## Sunday, January 29, 2017

### Base 12 Arithmetic: The Dozenal Society

Base 12 Arithmetic: The Dozenal Society

Introduction

Base 12 is a numerical base system that is based on the powers of 12 instead of 10.  The number line would like:

 … 12^3 (1728) 12^2 (144) 12^2 (12) Dozens 12^1 (1) ; 12^-1 (1/12) 12^-2 (1/144) 12^-3 (1/1728) …

For example, the number 49_12 in base 12 would read 4 dozen, 9 singles.  Its decimal equivalent is 4*12 + 9 = 57.

The number 24;53_12 in base 12 would be equivalent to 2 * 12 + 4 + 5/12 + 3/144 = 455/16 = 28.4375 in decimal.  Note that the semicolon is used as the fractional point.

Converting from base 10 to base 12 involves dividing the number by the powers of 12.  For example, 188 in decimal,

188/144 = 1 R11
188 – 1 * 144 = 44
44/12 = 3 R8

The dozenal equivalent is 138_12  (1 * 144 + 3 * 12 + 8).

There are a number of advocates for replacing Base 10 arithmetic with Base 12 arithmetic.  Two of the prominent groups are:

The Dozenal Society of America (Website: http://www.dozenal.org/   )

The Dozenal Society of Great Britain (Website:  http://www.dozenalsociety.org.uk/index.html )

I also found a number of videos on YouTube who extols the virtues of Base 12, including:

So why base 12 instead of base 10?  The Dozenal Society of America argues that the dozenal way of counting was arrived by societies worldwide:

*  Bakers sold baked goods in twelves
*  Rulers (non-metric) are usually a foot long divided into 12 sections (inches) – used by carpenters
* Historically, pharmacists and jewelers divided a pound into 12 ounces (the pound in this sense is called an Apothecary pound)

I will add that we have 12 months in a year, non-military clocks have work in 12 hours (with an AM/PM indicator), and in most astrological practices where Ophiuchus the Serpent Bearer isn’t included, there are 12 signs of the zodiac.

The factors of 10 are 1, 2, 5, and 10.  The factors of 12 are 1, 2, 3, 4, 6, and 12.  Base 12 advocates argue that the increase in number of factors would make more fractions easier to represent in dozenal than in decimal.  Examples include 0;2 for 1/6, 0;3 for 1/4, and 0;4 for 1/3.

Base 12 would also facilitate counting, instead of using two hands and their fingers, people would count using the twelve phalanges of a single hand, where the thumb is used as a counter.   One hand can represent singles while the other represents dozens.

Symbols

Thankfully, the symbols for digits 0 through 9 are retained. Unlike hexadecimal representation (Base 16), there is no uniform consensus of how ten and eleven are represented in Base 12.  Symbols include:

*  A for 10 and B for 11.  These symbols would align with the hexadecimal representation.

*  An upside down 2 for 10 and upside down 3 (Ɛ ) for 11.  In Microsoft Alt codes, the   Ɛ is associated hexcode 0190.  These symbols are popular in Britain.  No offense, but personally, these are not my favorite as look confusing and too close to the 2 and 3 we have.

*  One suggestion is to use T for 10 and E for 11.  In this video Bon Crowder explains the digits of Base 12:   https://www.youtube.com/watch?v=BJRYCwl5Rgw

* Another set of symbols are called dek (χ ) for 10 and el ( Ɛ except the bottom line is flat) for 11.  These two symbols have been suggested by William A. Dwiggins, and is used in The Dozenal Society of America’s publications.  For typing on the computer, the capital letter X stands for dek and capital letter E for el.

For the purposes of this blog and the tables presented, I will use X for 10 and E for 11.

Let’s compare how the basic operations addition and multiplication work in both decimal and dozenal.

Adding:  Base 10 vs. Base 12

Below is an adding table (jpeg image) for numbers 1 through 12 represented in both Base 10 and Base 12.

Multiplying:  Base 10 vs. Base 12

Below is a multiplication table (jpeg image) for numbers 1 through 12 represented in both Base 10 and Base 12.

Fractions: Base 10 versus Base 12

Here are some common fractions listed below in both decimal and dozenal systems.  Note the interesting patterns for 0.1 to 0.9 (the table to the right).

Dozenal Representation of Pi

The numerical constant π to 30 places in base 12 is:

π = 3.18480 9493E 91866 4573X 6211E E1515 51X05…

You can find representations of π to 100 places here:  http://turner.faculty.swau.edu/mathematics/materialslibrary/pi/pibases.html

Conclusion

What do you think, would a dozenal, base 12 system work for you?  Should it replace base 10 in everyday mathematics?  Personally I find value in both systems and the ability to quickly double and halve is partially base 10 has survived as the dominant base system for centuries.

In the upcoming week, I work on a program to convert to and from base 10 to 12 integers.  I am thinking about either keep the X (dek) and E (el) or using A for 10 and B for 11.

Eddie

Sources

Schiffman, Jay.  “Fundamental Operations in the Duodecimal System”.  The Dozenal Society of America.  1982 (1192_12
Retrieved January 27, 2017

Zirkel, Gene. “A Brief Introduction to Dozenal Counting”.  The Dozenal Society of America.  1995 (11X3_12
Retrieved January 27, 2017

This blog is property of Edward Shore, 2017  (1201_12)