Reference Mathematics Base

Base 12 Mathematics

Introduction

I'm not a mathematician but I do have an interest in Base 12 Mathematics.

Assuming we are using a non-zero base and the following 12 characters are the representative numeric digits;

123456789XYZ

The following are the basic sequential, addition, subtraction, multiplication and division matrices for reference.

NOTES:

  • Subtraction and Division are evaluated column by row. So 1/7 for example, would equal the column 1 divided by the row 7, referencing the cell value .186X35
  • Characters underlined represent infinite precision, so .186X35 is a shortened version of .186X35186X15186X35 ... for example.


Sequential Matrix

= 1 2 3 4 5 6 7 8 9 X Y Z
1 1 2 3 4 5 6 7 8 9 X Y Z
2 11 12 13 14 15 16 17 18 19 1X 1Y 2Z
3 21 22 23 24 25 26 27 28 29 2X 2Y 3Z
4 31 32 33 34 35 36 37 38 39 3X 3Y 4Z
5 41 42 43 44 45 46 47 48 49 4X 4Y 5Z
6 51 52 53 54 55 56 57 58 59 5X 5Y 6Z
7 61 62 63 64 65 66 67 68 69 6X 6Y 7Z
8 71 72 73 74 75 76 77 78 79 7X 7Y 8Z
9 81 82 83 84 85 86 87 88 89 8X 8Y 9Z
X 91 92 93 94 95 96 97 98 99 9X 9Y XZ
Y X1 X2 X3 X4 X5 X6 X7 X8 X9 XX XY YZ
Z Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 YX YY YZ


Addition Matrix

+ 1 2 3 4 5 6 7 8 9 X Y Z
1 2 3 4 5 6 7 8 9 X Y Z 11
2 3 4 5 6 7 8 9 X Y Z 11 12
3 4 5 6 7 8 9 X Y Z 11 12 13
4 5 6 7 8 9 X Y Z 11 12 13 14
5 6 7 8 9 X Y Z 11 12 13 14 15
6 7 8 9 X Y Z 11 12 13 14 15 16
7 8 9 X Y Z 11 12 13 14 15 16 17
8 9 X Y Z 11 12 13 14 15 16 17 18
9 X Y Z 11 12 13 14 15 16 17 18 19
X Y Z 11 12 13 14 15 16 17 18 19 1X
Y Z 11 12 13 14 15 16 17 18 19 1X 1Y
Z 11 12 13 14 15 16 17 18 19 1X 1Y 2Z


Subtraction Matrix

- 1 2 3 4 5 6 7 8 9 X Y Z
1 -1 1 2 3 4 5 6 7 8 9 X Y
2 -2 -1 1 2 3 4 5 6 7 8 9 X
3 -3 -2 -1 1 2 3 4 5 6 7 8 9
4 -4 -3 -2 -1 1 2 3 4 5 6 7 8
5 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
6 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6
7 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5
8 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4
9 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3
X -X -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2
Y -Y -X -9 -8 -7 -6 -5 -4 -3 -2 -1 1
Z -Z -Y -X -9 -8 -7 -6 -5 -4 -3 -2 -1


Multiplication Matrix

* 1 2 3 4 5 6 7 8 9 X Y Z
1 1 2 3 4 5 6 7 8 9 X Y Z
2 2 4 6 8 X Z 12 14 16 18 1X 2Z
3 3 6 9 Z 13 16 19 2Z 23 26 29 3Z
4 4 8 Z 14 18 2Z 24 28 2Z 34 38 4Z
5 5 X 13 18 21 26 2Y 34 39 42 47 5Z
6 6 Z 16 2Z 26 3Z 36 4Z 46 5Z 56 6Z
7 7 12 19 24 2Y 36 41 48 53 5X 65 7Z
8 8 14 2Z 28 26 4Z 48 54 6Z 68 74 8Z
9 9 16 23 2Z 39 46 53 6Z 69 76 83 9Z
X X 18 26 34 42 5Z 5X 68 76 84 92 XZ
Y Y 1X 29 38 47 56 65 74 83 92 X1 YZ
Z Z 2Z 3Z 4Z 5Z 6Z 7Z 8Z 9Z XZ YZ ZZ


Division Matrix

/ 1 2 3 4 5 6 7 8 9 X Y Z
1 1 2 3 4 5 6 7 8 9 X Y Z
2 .6 1 1.6 2 2.6 3 3.6 4 4.6 5 5.6 6
3 .4 .8 1 1.4 1.8 2 2.4 2.8 3 3.4 3.8 4
4 .3 .6 .9 1 1.3 1.6 1.9 2 2.3 2.6 2.9 3
5 .2497 .4972 .7249 .9724 1 1.2497 1.4972 1.7249 1.9724 2 2.2497 2.4972
6 .2 .4 .6 .8 .X 1 1.2 1.4 1.6 1.8 1.X 2
7 .186X35 .35186X .5186X3 .6X3518 .86X351 .X35186 1 1.186X35 1.35186X 1.5186X3 1.6X3518 1.86X351
8 .16 .3 .46 .6 .76 .9 .X6 1 1.16 1.3 1.46 1.6
9 .14 .28 .4 .54 .68 .8 .94 .X8 1 1.14 1.28 1.4
X .12497 .2497 .37249 .4972 .6 .7249 .84972 .9724 .X9724 1 1.12497 1.2497
Y .1 .2 .3 .4 .5 .6 .7 .8 .9 .X 1 1.1
Z .1 .2 .3 .4 .5 .6 .7 .8 .9 .X .Y 1


Calculation

I performed all the conversions for these tables manually using LibreOffice Calc and a handheld calculator. I did reference some base 12 fraction tables I did find online to verify my answers, but I made great effort in trying to identify how to get these answers by my own agency.

The addition, subtraction and multiplication tables were quite simple to figure out. It was the division table that provided a challenge.


x/5 Division Calculation

Lets take 1/5 for example. On a calculator, which uses our standard base 10 mathematics the answer we get is

1 / 5 = 0.2

TODO