FACTOID # 27: If you're itching to live in a trailer park, hitch up your home and head to South Carolina, where a whopping 18% of residences are mobile homes.
 
 Home   Encyclopedia   Statistics   States A-Z   Flags   Maps   FAQ   About 
   
 
WHAT'S NEW
RELATED ARTICLES
People who viewed "Duodecimal" also viewed:
 

SEARCH ALL

FACTS & STATISTICS    Advanced view

Search encyclopedia, statistics and forums:

 

 

(* = Graphable)

 

 


Encyclopedia > Duodecimal
Numeral systems by culture
Hindu-Arabic numerals
Indian
Eastern Arabic
Khmer
Indian family
Brahmi
Thai
East Asian numerals
Chinese
Counting rods
Japanese
Korean 
Alphabetic numerals
Abjad
Armenian
Cyrillic
Ge'ez
Hebrew
Greek (Ionian)
Āryabhaṭa
 
Other systems
Attic
Babylonian
Egyptian
Etruscan
Mayan
Roman
Urnfield
List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64
1, 3, 9, 12, 20, 24, 30, 36, 60, more…
v  d  e

The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. The number ten may be written as 'A', and the number eleven as 'B'. The number twelve (that is, the number twelve in the base 10 numerical system) is written as '10'. This article is about different methods of expressing numbers with symbols. ... I like cream cheese, it tastes good on toast. ... The Eastern Arabic numerals (also called Eastern Arabic numerals, Arabic-Indic numerals, Arabic Eastern Numerals) are the symbols (glyphs) used to represent the Hindu-Arabic numeral system in conjunction with the Arabic alphabet in Egypt, Iran, Pakistan and parts of India, and also in the no longer used Ottoman Turkish... Khmer numerals are the numerals used in the Khmer language of Cambodia. ... India has produced many numeral systems. ... The Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). ... The counting rods (Traditional Chinese: , Simplified Chinese: , pinyin: chou2) were used by ancient Chinese before the invention of the abacus. ... The Abjad numerals are a decimal numeral system which was used in the Arabic-speaking world prior to the use of the Hindu-Arabic numerals from the 8th century, and in parallel with the latter until Modern times. ... Cyrillic numerals was a numbering system derived from the Cyrillic alphabet, used by South and East Slavic peoples. ... Note: This article contains special characters. ... The system of Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet. ... Attic numerals were used by ancient Greeks, possibly from the 7th century BC. They were also known as Herodianic numerals because they were first described in a 2nd century manuscript by Herodian. ... 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. ... The Etruscan numerals were used by the ancient Etruscans. ... Mayan numerals. ... Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ... During the beginning of the Urnfield culture, around 1200 BC, a series of votive sickles of bronze with marks that have been interpreted as a numeral system, appeared in Central Europe. ... This is a list of numeral system topics, by Wikipedia page. ... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ... The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ... For other uses, see Decimal (disambiguation). ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... Quaternary is the base four numeral system. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ... Base32 is a derivation of Base64 with the following additional properties: The resulting character set is all uppercase, which can often be beneficial when using a case-sensitive filesystem. ... In computing, base64 is a data encoding scheme whereby binary-encoded data is converted to printable ASCII characters. ... The unary numeral system is the simplest numeral system to represent natural numbers: in order to represent a number N, an arbitrarily chosen symbol is repeated N times. ... Ternary or trinary is the base-3 numeral system. ... Nonary is a base 9 numeral system, typically using the digits 0-8, but not the digit 9. ... The vigesimal or base-20 numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). ... As there are 24 hours in a day a numbering system based upon 24, and as the base 12 is convenient here some examples of the base 24 (quadrovigesimal) system. ... Base 30 or trigesimal is a positional numeral system using 30 as the radix. ... Base 36 is a positional numeral system using 36 as the radix. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... In mathematics, the base or radix is the number of various unique symbols (digits), including zero, that a positional numeral system uses to represent numbers in a given counting system. ... This article is about different methods of expressing numbers with symbols. ... Look up twelve in Wiktionary, the free dictionary. ... This article is about the number 10. ... 11 (eleven) is the natural number following 10 and preceding 12. ... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to...


The number 12 has six factors, of which two are prime (2 and 3). It is a more convenient number system for computing fractions than other common number systems such as the decimal, vigesimal, binary and hexadecimal systems. In mathematics, a prime number (or a prime) is a natural number greater than 1 which has exactly two distinct natural number divisors: 1 and itself. ... For other uses, see Decimal (disambiguation). ... The vigesimal or base-20 numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ...

Contents

Origin

In this section, numerals are based on decimal places. For example, 10 means ten, 12 means twelve.

Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Kahugu), the Nimbia dialect of Gwandara[1]; the Chepang language of Nepal[2] 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. Digit may refer to: A finger or a toe Numerical digit, as used in mathematics or computer science Digit (unit), an ancient meterological unit Digit (magazine), an Indian information technology magazine This is a disambiguation page: a list of articles associated with the same title. ... This article is about the number 10. ... Look up twelve in Wiktionary, the free dictionary. ... Chepang is the commonly used name given to an ethnic group living in central and southern Nepal. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Minicoy Island (Maliku) Minicoy Island or Maliku is the second largest and the southern-most island of the Laccadive Archipelago north of the Maldives. ... Tolkien redirects here. ... Elvish languages are constructed languages used typically by elves in a fantasy setting. ...


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. The Germanic languages are a group of related languages constituting a branch of the Indo-European (IE) language family. ... The English language is a West Germanic language that originates in England. ... This article or section does not cite any references or sources. ...


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. Measurement is the determination of the size or magnitude of something. ... This article is about the concept of time. ... Central New York City. ... For other uses, see Zodiac (disambiguation). ... The hour (symbol: h) is a unit of time. ... The Chinese calendar is a lunisolar calendar, incorporating elements of a lunar calendar with those of a solar calendar. ... The Earthly Branches (Chinese: ; pinyin: dìzhī; or Chinese: ; pinyin: shíèrzhī; literally twelve branches) provide one Chinese system for reckoning time. ...


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. An inch (plural: inches; symbol or abbreviation: in or, sometimes, ″ - a double prime) is the name of a unit of length in a number of different systems, including English units, Imperial units, and United States customary units. ... A foot (plural: feet or foot;[1] symbol or abbreviation: ft or, sometimes, ′ – a prime) is a unit of length, in a number of different systems, including English units, Imperial units, and United States customary units. ... Troy ounce is a traditional unit of gold weight. ... Look up pound in Wiktionary, the free dictionary. ... For silver pennies produced after 1820 see Maundy money. ... This article is about coinage. ... Dozen is another word for the number twelve. ... Categories: | | ... 144 is the whole number following 143 and preceding 145. ... In 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. ... A great gross is equal to a dozen gross, i. ... 1728 is the natural number following 1727 and preceding 1729. ... In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. ... Uncia can mean: Uncia (coin), an ancient Roman bronze coin Snow Leopard, Uncia uncia This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... This article is about Ounce (unit of mass). ... This article does not cite any references or sources. ... GBP redirects here. ... For the American band, see Charlemagne (band). ...


Places

In a duodecimal place system, ten can be written as A, eleven can be written as B, and twelve is written as 10. This article is about the number 10. ... 11 (eleven) is the natural number following 10 and preceding 12. ...


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.
60 (sixty) is the natural number following 59 and preceding 61. ... 72 is the natural number following 71 and preceding 73. ... 144 is the whole number following 143 and preceding 145. ...


Comparison to other numeral systems

A duodecimal multiplication table
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, 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.
Times table redirects here. ... This article is about the number one. ... This article does not cite any references or sources. ... This article is about the number. ... This article discusses the number Four. ... Look up six in Wiktionary, the free dictionary. ... Look up twelve in Wiktionary, the free dictionary. ... In mathematics, a prime number (or a prime) is a natural number greater than 1 which has exactly two distinct natural number divisors: 1 and itself. ... This article is about the number one. ... This article does not cite any references or sources. ... Look up five in Wiktionary, the free dictionary. ... This article is about the number 10. ... This article discusses the number Four. ... 20 (twenty) is the natural number following 19 and preceding 21. ... Look up eight in Wiktionary, the free dictionary. ... 16 (sixteen) is the natural number following 15 and preceding 17. ...


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: The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ...


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 + 0.583333333333... + 0.055555555555...


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 = 0.583333333333... 0.08 = 0.055555555555... -------------------------------------------- 123,456.78 = 296,130.638888888888... 

That is, (dozenal) 123,456.78 equals (decimal) 296,130.638888888888... ≈ 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 + 0.849724972497249724972497... + 0.0B62A68781B05915343A0B62...


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 = 0.849724972497249724972497... 0.08 = 0.0B62A68781B05915343A0B62... -------------------------------------------------------- 123,456.78 = 5B,540.943A0B62A68781B05915343A... 

That is, (decimal) 123,456.78 equals (dozenal) 5B,540.943A0B62A68781B05915343A... ≈ 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 0.083 0.01 0.00694
200,000 497,664 20,000 41,472 2,000 3,456 200 288 20 24 2 2 0.2 0.16 0.02 0.0138
300,000 746,496 30,000 62,208 3,000 5,184 300 432 30 36 3 3 0.3 0.25 0.03 0.02083
400,000 995,328 40,000 82,944 4,000 6,912 400 576 40 48 4 4 0.4 0.3 0.04 0.027
500,000 1,244,160 50,000 103,680 5,000 8,640 500 720 50 60 5 5 0.5 0.416 0.05 0.03472
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 0.0416
700,000 1,741,824 70,000 145,152 7,000 12,096 700 1008 70 84 7 7 0.7 0.583 0.07 0.04861
800,000 1,990,656 80,000 165,888 8,000 13,824 800 1152 80 96 8 8 0.8 0.6 0.08 0.05
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 0.83 0.0A 0.0694
B00,000 2,737,152 B0,000 228,096 B,000 19,008 B00 1,584 B0 132 B 11 0.B 0.916 0.0B 0.07638

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 0.12497 0.01 0.015343A0B62A68781B059
200,000 97,8A8 20,000 B,6A8 2,000 1,1A8 200 148 20 18 2 2 0.2 0.2497 0.02 0.02A68781B05915343A0B6
300,000 125,740 30,000 15,440 3,000 1,8A0 300 210 30 26 3 3 0.3 0.37249 0.03 0.043A0B62A68781B059153
400,000 173,594 40,000 1B,194 4,000 2,394 400 294 40 34 4 4 0.4 0.4972 0.04 0.05915343A0B62A68781B0
500,000 201,428 50,000 24,B28 5,000 2,A88 500 358 50 42 5 5 0.5 0.6 0.05 0.07249
600,000 24B,280 60,000 2A,880 6,000 3,580 600 420 60 50 6 6 0.6 0.7249 0.06 0.08781B05915343A0B62A6
700,000 299,114 70,000 34,614 7,000 4,074 700 4A4 70 5A 7 7 0.7 0.84972 0.07 0.0A0B62A68781B05915343
800,000 326,B68 80,000 3A,368 8,000 4,768 800 568 80 68 8 8 0.8 0.9724 0.08 0.0B62A68781B05915343A
900,000 374,A00 90,000 44,100 9,000 5,260 900 630 90 76 9 9 0.9 0.A9724 0.09 0.10B62A68781B05915343A

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 0.3 0.4 0.25 0.3 0.2 0.2497 0.16 0.2 0.142857 0.186A35
^−2 0.25 0.3 0.1 0.14 0.0625 0.09 0.04 0.05915343A0
B62A68781B
0.027 0.04 0.0204081632653
06122448979591
836734693877551
0.02B322547A05A
644A9380B908996
741B615771283B
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 0.1 0.14 0.1 0.12497 0.09 0.1 0.083 0.1
^−2 0.015625 0.023 0.012345679 0.0194 0.01 0.015343A0B6
2A68781B059
0.00826446280
99173553719
0.0123456789B 0.00694 0.01

Fractions and irrational numbers

Fractions

Duodecimal fractions may be simple: For other uses, see Fraction (disambiguation). ...

  • 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 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. A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... An irreducible fraction (or fraction in lowest terms) is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent fraction. ... This article is about the concept in number theory. ...


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. In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... A recurring decimal is an expression representing a real number in the decimal numeral system, in which after some point the same sequence of digits repeats infinitely many times. ...


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.). Look up twelve in Wiktionary, the free dictionary. ... In mathematics, a prime number (or a prime) is a natural number greater than 1 which has exactly two distinct natural number divisors: 1 and itself. ... 11 (eleven) is the natural number following 10 and preceding 12. ... 13 (thirteen) is the natural number after 12 and before 14. ... A composite number is a positive integer which has a positive divisor other than one or itself. ... This article is about the number. ... Rounding is the process of reducing the number of significant digits in a number. ... This article is about the number. ... Look up five in Wiktionary, the free dictionary. ... This article does not cite any references or sources. ... In mathematics, a power of two is any of the nonnegative integer powers of the number two; in other words, two times itself a certain number of times. ...

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... = 0.3 0.4 3 1/3
1/4 2 0.25 0.3 2 1/4
1/5 5 0.2 0.24972497... = 0.2497 5 1/5
1/6 2, 3 0.16 0.2 2, 3 1/6
1/7 7 0.142857 0.186A35 7 1/7
1/8 2 0.125 0.16 2 1/8
1/9 3 0.1 0.14 3 1/9
1/10 2, 5 0.1 0.12497 2, 5 1/A
1/11 11 0.09 0.1 B 1/B
1/12 2, 3 0.083 0.1 2, 3 1/10
1/13 13 0.076923 0.0B 11 1/11
1/14 2, 7 0.0714285 0.0A35186 2, 7 1/12
1/15 3, 5 0.06 0.09724 3, 5 1/13
1/16 2 0.0625 0.09 2 1/14
1/17 17 0.0588235294117647 0.08579214B36429A7 15 1/15
1/18 2, 3 0.05 0.08 2, 3 1/16
1/19 19 0.052631578947368421 0.076B45 17 1/17
1/20 2, 5 0.05 0.07249 2, 5 1/18
1/21 3, 7 0.047619 0.06A3518 3, 7 1/19
1/22 2, 11 0.045 0.06 2, B 1/1A
1/23 23 0.0434782608695652173913 0.06316948421 1B 1/1B
1/24 2, 3 0.0416 0.06 2, 3 1/20
1/25 5 0.04 0.05915343A0B6 5 1/21
1/26 2, 13 0.0384615 0.056 2, 11 1/22
1/27 3 0.037 0.054 3 1/23
1/28 2, 7 0.03571428 0.05186A3 2, 7 1/24
1/29 29 0.0344827586206896551724137931 0.04B7 25 1/25
1/30 2, 3, 5 0.03 0.04972 2, 3, 5 1/26
1/31 31 0.032258064516129 0.0478AA093598166B74311B28623A55 27 1/27
1/32 2 0.03125 0.046 2 1/28
1/33 3, 11 0.03 0.04 3, B 1/29
1/34 2, 17 0.02941176470588235 0.0429A708579214B36 2, 15 1/2A
1/35 5, 7 0.0285714 0.0414559B3931 5, 7 1/2B
1/36 2, 3 0.027 0.04 2, 3 1/30

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 trascendental 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. In 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. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, a transcendental number is any irrational number that is not an algebraic number, i. ...

Algebraic irrational number In decimal In duodecimal / dozenal
√2 (the length of the diagonal of a unit square) 1.41421356237309... (≈ 1.414) 1.4B79170A07B857... (≈ 1.5)
√3 (the length of the diagonal of a unit cube, or twice the height of an equilateral triangle of unit side) 1.73205080756887... (≈ 1.732) 1.894B97BB968704... (≈ 1.895)
√5 (the length of the diagonal of a 1×2 rectangle) 2.2360679774997... (≈ 2.236) 2.29BB132540589... (≈ 2.2A)
φ (phi, the golden ratio = (1+√5)2) 1.6180339887498... (≈ 1.618) 1.74BB6772802A4... (≈ 1.75)
Trascendental irrational number In decimal In duodecimal / dozenal
π (pi, the ratio of circumference to diameter) 3.1415926535897932384626433
8327950288419716939937510...
(≈ 3.1416)
3.184809493B918664573A6211B
B151551A05729290A7809A492...
(≈ 3.1848)
e (the base of the natural logarithm) 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 constant (the status of which as a rational or irrational number is not yet known), are: The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ... For other uses, see Square. ... The square root of 3 is equal to the length across the flat sides of a regular hexagon with sides of length 1. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Height is the measurement of distance between a specified point and a corresponding plane of reference. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. ... A diagonal can refer to a line joining two nonadjacent vertices of a polygon or polyhedron, or in contexts any upward or downward sloping line. ... A 5 by 4 rectangle In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ... Not to be confused with Golden mean (philosophy), the felicitous middle between two extremes, Golden numbers, an indicator of years in astronomy and calendar studies, or the Golden Rule. ... When a circles diameter is 1, its circumference is Ï€. Pi or Ï€ is the ratio of a circles circumference to its diameter in Euclidean geometry, approximately 3. ... The circumference is the distance around a closed curve. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ... The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is an irrational constant approximately equal to 2. ... The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is γ ≈ 0. ...

Number In decimal In duodecimal / dozenal
γ (the limiting difference between the harmonic series and the natural logarithm) 0.57721566490153... (~ 0.577) 0.6B15188A6760B3... (~ 0.7)

The Euler-Mascheroni constant is a mathematical constant, used mainly in number theory, and is defined as the limiting difference between the harmonic series and the natural logarithm: Its approximate value is γ ≈ 0. ... See harmonic series (music) for the (related) musical concept. ...

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. 1935 (MCMXXXV) was a common year starting on Tuesday (link will display full calendar). ...


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, 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 for its resemblance to the Roman numeral X, and Image:Scripte.png as the first letter of the word "eleven". In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ... The vigesimal or base-20 numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). ... A script letter X An glyph from a computer typeface, modified by Daniel P. B. Smith with Photoshop to give the desired appearance. ... A script letter E An glyph from a computer typeface, modified by Daniel P. B. Smith with Photoshop to give the desired appearance. ... A script letter X An glyph from a computer typeface, modified by Daniel P. B. Smith with Photoshop to give the desired appearance. ... A script letter E An glyph from a computer typeface, modified by Daniel P. B. Smith with Photoshop to give the desired appearance. ...


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). Categories: Stub ... This article is about the typographical symbol. ... Number sign is one name for the symbol #, and is the preferred Unicode name for the codepoint represented by that glyph. ... For other uses, see Telephone (disambiguation). ... A dial is a generally a flat disk, often with numbers or similar markings on it, used for displaying the setting or output of a timepiece, radio or measuring instrument In telephony and telecommunications in connection with a telephone, a dial refers, in older telephones, to a rotating disk with...


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. Multiplication Rock was the mathematical arm of Schoolhouse Rock, developed in the early 1970s. ...


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: This page is a candidate for speedy deletion. ...

The duodecimal tables are easy to master, easier than the decimal ones; and in elementary teaching they would be so much more interesting, since young children would find more fascinating things to do with twelve rods or blocks than with ten. Anyone having these tables at command will do these calculations more than one-and-a-half times as fast in the duodecimal scale as in the decimal. This is my experience; I am certain that even more so it would be the experience of others.
 
— A. C. Aitken, in The Listener, January 25th, 1962 [1]
But the final quantitative advantage, in my own experience, is this: in varied and extensive calculations of an ordinary and not unduly complicated kind, carried out over many years, I come to the conclusion that the efficiency of the decimal system might be rated at about 65 or less, if we assign 100 to the duodecimal.
 
— A. C. Aitken, The Case Against Decimalisation (Edinburgh / London: Oliver & Boyd, 1962) [2]

See also

A senary numeral system is a base-six numeral system. ... As there are 24 hours in a day a numbering system based upon 24, and as the base 12 is convenient here some examples of the base 24 (quadrovigesimal) system. ... Base 36 is a positional numeral system using 36 as the radix. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... 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. ...

References

  1. ^ Matsushita, Shuji (1998), "Decimal vs. Duodecimal: An interaction between two systems of numeration", 2nd Meeting of the AFLANG, October 1998, Tokyo, <http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html>. Retrieved on 17 March 2008 
  2. ^ Mazaudon, Martine (2002), "Les principes de construction du nombre dans les langues tibéto-birmanes", in François, Jacques, La Pluralité, Leuven: Peeters, pp. 91-119, ISBN 9042912952, <http://halshs.archives-ouvertes.fr/docs/00/16/68/91/PDF/numerationTB_SLP.pdf> 

External links


  Results from FactBites:
 
Duodecimal - Wikipedia, the free encyclopedia (1506 words)
In a duodecimal place system, ten is written as A, eleven is written as B, twelve is written as 10.
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.
duodecimal - definition of duodecimal in Encyclopedia (826 words)
Use of the base 12 number system is not common, but at least one example of duodecimal numerals is in use in the Chepang language of Nepal.
Being a versatile denominator in fraction may explain why we have 12 inches in a foot, 12 ounces in a troy pound, 12 old British pence in a shilling, 12 items in a dozen, 12 dozens in a gross, 12 gross in a great gross, etc.
Advocates of duo-decimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.
  More results at FactBites »

 
 

COMMENTARY     


Share your thoughts, questions and commentary here
Your name
Your comments

Want to know more?
Search encyclopedia, statistics and forums:

 


Press Releases |  Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms, 1022, m