The
duodecimal system (also known as
base-12 or
dozenal) is a
numeral systemA numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
using
twelve12 is the natural number following 11 and preceding 13.The word "twelve" is a native English word that presumably arises from the Germanic compound twa-lif "two-leave", meaning that two is...
as its
baseIn arithmetic, the radix or base refers to the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b...
. In this system, the number
ten10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
may be written as 'A' or 'X', and the number
eleven11 is the natural number following 10 and preceding 12. It is the first number which cannot be represented by a human counting his or her eight fingers and two thumbs additively...
as 'B' or 'E' (another common notation, introduced by Sir
Isaac PitmanSir Isaac Pitman , knighted in 1894, developed the most widely used system of shorthand, known now as Pitman shorthand. He first proposed this in Stenographic Soundhand in 1837. Pitman was a qualified teacher and taught at a private school he founded in Wotton-under-Edge...
, 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 tenThe decimal numeral system has ten as its base. It is the most widely used numeral base.- Decimal notation :...
numerical system) is instead written as '10' in duodecimal (meaning "1
dozenDozen is another word for the number twelve. The dozen may be one of the earliest primitive groupings, perhaps because there are approximately a dozen cycles of the moon or months in a cycle of the sun or year. The dozen is convenient because its multiples and divisors are convenient: 12 = 2...
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
grossA gross is equal to a dozen dozen, i.e. 12 × 12 = 144.It can be used in duodecimal counting. The use of gross likely originated from the fact that 144 can be counted on the fingers using the fingertips and first two joints of each finger when marked by the thumb of one hand. The other hand...
", '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 numberA highly composite number is a positive integer with more divisors than any smaller positive integer....
, is the smallest number with four non-trivial
factorIn number theory, integer factorization or prime factorization is the breaking down of a composite number into smaller non-trivial divisors, which when multiplied together equal the original integer....
s (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the
subitizingSubitizing, coined in 1949 by E.L. Kaufman et al. refers to the rapid, accurate, and confident judgments of number performed for small numbers of items. The term is derived from the Latin adjective subitus and captures a feeling of immediately knowing how many items lie within the visual scene,...
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
primeIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
, which means the
reciprocalsIn mathematics, a multiplicative inverse or reciprocal for a number x, denoted by
1⁄
x or x
−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of x is also called the reciprocal of x...
of all
3-smoothIn number theory, a positive integer is called B-smooth if none of its prime factors are greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth since none of its prime factors are greater than 5...
numbers (such as 2, 3, 4, 6, 8, 9...) have a terminating representation in duodecimal. In particular, the five most elementary fractions , 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 multipleIn arithmetic and number theory, the least common multiple or lowest common multiple or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b. Since it is a multiple, it can be divided by a and b without a remainder...
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
decimalThe decimal numeral system has ten as its base. It is the most widely used numeral base.- Decimal notation :...
,
vigesimalThe vigesimal or base- numeral system is based on twenty .- Places :...
,
binaryThe binary numeral system, or base-2 number system represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
,
octalThe octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. Numerals can be made from binary numerals by grouping consecutive binary digits into groups of three...
and
hexadecimalIn mathematics and computer science, hexadecimal is a numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen.Its primary use is as a...
systems, although the
sexagesimalSexagesimal is a numeral system with sixty as the base. It originated with the ancient Sumerians in the 2000s BCE, was transmitted to the Babylonians, and is still used—in modified form—for measuring time, angles, and geographic coordinates....
system (where the reciprocals of all
5-smoothThe numbers that evenly divide the powers of 60 arise in several areas of mathematics and its applications, and have different names coming from these different areas of study...
numbers terminate) does better in this respect (but at the cost of an unwieldily large multiplication table).
Origin
- In this section, numerals are based on decimal places
Digit may refer to:* Digit , one of several most distal parts of a limb* Phone number, slang as digit, as in "Let me get your digits so I can call you tonight."* Numerical digit, as used in mathematics or computer science...
. For example, 10 means ten10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
, 12 means twelve12 is the natural number following 11 and preceding 13.The word "twelve" is a native English word that presumably arises from the Germanic compound twa-lif "two-leave", meaning that two is...
.
Languages using duodecimal number systems are uncommon. Languages in the
NigeriaNigeria , officially the Federal Republic of Nigeria, is a federal constitutional republic comprising thirty-six states and one Federal Capital Territory. The country is located in West Africa and shares land borders with the Republic of Benin in the west, Chad and Cameroon in the east, and Niger...
n Middle Belt such as Janji, Gbiri-Niragu (Kahugu), the Nimbia dialect of Gwandara; the
ChepangChepang is the commonly used name given to an indigenous ethnic group living in central and southern Nepal.The language is also known as Chepang but is called Chyo-bang by the people themselves.Some Bahun Chettri castes call these people the "Praja" meaning "political subjects"...
language of
NepalNepal , officially the Federal Democratic Republic of Nepal, is a landlocked country in South Asia and the world's youngest republic. It is bordered to the north by the People's Republic of China, and to the south, east, and west by the Republic of India...
and the
Mahl languageMahl or Mahal, also known locally as Maliku Bas, is an Indo-Aryan language spoken by the people of Minicoy Island , in the Union Territory of Lakshadweep, India. It is a variant of Dhivehi, the official language of Maldives....
of Minicoy Island in
IndiaIndia, officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the south, the Arabian Sea on the west, and the Bay of Bengal...
are known to use duodecimal numerals. In fiction,
J. R. R. TolkienJohn Ronald Reuel Tolkien, CBE was an English writer, poet, philologist, and university professor, best known as the author of the classic high fantasy works The Hobbit, The Lord of the Rings and The Silmarillion.Tolkien was Rawlinson and Bosworth Professor of Anglo-Saxon at Oxford from...
's
Elvish languagesJ. R. R. Tolkien's fantasy fiction contains several languages for Elves. The author, a philologist by profession, spent much time on these constructed languages. His interest was primarily philological, and the languages were the first thing Tolkien created for his secondary world...
used duodecimal.
Germanic languagesThe Germanic languages are a group of related languages that constitute a branch of the Indo-European language family. The common ancestor of all the languages in this branch is Proto-Germanic, spoken in approximately the mid-1st millennium BC in Iron Age northern Europe...
have special words for 11 and 12, such as
eleven and
twelve in
EnglishEnglish is a West Germanic language that developed in England during the Anglo-Saxon era. As a result of the military, economic, scientific, political, and cultural influence of the British Empire during the 18th, 19th, and early 20th centuries, and of the United States since the mid 20th century,...
, 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
timeTime is a component of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify the motions of objects...
in many
civilizationA civilization is a complex society or culture group characterized by dependence upon agriculture, long-distance trade, state form of government, occupational specialization, population, and class stratification.-Definition:...
s are duodecimal. There are twelve signs of the
zodiacIn astronomy, the zodiac is the ring of constellations that lines the ecliptic, which is the apparent path of the Sun across the sky over the course of the year. The Moon and planets also lie within the ecliptic, and so are also within the constellations of the zodiac. In astrology, the zodiac...
, twelve months in a year, and twelve European
hourThe hour is a unit of time. It is not an SI unit but is accepted for use with the SI.-Definition:In modern usage, an hour is a unit of measurement of time of the duration of 60 minutes, or 3600 seconds...
s in a day or night. Traditional
Chinese calendarThe Chinese calendar is lunisolar, incorporating elements of a lunar calendar with those of a solar calendar. It is not exclusive to China, but followed by many other Asian cultures. It is often referred to as the Chinese calendar because it was first perfected by the Chinese around 500 BCE...
s, clocks, and compasses are based on the twelve
Earthly BranchesThe Earthly Branches provide one Chinese system for reckoning time.This system was built from observations of the orbit of Jupiter. Chinese astronomers divided the celestial circle into 12 sections to follow the orbit of Suìxīng . Astronomers rounded the orbit of Suixing to 12 years...
.
Being a versatile denominator in fractions may explain why we have 12
inchAn inch is the name of a unit of length in a number of different systems, including Imperial units, and United States customary units. There are 36 inches in a yard and 12 inches in a foot...
es in an imperial foot, 12 ounces in a
troyTroy weight is a system of units of mass customarily used for precious metals, black powder, and gemstones.Named after Troyes, France, the troy system of weights was known to exist in medieval times, at the celebrated fair at Troyes in North Eastern France...
poundThe pound or pound-mass is a unit of mass used in the imperial, United States customary and other systems of measurement...
, 12
old British penceThe penny, originally a coin of 1.3 to 1.5 g pure silver, was introduced around the year 785 by King Offa of Mercia. Coins of the same value were in circulation continuously until decimalisation in 1971, at which time a new penny was introduced worth 2.4 times the value of the old coin...
in a
shillingThe shilling is a unit of currency used in current and former English Commonwealth countries and still used in countries which have become republics, such as Kenya. The word shilling comes from schilling, an accounting term that dates back to Anglo-Saxon times where it was deemed to be the value of...
, 12 items in a
dozenDozen is another word for the number twelve. The dozen may be one of the earliest primitive groupings, perhaps because there are approximately a dozen cycles of the moon or months in a cycle of the sun or year. The dozen is convenient because its multiples and divisors are convenient: 12 = 2...
, 12 dozens in a
grossA gross is equal to a dozen dozen, i.e. 12 × 12 = 144.It can be used in duodecimal counting. The use of gross likely originated from the fact that 144 can be counted on the fingers using the fingertips and first two joints of each finger when marked by the thumb of one hand. The other hand...
(
144144 is the natural number following 143 and preceding 145. 144 is a dozen dozens, or one gross.-In mathematics:It is the twelfth Fibonacci number, and the largest one to also be a square, as the square of 12 , following 89 and preceding 233.144 is the smallest number with exactly 15 divisors.144 is...
,
squareIn mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3. Square...
of 12), 12 gross in a great gross (
17281728 is the natural number following 1727 and preceding 1729. 1728 is a dozen gross, or one great gross.-In mathematics:It is the cube of 12 and, as such, is important in the duodecimal number system. 1728 occurs in the algebraic formula for the j-invariant of an elliptic curve...
,
cubeIn arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself three times:This is also the volume formula for a geometric cube with sides of length n, giving rise to the name. The inverse operation of finding a number whose cube is n is called...
of 12), 24 (12 * 2) hours in a day, etc. The Romans used a fraction system based on 12, including the
unciaUncia can refer to:* Uncia , an ancient Roman bronze coin* Uncia , an ancient Roman unit of length* Uncia , a Bolivian tin mine...
which became both the English words
ounceThe ounce is a unit of mass with several definitions, the most commonly used of which are equal to approximately 30 grams. The ounce is used in a number of different systems, including various systems of mass that form part of the imperial and United States customary systems...
and
inch. Pre-
decimalisationDecimal Day was the day the United Kingdom and Ireland decimalised their currencies.-The old system:Under the old currency of pounds, shillings and pence, the pound was made up of 240 pence , with twelve pence in a shilling and 20 shillings in a...
, the
United KingdomThe United Kingdom of Great Britain and Northern Ireland is a sovereign state located off the northwestern coast of continental Europe. It is an island country, spanning an archipelago including Great Britain, the northeastern part of Ireland, and many small islands...
and
Republic of IrelandIreland is a country in north-western Europe. The modern sovereign state occupies about five-sixths of the island of Ireland, which was partitioned on 3 May 1921. It is a parliamentary democracy and a republic...
used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the
pound sterlingThe pound sterling , often simply called the pound, is the currency of the United Kingdom, its Crown dependencies and the British Overseas Territories of South Georgia and the South Sandwich Islands and British Antarctic Territory...
or
Irish poundThe Irish pound or punt Éireannach was the currency of Ireland until 2002. Its ISO 4217 code was IEP, and the usual notation was the prefix £, or IR£ where confusion might have arisen with the pound sterling or other pounds...
), and
CharlemagneCharlemagne was King of the Franks from 768 to his death. He expanded the Frankish kingdoms into a Frankish Empire that incorporated much of Western and Central Europe...
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,
ten10 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
can be written as A,
eleven11 is the natural number following 10 and preceding 12. It is the first number which cannot be represented by a human counting his or her eight fingers and two thumbs additively...
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
6060 is the natural number following 59 and preceding 61. Being three times twenty, 60 is called "three score" in some older literature.-In mathematics:...
(= five times twelve), duodecimal 60 is equivalent to decimal
7272 is the natural number following 71 and preceding 73. It is half a gross or 6 dozen .-In mathematics:...
(= six times twelve = half a gross), duodecimal 100 has the same value as decimal
144144 is the natural number following 143 and preceding 145. 144 is a dozen dozens, or one gross.-In mathematics:It is the twelfth Fibonacci number, and the largest one to also be a square, as the square of 12 , following 89 and preceding 233.144 is the smallest number with exactly 15 divisors.144 is...
(= twelve times twelve = one gross), etc.
Comparison to other numeral systems
A duodecimal multiplication tableIn mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system....
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | 10 |
|---|
| 2 |
4 |
6 |
8 |
A |
10 |
12 |
14 |
16 |
18 |
1A |
20 |
| 3 |
6 |
9 |
10 |
13 |
16 |
19 |
20 |
23 |
26 |
29 |
30 |
| 4 |
8 |
10 |
14 |
18 |
20 |
24 |
28 |
30 |
34 |
38 |
40 |
| 5 |
A |
13 |
18 |
21 |
26 |
2B |
34 |
39 |
42 |
47 |
50 |
| 6 |
10 |
16 |
20 |
26 |
30 |
36 |
40 |
46 |
50 |
56 |
60 |
| 7 |
12 |
19 |
24 |
2B |
36 |
41 |
48 |
53 |
5A |
65 |
70 |
| 8 |
14 |
20 |
28 |
34 |
40 |
48 |
54 |
60 |
68 |
74 |
80 |
| 9 |
16 |
23 |
30 |
39 |
46 |
53 |
60 |
69 |
76 |
83 |
90 |
| A |
18 |
26 |
34 |
42 |
50 |
5A |
68 |
76 |
84 |
92 |
A0 |
| B |
1A |
29 |
38 |
47 |
56 |
65 |
74 |
83 |
92 |
A1 |
B0 |
| 10 |
20 |
30 |
40 |
50 |
60 |
70 |
80 |
90 |
A0 |
B0 |
100 |
The number 12 has six factors, which are
1{| class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 1|-| colspan="2" | |-| Cardinal| 1
one|-| Ordinal| 1st
first|-| Numeral system| unary|-| Factorization| |-...
,
2class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 2|-| colspan="2" | |-| Ordinal number| 2nd
second|-| Numeral system| binary|-| Factorisation| prime|-| Gaussian integer factorisation...
,
3class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 3|-| colspan="2" | |-| Cardinal| 3
three|-| Ordinal| 3rd
third|-| Numeral system| ternary|-| Factorization| prime|-...
,
4| class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 4|-| colspan="2" | |-| Cardinal| 4
four|-| Ordinal| 4th
fourth|-| Numeral system| quaternary|-| Factorization| |-...
,
6| class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 6|-| colspan="2" | |-| Cardinal| 6
six|-| Ordinal| 6th
sixth|-| Numeral system| senary|-| Factorization| |-| Divisors...
, and
1212 is the natural number following 11 and preceding 13.The word "twelve" is a native English word that presumably arises from the Germanic compound twa-lif "two-leave", meaning that two is...
, of which 2 and 3 are
primeIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
. The decimal system has only four factors, which are
1{| class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 1|-| colspan="2" | |-| Cardinal| 1
one|-| Ordinal| 1st
first|-| Numeral system| unary|-| Factorization| |-...
,
2class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 2|-| colspan="2" | |-| Ordinal number| 2nd
second|-| Numeral system| binary|-| Factorisation| prime|-| Gaussian integer factorisation...
,
5{| class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 5|-| colspan="2" | |-| Cardinal| 5
five|-| Ordinal| 5th
fifth|-| Numeral system| quinary|-| Factorization| prime|-| Divisors...
, and
1010 is an even natural number following 9 and preceding 11.-In mathematics:Ten is a composite number, its proper divisors being , and...
; of which 2 and 5 are prime. Vigesimal adds two factors to those of ten, namely
4| class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 4|-| colspan="2" | |-| Cardinal| 4
four|-| Ordinal| 4th
fourth|-| Numeral system| quaternary|-| Factorization| |-...
and
2020 is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score.-In mathematics:*An icosahedron has 20 faces. A dodecahedron has 20 vertices....
, 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| class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 8|-| colspan="2" | |-| Cardinal| 8
eight|-| Ordinal| 8th
eighth|-| Numeral system| octal|-| Factorization| |-| Divisors...
and
1616 is the natural number following 15 and preceding 17. 16 is a composite number, and a square number, being 4
2 = 4 × 4. It is the smallest number with exactly five divisors, its proper divisors being , , and ....
to those of 2, but no additional prime.
TrigesimalBase 30 or trigesimal is a positional numeral system using 30 as the radix. Digits in this base can be represented using the Arabic numerals 0-9 and the Latin letters A-T....
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).
SexagesimalSexagesimal is a numeral system with sixty as the base. It originated with the ancient Sumerians in the 2000s BCE, was transmitted to the Babylonians, and is still used—in modified form—for measuring time, angles, and geographic coordinates....
-- which the ancient Sumerians and
BabyloniaBabylonia was a civilization in Lower Mesopotamia , with Babylon as its capital. Babylonia emerged when Hammurabi created an empire out of the territories of the former kingdoms of Sumer and Akkad...
ns 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 positional notation). 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
| Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
| 100,000 |
248,832 |
10,000 |
20,736 |
1,000 |
1,728 |
100 |
144 |
10 |
12 |
1 |
1 |
0.1 |
| 0.01 |
|
| 200,000 |
497,664 |
20,000 |
41,472 |
2,000 |
3,456 |
200 |
288 |
20 |
24 |
2 |
2 |
0.2 |
| 0.02 |
|
| 300,000 |
746,496 |
30,000 |
62,208 |
3,000 |
5,184 |
300 |
432 |
30 |
36 |
3 |
3 |
0.3 |
0.25 |
0.03 |
|
| 400,000 |
995,328 |
40,000 |
82,944 |
4,000 |
6,912 |
400 |
576 |
40 |
48 |
4 |
4 |
0.4 |
| 0.04 |
|
| 500,000 |
1,244,160 |
50,000 |
103,680 |
5,000 |
8,640 |
500 |
720 |
50 |
60 |
5 |
5 |
0.5 |
| 0.05 |
|
| 600,000 |
1,492,992 |
60,000 |
124,416 |
6,000 |
10,368 |
600 |
864 |
60 |
72 |
6 |
6 |
0.6 |
0.5 |
0.06 |
|
| 700,000 |
1,741,824 |
70,000 |
145,152 |
7,000 |
12,096 |
700 |
1008 |
70 |
84 |
7 |
7 |
0.7 |
| 0.07 |
|
| 800,000 |
1,990,656 |
80,000 |
165,888 |
8,000 |
13,824 |
800 |
1152 |
80 |
96 |
8 |
8 |
0.8 |
| 0.08 |
|
| 900,000 |
2,239,488 |
90,000 |
186,624 |
9,000 |
15,552 |
900 |
1,296 |
90 |
108 |
9 |
9 |
0.9 |
0.75 |
0.09 |
0.0625 |
| A00,000 |
2,488,320 |
A0,000 |
207,360 |
A,000 |
17,280 |
A00 |
1,440 |
A0 |
120 |
A |
10 |
0.A |
| 0.0A |
|
| B00,000 |
2,737,152 |
B0,000 |
228,096 |
B,000 |
19,008 |
B00 |
1,584 |
B0 |
132 |
B |
11 |
0.B |
| 0.0B |
|
Decimal to Dozenal digit conversion
| Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
| 100,000 |
49,A54 |
10,000 |
5,954 |
1,000 |
6B4 |
100 |
84 |
10 |
A |
1 |
1 |
0.1 |
| 0.01 |
|
| 200,000 |
97,8A8 |
20,000 |
B,6A8 |
2,000 |
1,1A8 |
200 |
148 |
20 |
18 |
2 |
2 |
0.2 |
| 0.02 |
|
| 300,000 |
125,740 |
30,000 |
15,440 |
3,000 |
1,8A0 |
300 |
210 |
30 |
26 |
3 |
3 |
0.3 |
| 0.03 |
|
| 400,000 |
173,594 |
40,000 |
1B,194 |
4,000 |
2,394 |
400 |
294 |
40 |
34 |
4 |
4 |
0.4 |
| 0.04 |
|
| 500,000 |
201,428 |
50,000 |
24,B28 |
5,000 |
2,A88 |
500 |
358 |
50 |
42 |
5 |
5 |
0.5 |
0.6 |
0.05 |
|
| 600,000 |
24B,280 |
60,000 |
2A,880 |
6,000 |
3,580 |
600 |
420 |
60 |
50 |
6 |
6 |
0.6 |
| 0.06 |
|
| 700,000 |
299,114 |
70,000 |
34,614 |
7,000 |
4,074 |
700 |
4A4 |
70 |
5A |
7 |
7 |
0.7 |
| 0.07 |
|
| 800,000 |
326,B68 |
80,000 |
3A,368 |
8,000 |
4,768 |
800 |
568 |
80 |
68 |
8 |
8 |
0.8 |
| 0.08 |
|
| 900,000 |
374,A00 |
90,000 |
44,100 |
9,000 |
5,260 |
900 |
630 |
90 |
76 |
9 |
9 |
0.9 |
| 0.09 |
|
Conversion of powers
| Exponent |
Powers of 2 |
Powers of 3 |
Powers of 4 |
Powers of 5 |
Powers of 6 |
Powers of 7 |
| Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
| ^6 |
64 |
54 |
729 |
509 |
4,096 |
2454 |
15,625 |
9,061 |
46,656 |
23,000 |
117,649 |
58,101 |
| ^5 |
32 |
28 |
243 |
183 |
1,024 |
714 |
3,125 |
1,985 |
7,776 |
4,600 |
16,807 |
9,887 |
| ^4 |
16 |
14 |
81 |
69 |
256 |
194 |
625 |
441 |
1,296 |
900 |
2,401 |
1,481 |
| ^3 |
8 |
8 |
27 |
23 |
64 |
54 |
125 |
A5 |
216 |
160 |
343 |
247 |
| ^2 |
4 |
4 |
9 |
9 |
16 |
14 |
25 |
21 |
36 |
30 |
49 |
41 |
| ^1 |
2 |
2 |
3 |
3 |
4 |
4 |
5 |
5 |
6 |
6 |
7 |
7 |
| ^−1 |
0.5 |
0.6 |
0. |
0.4 |
0.25 |
0.3 |
0.2 |
| 0.1 |
0.2 |
0. |
|
| ^−2 |
0.25 |
0.3 |
0. |
0.14 |
0.0625 |
0.09 |
0.04 |
| 0.02 |
0.04 |
0. |
|
| Exponent |
Powers of 8 |
Powers of 9 |
Powers of 10 |
Powers of 11 |
Powers of 12 |
| Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
Dec. |
Doz. |
| ^6 |
262,144 |
107,854 |
531,441 |
217,669 |
1,000,000 |
402,854 |
1,771,561 |
715,261 |
2,985,984 |
1,000,000 |
| ^5 |
32,768 |
16,B68 |
59,049 |
2A,209 |
100,000 |
49,A54 |
161,051 |
79,24B |
248,832 |
100,000 |
| ^4 |
4,096 |
2,454 |
6,561 |
3,969 |
10,000 |
5,954 |
14,641 |
8,581 |
20,736 |
10,000 |
| ^3 |
512 |
368 |
729 |
509 |
1,000 |
6B4 |
1,331 |
92B |
1,728 |
1,000 |
| ^2 |
64 |
54 |
81 |
69 |
100 |
84 |
121 |
A1 |
144 |
100 |
| ^1 |
8 |
8 |
9 |
9 |
10 |
A |
11 |
B |
12 |
10 |
| ^−1 |
0.125 |
0.16 |
0. |
0.14 |
0.1 |
| 0. |
| 0.08 |
0.1 |
| ^−2 |
0.015625 |
0.023 |
0. |
0.0194 |
0.01 |
| 0. |
| 0.0069 |
0.01 |
Fractions
Duodecimal
fractionA fraction is a number that can represent part of a whole.The earliest fractions were reciprocals of integers, symbols representing one half, one third, one quarter, and so on...
s may be simple:
- 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)
| Examples in duodecimal |
Decimal equivalent |
| 1 × (5 / 8) = 0.76 |
1 × (5 / 8) = 0.625 |
| 100 × (5 / 8) = 76 |
144 × (5 / 8) = 90 |
| 576 / 9 = 76 |
810 / 9 = 90 |
| 400 / 9 = 54 |
576 / 9 = 64 |
| 1A.6 + 7.6 = 26 |
22.5 + 7.5 = 30 |
As explained in recurring decimals, whenever an
irreducible fractionAn irreducible fraction is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent vulgar fraction...
is written in
radix pointIn mathematics and computing, a radix point is the symbol used in numerical representations to separate the integer part of a number from its fractional part . "Radix point" is a general term that applies to all number bases...
notation in any base, the fraction can be expressed exactly (terminates) if and only if all the
prime factorIn number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization....
s 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.
Recurring digits
Arguably, factors of 3 are more commonly encountered in real-life
divisionright|thumb|200px|In mathematics, especially in elementary arithmetic, division is the arithmetic operation that is the inverse of multiplication...
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
1212 is the natural number following 11 and preceding 13.The word "twelve" is a native English word that presumably arises from the Germanic compound twa-lif "two-leave", meaning that two is...
(twelve) is between two
prime numberIn mathematics, a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. The first twenty-six prime numbers are:An infinitude of prime numbers exists, as demonstrated by Euclid around 300 BC. The number 1 is by definition not a prime number...
s,
1111 is the natural number following 10 and preceding 12. It is the first number which cannot be represented by a human counting his or her eight fingers and two thumbs additively...
(eleven) and
1313 is the natural number after 12 and before 14. It is the smallest integer with eight letters in its spelled out name in English. It is also the age at which children officially become teenagers....
(thirteen), whereas ten is adjacent to
composite numberA composite number is a positive integer which has a positive divisor other than one or itself. In other words, if n > 0 is an integer and there are integers 1 < a, b < n such that n = a × b then n is composite. By definition, every integer greater than one is either a prime number or a composite...
9{| class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 9|-| colspan="2" | |-| Cardinal| 9
nine|-| Ordinal| 9th
ninth|-| Numeral system| nonary|-| Factorization| |-...
. 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
roundingRounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation; for example, replacing US$ 23.4476 by US$ 23.45, or the fraction 312/937 by 1/3, or by 1.41....
, 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
3class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 3|-| colspan="2" | |-| Cardinal| 3
three|-| Ordinal| 3rd
third|-| Numeral system| ternary|-| Factorization| prime|-...
in its factorization, while only one out of every five contains the prime factor
5{| class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 5|-| colspan="2" | |-| Cardinal| 5
five|-| Ordinal| 5th
fifth|-| Numeral system| quinary|-| Factorization| prime|-| Divisors...
. 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
2class="infobox" style="width: 20em;"|-! colspan="2" align="center" style="font: 10em times; background:#ccc;" | 2|-| colspan="2" | |-| Ordinal number| 2nd
second|-| Numeral system| binary|-| Factorisation| prime|-| Gaussian integer factorisation...
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 twoIn mathematics, a power of two is any of the integer powers of the number two; in other words, two multiplied by itself a certain number of times. Note that one is a power of two...
will have a shorter, more convenient terminating representation in dozenal than in decimal (e.g., 1/(2
2) = 0.25
dec = 0.3
doz; 1/(2
3) = 0.125
dec = 0.16
doz; 1/(2
4) = 0.0625
dec = 0.09
doz; 1/(2
5) = 0.03125
dec = 0.046
doz; etc.).
Decimal base
Prime factors of the base: 2, 5 |
Duodecimal / Dozenal base
Prime factors of the base: 2, 3 |
| Fraction |
Prime factors
of the denominator |
Positional representation |
Positional representation |
Prime factors
of the denominator |
Fraction |
| 1/2 |
2 |
0.5 |
0.6 |
2 |
1/2 |
| 1/3 |
3 |
0.3333... = 0. |
0.4 |
3 |
1/3 |
| 1/4 |
2 |
0.25 |
0.3 |
2 |
1/4 |
| 1/5 |
5 |
0.2 |
0.24972497... = 0. |
5 |
1/5 |
| 1/6 |
2, 3 |
0.1 |
0.2 |
2, 3 |
1/6 |
| 1/7 |
7 |
0. |
0. |
7 |
1/7 |
| 1/8 |
2 |
0.125 |
0.16 |
2 |
1/8 |
| 1/9 |
3 |
0. |
0.14 |
3 |
1/9 |
| 1/10 |
2, 5 |
0.1 |
0.1 |
2, 5 |
1/A |
| 1/11 |
11 |
0. |
0. |
B |
1/B |
| 1/12 |
2, 3 |
0.08 |
0.1 |
2, 3 |
1/10 |
| 1/13 |
13 |
0. |
0. |
11 |
1/11 |
| 1/14 |
2, 7 |
0.0 |
0.0 |
2, 7 |
1/12 |
| 1/15 |
3, 5 |
0.0 |
0.0 |
3, 5 |
1/13 |
| 1/16 |
2 |
0.0625 |
0.09 |
2 |
1/14 |
| 1/17 |
17 |
0. |
0. |
15 |
1/15 |
| 1/18 |
2, 3 |
0.0 |
0.08 |
2, 3 |
1/16 |
| 1/19 |
19 |
0. |
0. |
17 |
1/17 |
| 1/20 |
2, 5 |
0.05 |
0.0 |
2, 5 |
1/18 |
| 1/21 |
3, 7 |
0. |
0.0 |
3, 7 |
1/19 |
| 1/22 |
2, 11 |
0.0 |
0.0 |
2, B |
1/1A |
| 1/23 |
23 |
0. |
0. |
1B |
1/1B |
| 1/24 |
2, 3 |
0.041 |
0.06 |
2, 3 |
1/20 |
| 1/25 |
5 |
0.04 |
0. |
5 |
1/21 |
| 1/26 |
2, 13 |
0.0 |
0.0 |
2, 11 |
1/22 |
| 1/27 |
3 |
0. |
0.054 |
3 |
1/23 |
| 1/28 |
2, 7 |
0.03 |
0.0 |
2, 7 |
1/24 |
| 1/29 |
29 |
0. |
0. |
25 |
1/25 |
| 1/30 |
2, 3, 5 |
0.0 |
0.0 |
2, 3, 5 |
1/26 |
| 1/31 |
31 |
0. |
0. |
27 |
1/27 |
| 1/32 |
2 |
0.03125 |
0.046 |
2 |
1/28 |
| 1/33 |
3, 11 |
0. |
0.0 |
3, B |
1/29 |
| 1/34 |
2, 17 |
0.0 |
0.0 |
2, 15 |
1/2A |
| 1/35 |
5, 7 |
0.0 |
0. |
5, 7 |
1/2B |
| 1/36 |
2, 3 |
0.02 |
0.04 |
2, 3 |
1/30 |
Irrational numbers
As for
irrational numberIn mathematics, an irrational number is any real number that is not a rational number—that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. Informally, this means numbers that cannot be represented as simple fractions...
s, none of them has a finite representation in
any of the
rationalIn mathematics, a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer corresponds to a rational number. The set of all rational numbers is usually denoted .Formally each 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 × 1/10
-2 + 2 × 1/10
-1 + 3 × 1/10
0 + 4 × 1/10
1 + 5 × 1/10
2 + 6 × 1/10
3 (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 repetition; 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
algebraicIn mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational coefficients...
and
transcendentalIn mathematics, a transcendental number is a number that is not algebraic, that is, not a solution of a non-constant polynomial equation with rational coefficients....
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.
| Algebraic irrational number |
In decimal |
In duodecimal / dozenal |
| √2 The square root of 2, also known as Pythagoras' constant,is the positive real number that, when multiplied by itself, gives the number 2....
(the length of the diagonalA diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
of a unit squareIn geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
) |
1.41421356237309... (≈ 1.414) |
1.4B79170A07B857... (≈ 1.5) |
√3The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted byThe first sixty significant digits of its decimal expansion are:The rounded value of 1.732 is correct to within 0.01% of the actual value....
(the length of the diagonal of a unit cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
, or twice the heightHeight is the measurement of vertical distance, but has two meanings in common use. It can either indicate how "tall" something is, or how "high up" it is. For example one could say "That is a tall building", or "That airplane is high up in the sky"...
of an equilateral triangleIn geometry, an equilateral triangle is a triangle in which all three sides are equal. In traditional or Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°...
of unit side) |
1.73205080756887... (≈ 1.732) |
1.894B97BB968704... (≈ 1.895) |
√5The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. This number appears in the formula for the golden ratio. It can be denoted in surd form as:...
(the length of the diagonalA diagonal is a line joining two nonconsecutive vertices of a polygon or polyhedron. Informally, any sloping line is called diagonal. The word "diagonal" derives from the Greek διαγώνιος , from dia- and gonia ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a...
of a 1×2 rectangleIn Euclidean geometry, the term rectangle normally refers to a quadrilateral with four right angles. This is a simple rectangle. A simple rectangle with vertices ABCD would be denoted as ....
) |
2.2360679774997... (≈ 2.236) |
2.29BB132540589... (≈ 2.2A) |
| φ In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887...
(phi, the golden ratio = ) |
1.6180339887498... (≈ 1.618) |
1.74BB6772802A4... (≈ 1.75) |
| Transcendental irrational number |
In decimal |
In duodecimal / dozenal |
πPi or π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. The symbol π was first proposed by the Welsh mathematician William Jones in 1706...
(pi, the ratio of circumferenceThe circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....
to diameterIn geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
) |
3.1415926535897932384626433
8327950288419716939937510...
(≈ 3.1416) |
3.184809493B918664573A6211B
B151551A05729290A7809A492...
(≈ 3.1848) |
eThe mathematical constant e is the unique real number such that the value of the derivative of the function f = e x at the point x = 0 is exactly 1. The function e x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm...
(the base of the natural logarithmThe natural logarithm is the logarithm to the base e, where e is an irrational constant approximately equal to 2.718 281828...
) |
2.718281828459045... (≈ 2.718) |
2.8752360698219B8... (≈ 2.875) |
The first few digits of the decimal and dozenal representation of another important number, the
Euler-Mascheroni constantThe Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
(the status of which as a rational or irrational number is not yet known), are:
| Number |
In decimal |
In duodecimal / dozenal |
| γ The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
(the limiting difference between the harmonic seriesIn mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics, in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...
and the natural logarithm) |
0.57721566490153... (~ 0.577) |
0.6B15188A6760B3... (~ 0.7) |
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
hexadecimalIn mathematics and computer science, hexadecimal is a numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen.Its primary use is as a...
notation and
vigesimalThe vigesimal or base- numeral system is based on twenty .- Places :...
notation (or 'T' and 'E' for ten and eleven), he suggested in his book and used a script X and a script E, (
U+Unicode is a computing industry standard allowing computers to consistently represent and manipulate text expressed in most of the world's writing systems...
1D4B3) and
(U+2130), 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 for its resemblance to the Roman numeral X, and
as the first letter of the word "eleven".
Another popular notation, introduced by Sir
Isaac PitmanSir Isaac Pitman , knighted in 1894, developed the most widely used system of shorthand, known now as Pitman shorthand. He first proposed this in Stenographic Soundhand in 1837. Pitman was a qualified teacher and taught at a private school he founded in Wotton-under-Edge...
, is to use a rotated 2 to represent ten and a rotated or horizontally flipped 3 (which again resembles
) 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
asteriskAn asterisk is a typographical symbol or glyph. It is so called because it resembles a conventional image of a star. Computer scientists and mathematicians often pronounce it as star...
* for ten and a
hashNumber sign is a name for the symbol #, which is used for a variety of purposes including the designation of a number . "Number sign" is the preferred Unicode name for the code point...
# 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
telephoneThe telephone is a telecommunications device that transmits and receives sound, most commonly the human voice. It is one of the most common household appliances in the developed world, and has long been considered indispensable to business, industry and government...
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). Problems with these symbols are evident, most notably that most of them do not fit on most
calculatorA calculator is a device that is used for performing mathematical calculations. It differs from a computer by having a limited problem solving ability and an interface optimized for interactive calculation rather than programming...
displays (
being an exception, although "E" is used on calculators to indicate an
error messageAn error message is a message displayed when an unexpected condition occurs, usually on a computer or other device. Error messages are often displayed using dialog boxes. Error messages are used when user intervention is required, indicate that a desired operation has failed, or give very...
). However, 10 and 11 do fit, both within a single digit (11 fits as is, while the 10 has to be tilted sideways, resulting in a character that resembles an O with a
macronA macron, from the Greek , meaning "long", is a diacritic placed above a vowel . It was originally used to mark a long syllable in Græco-Roman metrics, but now also indicates that the vowel is long...
, ō or
0). A and B also fit (although B must be represented as lowercase "b" and as such, 6 must have a bar over it to distinguish the two figures) and are used on calculators for bases higher than ten.
In
'Little Twelvetoes', American television series
Schoolhouse Rock!Schoolhouse Rock! is an American series of animated musical educational short films that aired during the Saturday morning children's programming on the U.S. television network ABC. The topics covered included grammar, science, economics, history, mathematics, and civics...
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. ("Dek" is from the prefix "deca," "el" being short for "eleven" and "doh" an apparent shortening of "dozen.")
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 AitkenAlexander Craig Aitken, FRS FRSE FRSL was one of New Zealand's greatest mathematicians. He studied for a PhD at the University of Edinburgh, where his dissertation, "Smoothing of Data", was considered so impressive that he was awarded a DSc in 1926, and was elected a fellow of the Royal Society...
was an outspoken advocate of the advantages and superiority of duodecimal over decimal:
In
Leo FrankowskiLeo Frankowski was an American writer of science fiction novels. He lived in Russia for four years with his now ex-wife and adopted teenage daughter, but...
's
Conrad StargardConrad Stargard is the protagonist and title character in a series of time travel novels written by Leo Frankowski. In them, a Polish engineer named Conrad Schwartz is sent back in time to the 13th century where he has to establish himself and cope with various crises including the eventual...
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
In mathematics, a senary numeral system is a base- numeral system. The name heximal is also valid for such a numeral system, but is deprecated to avoid confusion with the more often used hexadecimal number base, colloquially known as 'hex'....
(base 6)
- Quadrovigesimal
The base- system is a numeral system with 24 as its base.There are 24 hours in a day, so our time keeping system includes a base-24 component...
(base 24)
- Hexatridecimal
Base 36 is a positional numeral system using 36 as the radix. The choice of 36 is convenient in that the digits can be represented using the Arabic numerals 0-9 and the Latin letters A-Z. Base 36 is therefore the most compact case-insensitive alphanumeric numeral system using ASCII characters,...
(base 36)
- Sexagesimal
Sexagesimal is a numeral system with sixty as the base. It originated with the ancient Sumerians in the 2000s BCE, was transmitted to the Babylonians, and is still used—in modified form—for measuring time, angles, and geographic coordinates....
(base 60)
- Babylonian numerals
Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record....
External links