Base Six

The base not taken

Unfortunately, the world has adopted base two (for computer hardware) and base ten (for most human calculations).

Advantages of base six

Base six has advantages over BaseTwo, BaseFour, BaseEight, BaseTen, and BaseTwelve.

More common numbers have exact decimal representations in base six than in base two, four, eight, or ten.
The same numbers have exact decimal representations in base six as in base twelve, but base six requires fewer symbols and less memorization.
It is probably easier to learn to count in base six than in base ten.
Base six' times tables are probably easier to memorize than base ten's times tables.
A base six version of Avagadro's number could be very simple: 10^100 (base 5+1) is close to the number of nucleons in a gram of rest mass.

Finger hardware

Base six arithmetic can be done on one's fingers: Each hand can have between 0 and 5 digits up, representing numbers between 00 (base 5+1) and 55 (base 5+1). This is not as efficient as using roman numerals, which can represent numbers between 00 (base 9+1) and 99 (base 9+1). Both are more efficent than most people's finger math, which can only represent numbers between 00 and 10 (base 9+1).

Computer hardware

Hardware could be built with native base six support, but it might require (at the transistor or gate level) a combination of two-state and three-state logic.

A times table in base six:


        X |  0   1   2   3   4   5  10    11  12  13  14  15  20

        ---+---------------------------    ----------------------

| 0 0 0 0 0 0 0 0 0 0 0 0 0
| 0 1 2 3 4 5 10 11 12 13 14 15 20
| 0 2 4 10 12 14 20 22 24 30 32 34 40
| 0 3 10 13 20 23 30 33 40 43 50 53 100
| 0 4 12 20 24 32 40 44 52 100 104 112 120
| 0 5 14 23 32 41 50 55 104 113 122 131 140
1. | 0 10 20 30 40 50 100 110 120 130 140 150 200

| 0 11 22 33 44 55 110 121 132 143 154 205 220
| 0 12 24 40 52 104 120 132 144 200 212 224 240
| 0 13 30 43 100 113 130 143 200 213 230 243 300
| 0 14 32 50 104 122 140 154 212 230 244 302 320
| 0 15 34 53 112 131 150 205 224 243 302 321 340
| 0 20 40 100 120 140 200 220 240 300 320 340 400

Using base six, there are 441 cells.
Note that the 0x, 1x, 10x, and parts of the 11x rows and columns are trivial (= 203 cells, leaving 234 harder cells).
The 2x, 3x, and parts of the 5x rows and columns are easy. This leaves 1 harder cell in the first 100, 24 harder cells in the next 200, and 41 harder cells in the last 100 cells.
This means that only 1/4 of the cells are hard to memorize.
Exercise for the reader: Is the same size base 9+1 times table easier or harder to memorize?

Discussion:

You can argue the same - more or less - about any number system.

Base 8 for example has the advantage of being trivial to convert to binary and still smaller than ten (and much more uniform in the multiplication table as for all powers of 2).
Base 10 has the advantage of being composed of 2 and 5 both important numbers (5 in particular being the number of digits on ones hand so counting multiples of 5 is intuitive, even my 5 year old has picked that up without being told).
Base 2 has the advantage that you can count up to 1023 with your fingers (each finger one bit). Needs some experience but I do this when I want to count something while doing something else - no danger of forgetting where you were. Naming number would presumably need some grouping e.g. of 5 bits together - that's only 32 names. I'd propose 5-sylabic words composed of i and o, e.g. Nonologono für 0 and Wonofogoni for 1 - but that's just and idea made up on the spot.
- I think that has a high risk of flipping someone off at number four...

I don't think that the size and simplicity of the multiplication table are key features of a system. In the end all your candidate systems have multiplication tables that are fairly easy to remember compared to the base 60 system of the babylonians (57*56/2 non-trivial entries). I don't think it is necessary to make it simpler.

And you don't just memorize the entries. The structure and interrelation of the matrix entries tell you something about the numbers (number theoretical properties). I hope children don't just do rote memoization of the table but discover and learn the patterns behind the numbers. At least I did. From this point of view any system with a base composed of more than one prime might be advantageous as it can teach you more number theoretical properties while learning the multiplication tables.

And I didn't even start to argue about the advantages of staying with one system - there have been failed tries...

-- GunnarZarncke

As far as human-friendly, I propose base 12. Let's look at base divisors:


        Base 6:  2, 3, 6

        Base 8:  2, 4, 8

        Base 10: 2, 5

        Base 12: 2, 3, 4, 6

        Base 16: 2, 6, 8

        Base 30: 2, 3, 5, 6

        Base 60: 2, 3, 4, 5, 6

I give higher weights to the lower divisors such that being divisible by 3 is more important than being divisible by 8. 60 would be the hands-down favorite if it wasn't so large. 12 is the best compromise.

CategoryMath