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).
Base 12 Advocates
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:
Numberphile: https://www.youtube.com/watch?v=U6xJfP7-HCc
ParchitaFM: https://www.youtube.com/watch?v=snz-81omkFQ
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)
http://www.dozenal.org/drupal/sites_bck/default/files/db31315_0.pdf
Retrieved January
27, 2017
Zirkel, Gene. “A Brief Introduction to Dozenal Counting”. The Dozenal Society of America. 1995 (11X3_12)
http://www.dozenal.org/drupal/sites_bck/default/files/db38206_0.pdf
Retrieved
January 27, 2017
This blog is
property of Edward Shore, 2017 (1201_12)