# Dictionary Definition

duodecimal adj : based on twelve; "the duodecimal
number system"

# User Contributed Dictionary

- Of a number, expressed in duodecimal.

### Noun

duodecimal- A number system that uses 12 as its base.

#### Synonyms

- base 12

# Extensive Definition

The duodecimal system (also known as base-12
or dozenal) is a numeral
system using twelve as its
base. In this system, the
number ten may be
written as 'A', and the number eleven as 'B'
(another common notation, introduced by Sir Isaac
Pitman, is to use a rotated '2' for ten and a reversed '3' for
eleven). The number twelve (that is, the number written as '12' in
the base ten numerical
system) is instead written as '10' in duodecimal (meaning "1
dozen and 0 units",
instead of "1 ten and 0 units"), whereas the digit string '12'
means "1 dozen and 2 units" (i.e. the same number that in decimal
is written as '14'). Similarly, in duodecimal '100' means "1
gross",
'1000' means "1 great gross",
and '0.1' means "1 twelfth" (instead of their decimal meanings "1
hundred", "1 thousand", and "1 tenth").

The number twelve, a highly
composite, is the smallest number with four non-trivial
factors (2, 3, 4, 6), and the smallest to include as factors
all four numbers (1 to 4) within the subitizing
range. As a result of this increased factorability of the radix and
its divisibility by a wide range of the most elemental numbers
(whereas ten has only two non-trivial factors: 2 and 5, with
neither 3 nor 4), duodecimal representations fit more easily than
decimal ones into many common patterns, as evidenced by the higher
regularity observable in the duodecimal multiplication table. Of
its factors, 2 and 3 are prime, which
means the reciprocals
of all 3-smooth
numbers (such as 2, 3, 4, 6, 8, 9...) have a terminating
representation in duodecimal. In particular, the five most
elementary fractions (, , , and ), all have a short terminating
representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9,
respectively), and twelve is the smallest radix with this feature
(since it is the least
common multiple of 3 and 4). This all makes it a more
convenient number system for computing fractions than most other
number systems in common use, such as the decimal, vigesimal, binary,
octal and hexadecimal systems,
although the sexagesimal system (where
the reciprocals of all 5-smooth
numbers terminate) does better in this respect (but at the cost of
an unwieldily large multiplication table).

## Origin

Languages using duodecimal number systems are
uncommon. Languages in the Nigerian Middle
Belt such as Janji, Gbiri-Niragu
(Kahugu), the Nimbia dialect of Gwandara; the
Chepang
language of Nepal and the
Mahl
language of Minicoy
Island in India are known to
use duodecimal numerals. In fiction, J. R. R.
Tolkien's Elvish
languages used duodecimal.

Germanic
languages have special words for 11 and 12, such as eleven and
twelve in English,
which are often misinterpreted as vestiges of a duodecimal system.
However, they are considered to come from Proto-Germanic
*ainlif and *twalif (respectively one left and two left), both of
which were decimal. Admittedly, the survival of such apparently
unique terms may be connected with duodecimal tendencies, but their
origin is not duodecimal.

Historically, units
of time in many civilizations are
duodecimal. There are twelve signs of the zodiac, twelve months in a year,
and twelve European hours
in a day or night. Traditional Chinese
calendars, clocks, and compasses are based on the twelve
Earthly
Branches.

Being a versatile denominator in fractions may
explain why we have 12 inches in an imperial foot,
12 ounces in a troy pound, 12
old British pence in a shilling, 12 items in a
dozen, 12 dozens in a
gross
(144,
square
of 12), 12 gross in a great gross
(1728,
cube of
12), 24 (12 * 2) hours in a day, etc. The Romans used a fraction
system based on 12, including the uncia which became both the
English words ounce and
inch. Pre-decimalisation,
Great
Britain used a mixed duodecimal-vigesimal currency system (12
pence = 1 shilling, 20 shillings or 240 pence to the pound
sterling), and Charlemagne
established a monetary system that also had a mixed base of twelve
and twenty, the remnants of which persist in many places.

## Places

In a duodecimal place system, ten can be written as A, eleven can be written as B, and twelve is written as 10.For alternative symbols, see the section
"Advocacy and 'dozenalism'" below.

According to this notation, duodecimal 50
expresses the same quantity as decimal 60 (= five
times twelve), duodecimal 60 is equivalent to decimal 72 (= six
times twelve = half a gross), duodecimal 100 has the same value as
decimal 144 (=
twelve times twelve = one gross), etc.

## Comparison to other numeral systems

The number 12 has six factors, which are 1, 2, 3, 4, 6, and
12,
of which 2 and 3 are prime. The
decimal system has only four factors, which are 1, 2, 5, and
10;
of which 2 and 5 are prime. Vigesimal adds two factors to those of
ten, namely 4 and 20, but no
additional prime factor. Although twenty has 6 factors, 2 of them
prime, similarly to twelve, it is also a much larger base (i.e.,
the digit set and the multiplication table are much larger) and
prime factor 5, being less common in the prime factorization of
numbers, is arguably less useful than prime factor 3. Binary has
only two factors, 1 and 2, the latter being prime. Hexadecimal has
five factors, adding 4, 8 and 16 to those
of 2, but no additional prime. Trigesimal is the
smallest system that has three different prime factors (all of the
three smallest primes: 2, 3 and 5) and it has eight factors in
total (1, 2, 3, 5, 6, 10, 15, and 30). Sexagesimal --
which the ancient Sumerians and
Babylonians among
others actually used -- adds the four convenient factors 4, 12 and
20 to this but no new prime factors.

## Conversion tables to and from decimal

To convert numbers between bases, one can use the
general conversion algorithm (see the relevant section under
radix). Alternatively, one can use digit-conversion tables. The
ones provided below can be used to convert any dozenal number
between 0.01 and BBB,BBB.BB to decimal, or any decimal number
between 0.01 and 999,999.99 to dozenal. To use them, we first
decompose the given number into a sum of numbers with only one
significant digit each. For example:

123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50
+ 6 + 0.7 + 0.08

This decomposition works the same no matter what
base the number is expressed in. Just isolate each non-zero digit,
padding them with as many zeros as necessary to preserve their
respective place values. If the digits in the given number include
zeroes (for example, 102,304.05), these are, of course, left out in
the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 +
0.05). Then we use the digit conversion tables to obtain the
equivalent value in the target base for each digit. If the given
number is in dozenal and the target base is decimal, we get:

(dozenal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6
+ 0.7 + 0.08 = (decimal) 248,832 + 41,472 + 5,184 + 576 + 60 + 6 +
0.58333333333... + 0.05555555555...

Now, since the summands are already converted to
base ten, we use the usual decimal arithmetic to perform the
addition and recompose the number, arriving at the conversion
result:

Dozenal -----> Decimal 100,000 = 248,832
20,000 = 41,472 3,000 = 5,184 400 = 576 50 = 60 + 6 = + 6 0.7 =
0.58333333333... 0.08 = 0.05555555555...
-------------------------------------------- 123,456.78 =
296,130.63888888888...

That is, (dozenal) 123,456.78 equals (decimal)
296,130.63888888888... ≈ 296,130.64

If the given number is in decimal and the target
base is dozenal, the method is basically same. Using the digit
conversion tables:

(decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6
+ 0.7 + 0.08 = (dozenal) 49,A54 + B,6A8 + 1,8A0 + 294 + 42 + 6 +
0.84972497249724972497... + 0.0B62...

However, in order to do this sum and recompose
the number, we now have to use the addition tables for dozenal,
instead of the addition tables for decimal most people are already
familiar with, because the summands are now in base twelve and so
the arithmetic with them has to be in dozenal as well. In decimal,
6 + 6 equals 12, but in dozenal it equals 10; so if we used decimal
arithmetic with dozenal numbers we would arrive at an incorrect
result. Doing the arithmetic properly in dozenal, we get the
result:

Decimal -----> Dozenal 100,000 = 49,A54 20,000
= B,6A8 3,000 = 1,8A0 400 = 294 50 = 42 + 6 = + 6 0.7 =
0.84972497249724972497... 0.08 = 0.0B62...
-------------------------------------------------------- 123,456.78
= 5B,540.943A...

That is, (decimal) 123,456.78 equals (dozenal)
5B,540.943A... ≈ 5B,540.94

### Dozenal to Decimal digit conversion

### Decimal to Dozenal digit conversion

### Conversion of powers

- 1/2 = 0.6
- 1/3 = 0.4
- 1/4 = 0.3
- 1/6 = 0.2
- 1/8 = 0.16
- 1/9 = 0.14

or complicated

- 1/5 = 0.24972497... recurring (easily rounded to 0.25)
- 1/7 = 0.186A35186A35... recurring (easily rounded to 0.187)
- 1/A = 0.124972497... recurring (rounded to 0.125)
- 1/B = 0.11111... recurring (rounded to 0.11)
- 1/11 = 0.0B0B... recurring (rounded to 0.0B)

As explained in recurring
decimals, whenever an irreducible
fraction is written in “decimal” notation, in any base, the
fraction can be expressed exactly (terminates) if and only if all
the prime
factors of its denominator are also prime factors of the base.
Thus, in base-ten (= 2×5) system, fractions whose
denominators are made up solely of multiples of 2 and 5 terminate:
¹⁄8 = ¹⁄(2×2×2), ¹⁄20 =
¹⁄(2×2×5), and ¹⁄500 =
¹⁄(2×2×5×5×5) can be
expressed exactly as 0.125, 0.05, and 0.002 respectively. ¹⁄3 and
¹⁄7, however, recur (0.333... and 0.142857142857...). In the
duodecimal (= 2×2×3) system, ¹⁄8 is exact; ¹⁄20
and ¹⁄500 recur because they include 5 as a factor; ¹⁄3 is exact;
and ¹⁄7 recurs, just as it does in decimal.

Because each place is more precise in the
duodecimal system, "decimals" can be written with greater accuracy.
For example, the square root of 2 (1.4142135... in decimal,
1.4B79170A07B86... in duodecimal) can be rounded to 1.5 in
duodecimal. This number is more precise than rounding to 1.41 in
decimal.

### Recurring digits

Arguably, factors of 3 are more commonly
encountered in real-life division
problems than factors of 5 (or would be, were it not for the
decimal system having influenced most cultures). Thus, in practical
applications, the nuisance of recurring
decimals is encountered less often when duodecimal notation is
used. Advocates of duodecimal systems argue that this is
particularly true of financial calculations, in which the twelve
months of the year often enter into calculations.

However, when recurring fractions do occur in
duodecimal notation, they are less likely to have a very short
period than in decimal notation, because 12 (twelve)
is between two prime
numbers, 11 (eleven)
and 13
(thirteen), whereas ten is adjacent to composite
number 9.
Nonetheless, having a shorter or longer period doesn't help the
main inconvenience that one does not get a finite representation
for such fractions in the given base (so rounding, which introduces
inexactitude, is necessary to handle them in calculations), and
overall one is more likely to have to deal with infinite recurring
digits when fractions are expressed in decimal than in duodecimal,
because one out of every three consecutive numbers contains the
prime factor 3 in its
factorization, while only one out of every five contains the prime
factor 5. All other
prime factors, except 2, are not shared by either ten or twelve, so
they do not influence the relative likeliness of encountering
recurring digits (any irreducible fraction that contains any of
these other factors in its denominator will recur in either base).
Also, the prime factor 2 appears
twice in the factorization of twelve, while only once in the
factorization of ten; which means that most fractions whose
denominators are powers of
two will have a shorter, more convenient terminating
representation in dozenal than in decimal (e.g., 1/(22) = 0.25 dec
= 0.3 doz; 1/(23) = 0.125 dec = 0.16 doz; 1/(24) = 0.0625 dec =
0.09 doz; 1/(25) = 0.03125 dec = 0.046 doz; etc.).

### Irrational numbers

As for irrational
numbers, none of them has a finite representation in any of the
rational-based
positional number systems (such as the decimal and duodecimal
ones); this is because a rational-based positional number system is
essentially nothing but a way of expressing quantities as a sum of
fractions whose denominators are powers of the base, and by
definition no finite sum of rational numbers can ever result in an
irrational number. For example, 123.456 = 1 × 103/10 + 2 × 102/10 +
3 × 10/10 + 4 × 1/10 + 5 × 1/102 + 6 × 1/103 (this is also the
reason why fractions that contain prime factors in their
denominator not in common with those of the base do not have a
terminating representation in that base). Moreover, the infinite
series of digits of an irrational number doesn't exhibit a pattern
of recursion; instead, the different digits succeed in a seemingly
random fashion. The following chart compares the first few digits
of the decimal and duodecimal representation of several of the most
important algebraic
and transcendental
irrational numbers. Some of these numbers may be perceived as
having fortuitous patterns, making them easier to memorize, when
represented in one base or the other.

The first few digits of the decimal and dozenal
representation of another important number, the Euler-Mascheroni
constant (the status of which as a rational or irrational
number is not yet known), are:

## Advocacy and "dozenalism"

The case for the duodecimal system was put forth at length in F. Emerson Andrews' 1935 book New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized either by the adoption of ten-based weights and measure or by the adoption of the duodecimal number system.Rather than the symbols 'A' for ten and 'B' for
eleven as used in hexadecimal notation and
vigesimal notation (or
'T' and 'E' for ten and eleven), he suggested in his book and used
a script X and a script E, Image:Scriptx.png
and Image:Scripte.png,
to represent the digits ten and eleven respectively, because, at
least on a page of Roman script, these characters were distinct
from any existing letters or numerals, yet were readily available
in printers' fonts. He chose Image:Scriptx.png
for its resemblance to the Roman numeral X, and Image:Scripte.png
as the first letter of the word "eleven".

Another popular notation, introduced by Sir
Isaac
Pitman, is to use a rotated 2 to represent ten and a rotated or
horizontally flipped 3 to represent eleven. This is the convention
commonly employed by the Dozenal Society of Great Britain and has
the advantage of being easily recognizable as digits because of
their resemblance in shape to existing digits. On the other hand,
the Dozenal Society of America adopted for some years the
convention of using an asterisk * for ten and a
hash
# for eleven. The reason was the symbol * resembles a
struck-through X while # resembles a doubly-struck-through 11, and
both symbols are already present in telephone dials. However, critics pointed out
these symbols do not look anything like digits. Some other systems
write 10 as ɸ (a combination of 1 and 0) and eleven as a cross of
two lines (+, x, or † for example).

In 'Little
Twelvetoes', American television series Schoolhouse
Rock! portrayed an alien child using base-twelve arithmetic,
using 'dek', 'el', and 'doh' as names for ten, eleven, and twelve,
and Andrews' script-X and script-E for the digit symbols.

The Dozenal Society of America and the Dozenal
Society of Great Britain promote widespread adoption of the
base-twelve system. They use the word dozenal instead of
"duodecimal" because the latter comes from Latin roots that express
twelve in base-ten terminology.

The renowned mathematician and mental calculator
Alexander
Craig Aitken was an outspoken advocate of the advantages and
superiority of duodecimal over decimal:

In Leo
Frankowski's Conrad
Stargard novels, Conrad introduces a duodecimal system of
arithmetic at the suggestion of a merchant, who is accustomed to
buying and selling goods in dozens and grosses, rather than tens or
hundreds. He then invents an entire system of weights and measures
in base twelve, including a clock with twelve hours in a day
(rather than twenty-four.)

## See also

- Senary (base 6)
- Quadrovigesimal (base 24)
- Hexatridecimal (base 36)
- Sexagesimal (base 60)
- Babylonian numerals

## References

## External links

- The origin of a duodecimal system (Japanese) — explains a possible origin of a duodecimal system in a language
- Dozenal Society of America
- Dozenal Society of Great Britain website
- Online Decimal-Dozenal Calculator

duodecimal in Danish: Duodecimal

duodecimal in German: Duodezimalsystem

duodecimal in Esperanto: Dekduuma sistemo

duodecimal in Spanish: Sistema duodecimal

duodecimal in French: Système duodécimal

duodecimal in Korean: 십이진법

duodecimal in Icelandic: Tylftakerfi

duodecimal in Hebrew: בסיס דואודצימלי

duodecimal in Dutch: Twaalftallig stelsel

duodecimal in Japanese: 十二進法

duodecimal in Polish: Dwunastkowy system
liczbowy

duodecimal in Portuguese: Sistema de numeração
duodecimal

duodecimal in Russian: Двенадцатеричная система
счисления

duodecimal in Slovenian: Dvanajstiški številski
sistem

duodecimal in Finnish:
Duodesimaalijärjestelmä

duodecimal in Thai: เลขฐานสิบสอง

duodecimal in Ukrainian: Дванадцяткова система
числення