12 + 0•1
13	(1)(1)		1•12 + 1•1
14	(1)(2)		1•12 + 2•1
15	(1)(3)		1•12 + 3•1
16	(1)(4)		1•12 + 4•1
17	(1)(5)		1•12 + 5•1
18	(1)(6)		1•12 + 6•1
19	(2)(-5)		2•12 + -5•1
20	(2)(-4)		2•12 + -4•1
21	(2)(-3)		2•12 + -3•1
22	(2)(-2)		2•12 + -2•1
23	(2)(-1)		2•12 + -1•1
24	(2)(0)		2•12 + 0•1
25	(2)(1)		2•12 + 1•1
... ... ...	... ... ...	... ... ...	... ... ...
70	(6)(-2)		7•12 + -2•1
71	(6)(-1)		7•12 + -1•1
72	(6)(0)		7•12 + 0•1
73	(1)(-6)(1)		1•144 + -6•12 + 1•1
74	(1)(-6)(2)		1•144 + -6•12 + 2•1
75	(1)(-6)(3)		1•144 + -6•12 + 3•1
76	(1)(-6)(4)		1•144 + -6•12 + 4•1
77	(1)(-6)(5)		1•144 + -6•12 + 5•1
78	(1)(-6)(6)		1•144 + -6•12 + 6•1
79	(1)(-5)(-5)		1•144 + -5•12 + -5•1
80	(1)(-5)(-4)		1•144 + -5•12 + -4•1
81	(1)(-5)(-3)		1•144 + -5•12 + -3•1
... ... ...	... ... ...	... ... ...	... ... ...
142	(1)(0)(-2)		1•144 + 0•12 + -2•1
143	(1)(0)(-1)		1•144 + 0•12 + -1•1
144	(1)(0)(0)		1•144 + 0•12 + 0•1
145	(1)(0)(1)		1•144 + 0•12 + 1•1

There are a couple of peculiarities about this system. Truly "balanced" systems really only function with an odd-numbered base. With an even base, there will always be an ambiguous way of expressing some numbers. E.g. We could write the decimal number 6 as either (6) or (1)(-6). In the system used here, the standard I chose is that the number "rolls over" to the next place value whenever it is closer than the halfway point than to 0. So counting upwards, we roll over to (1)(-5) after (6). And we add in the 144s place after 72, as seen in the above table. There are alternative ways that could have been chosen, but this is the most nearly "balanced" the system could be.

This type of number system was first outlined to me by a former neighbor of mine who wanted to create his own number system on this model. I only met him once, after he no longer lived across the street from me. He was visiting his former roommates, and I was introduced to him. All of the people that I knew him through have since moved, and I do not know his last name or how to track him down to give him proper credit, but the quiz was ultimately based on his idea. He used overbars to indicate negative digits, but said he wanted to replace them with his own symbols that had some sort of inherent meaning.

The actual symbols used here were my own, and were loosely inspired in their form by the Kaktovik Iñupiaq numerals, but are really independent of them in practice. I wanted them to have some indication of the number built into them with a sub-grouping of 3 for convenience of writing. I wanted the system to work such that inverting each digit would make a number negative. The zero digit was chosen as a combination of (1) and (-1), and to resemble the already familiar Indo-Arabic numeral for 0.