Duodecimal

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The duodecimal system (also known as base 12 or dozenal) is a positional notation numeral system using twelve as its base. In this system, the number ten may be written by a rotated "2" (2) and the number eleven by a rotated "3" (3). This notation was introduced by Sir Isaac Pitman,[1] these digit forms are available as Unicode characters on computerized systems since June 2015[2] as ↊ (Code point 218A) and ↋ (Code point 218B), respectively.[3] Other notations use "A", "T", or "X" for ten and "B" or "E" for eleven, the number twelve (that is, the number written as "12" in the base ten numerical system) is instead written as "10" in duodecimal (meaning "1 dozen and 0 units", instead of "1 ten and 0 units"), whereas the digit string "12" means "1 dozen and 2 units" (i.e. the same number that in decimal is written as "14"). Similarly, in duodecimal "100" means "1 gross", "1000" means "1 great gross", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth").

The number twelve, a superior highly composite number, is the smallest number with four non-trivial factors (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the subitizing range. As a result of this increased factorability of the radix and its divisibility by a wide range of the most elemental numbers (whereas ten has only two non-trivial factors: 2 and 5, and not 3, 4, or 6), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table, as a result, duodecimal has been described as the optimal number system.[4] Of its factors, 2 and 3 are prime, which means the reciprocals of all 3-smooth numbers (such as 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, ...) have a terminating representation in duodecimal. In particular, the five most elementary fractions (​ 12, ​ 13, ​ 23, ​ 14 and ​ 34) 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 (because it is the least common multiple of 3 and 4). This all makes it a more convenient number system for computing fractions than most other number systems in common use, such as the decimal, vigesimal, binary, octal and hexadecimal systems. Although the trigesimal and sexagesimal systems (where the reciprocals of all 5-smooth numbers terminate) do even better in this respect, this is at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.

Origin[edit]

In this section, numerals are based on decimal places[2]. 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 (Gure-Kahugu), Piti, and the Nimbia dialect of Gwandara;[5] the Chepang language of Nepal[6] and the Mahl language of Minicoy Island in India are known to use duodecimal numerals.

Germanic languages have special words for 11 and 12, such as eleven and twelve in English. However, they are considered to come from Proto-Germanic *ainlif and *twalif (respectively one left and two left), both of which were decimal.[7][8]

Historically, units of time in many civilizations are duodecimal. There are twelve signs of the zodiac, twelve months in a year, and the Babylonians had twelve hours in a day (although at some point this was changed to 24). Traditional Chinese calendars, clocks, and compasses are based on the twelve Earthly Branches. There are 12 inches in an imperial foot, 12 troy ounces in a troy pound, 12 old British pence in a shilling, 24 (12×2) hours in a day, and many other items counted by the dozen, gross (144, square of 12) or great gross (1728, cube of 12). The Romans used a fraction system based on 12, including the uncia which became both the English words ounce and inch. Pre-decimalisation, Ireland and the United Kingdom used a mixed duodecimal-vigesimal currency system (12 pence = 1 shilling, 20 shillings or 240 pence to the pound sterling or Irish pound), and Charlemagne established a monetary system that also had a mixed base of twelve and twenty, the remnants of which persist in many places.

Table of units from a base of 12
Relative
value
French unit
of length
English unit
of length
English
(Troy) unit
of weight
Roman unit
of weight
English unit
of mass
120 pied foot pound libra
12−1 pouce inch ounce uncia slinch
12−2 ligne line 2 scruples 2 scrupula slug
12−3 point point seed siliqua

The importance of 12 has been attributed to the number of lunar cycles in a year, and also to the fact that humans have 12 finger bones (phalanges) on one hand (three on each of four fingers),[9][10] it is possible to count to 12 with the thumb acting as a pointer, touching each finger bone in turn. A traditional finger counting system still in use in many regions of Asia works in this way, and could help to explain the occurrence of numeral systems based on 12 and 60 besides those based on 10, 20 and 5; in this system, the one (usually right) hand counts repeatedly to 12, displaying the number of iterations on the other (usually left), until five dozens, i. e. the 60, are full.[11][12]

Notations and pronunciations[edit]

Transdecimal symbols[edit]

In a duodecimal place system twelve is written as 10, but there are numerous proposals for how to write ten and eleven,[13] the simplified notations use only basic and easy to access letters such as T and E (for ten and eleven), X and Z, t and e, d and k, others use A and B or a and b as in the hexadecimal system. Some employ Greek letters such as δ (standing for Greek δέκα 'ten') and ε (for Greek ένδεκα 'eleven'), or τ and ε.[13] Frank Emerson Andrews, an early American advocate for duodecimal, suggested and used in his book New Numbers an X (from the Roman numeral for ten) and a script E (ℰ, U+2130).[14]

Dozenal gb 10.svg Dozenal gb 11.svg
Dozenal us 10.svg Dozenal us 11.svg

The Dozenal Society of Great Britain proposes a rotated digit two 2 (↊, U+218A) for ten and a reversed or rotated digit three 3 (↋, U+218B) for eleven.[13] This notation was introduced by Sir Isaac Pitman.[13][15]

Until 2015, the Dozenal Society of America (DSA) used Dozenal us 10.svg and Dozenal us 11.svg, the symbols devised by William Addison Dwiggins.[13][16] After the Pitman digits (↊, U+218A and ↋, U+218B) were added to Unicode in 2015[2][17], the DSA took a vote and then began publishing content using the Pitman digits instead.[18][19] They still use the letters X and E as the equivalent in ASCII text.

Other proposals are more creative or aesthetic, for example, Edna Kramer in her 1951 book The Main Stream of Mathematics used a six-pointed asterisk (sextile) ⚹ for ten and a hash (or octothorpe) # for eleven.[13] The symbols were chosen because they are available in typewriters and already present in telephone dials,[13] this notation was used in publications of the Dozenal Society of America in the period 1974–2008.[20][21] Many don't use any of the Hindu-Arabic symbols, under the principle of "separate identity."[13]

Base notation[edit]

There are also varying proposals of how to distinguish a duodecimal number from a decimal one, or one in a different base, they include italicizing duodecimal numbers ( 54 = 64 ), adding a "Humphrey point" (a semicolon ";" instead of a decimal point "." ) to duodecimal numbers ( 54; = 64. ) ( 54;0 = 64.0 ), or some combination of the two. More also add extra marking to one or more bases. Others use subscript or affixed labels to indicate the base, allowing for more than decimal and duodecimal to be represented:[19]

Common Base Abb. Letter Cardinal Decimal Duodecimal
binary bin b two 2 2
octal oct o eight 8 8
decimal dec d ten 10
dozenal (duodecimal) doz z twelve 12 10
hexadecimal hex x sixteen 16 14

This allows one to write "54z = 64d," "54twelve = 64ten" or "doz 54 = dec 64." In programming, binary, octal, and hexadecimal often use a similar scheme: a binary number starts with 0b, octal with 0o, and hexadecimal with 0x.

Pronunciation[edit]

The Dozenal Society of America suggests the pronunciation of ten and eleven as "dek" and "el", each order has its own name and the prefix e- is added for fractions,[16][22] the symbol corresponding to the decimal point or decimal comma, separating the whole number part from the fractional part, is the semicolon ";". The overall system is:[16]

Duodecimal Name Decimal Duodecimal fraction Name
1 one 1
10 do 12 0;1 edo
100 gro 144 0;01 egro
1,000 mo 1,728 0;001 emo
10,000 do-mo 20,736 0;000,1 edo-mo
100,000 gro-mo 248,832 0;000,01 egro-mo
1,000,000 bi-mo 2,985,984 0;000,001 ebi-mo
1,000,000,000 tri-mo 5,159,780,352 0;000,000,001 etri-mo

Multiple digits in this are pronounced differently. 12 is "one do two", 30 is "three do", 100 is "one gro", BA9 (ET9) is "el gro dek do nine", B8,65A,300 (E8,65T,300) is "el do eight bi-mo, six gro five do dek mo, three gro", and so on.[22]

Advocacy and "dozenalism"[edit]

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.

A duodecimal clockface as in the logo of the Dozenal Society of America, here used to denote musical keys

Both 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" to avoid the more overtly base-ten terminology. It should be noted that the etymology of 'dozenal' is itself also an expression based on base-ten terminology since 'dozen' is a direct derivation of the French word 'douzaine' which is a derivative of the French word for twelve, douze which is related to the old French word 'doze' from Latin 'duodecim'.

It has been suggested by some members of the Dozenal Society of America and Duodecimal Society of Great Britain that a more apt word would be 'uncial'. Uncial is a derivation of the Latin word 'one-twelfth' which is 'uncia' and also the base-twelve analogue of the Latin word 'one-tenth' which is 'decima'; in the same manner as decimal comes from the Latin word for one-tenth decima, (Latin for ten was decem), the direct analogue for a base-twelve system is uncial. An early use of this word can be found in Vol 1 Issue 2 of The Duodecimal Bulletin of the DSA dated June 1945 in which a submission on page 9 by a Pvt William S. Crosby titled "The Uncial Jottings of a Harried Infantryman", he includes the same argument for the word 'uncial', although not accepted by either of these two 'Uncial' societies, the use is beginning to grow.

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

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, "Twelves and Tens" in The Listener (January 25, 1962)[23]

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 (1962)[24]

In Jorge Luis Borges' short story Tlön, Uqbar, Orbis Tertius Herbert Ashe, a melancholy English engineer, working for the Southern Argentine Railway company, is converting a duodecimal number system to a hexadecimal system. He leaves behind on his death in 1937 a manuscript Orbis Tertius that posthumously identifies him as one of the anonymous authors of the encyclopaedia of Tlön.

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

In Lee Carroll's Kryon: Alchemy of the Human Spirit, a chapter is dedicated to the advantages of the duodecimal system. The duodecimal system is supposedly suggested by Kryon (a fictional entity believed in by New Age circles) for all-round use, aiming at better and more natural representation of nature of the Universe through mathematics. An individual article "Mathematica" by James D. Watt (included in the above publication) exposes a few of the unusual symmetry connections between the duodecimal system and the golden ratio, as well as provides numerous number symmetry-based arguments for the universal nature of the base-12 number system.[25]

In "Little Twelvetoes", American television series Schoolhouse Rock! portrayed an alien child using base-twelve arithmetic, using "dek", "el" and "doh" as names for ten, eleven and twelve, and Andrews' script-X and script-E for the digit symbols.[26]

In computing[edit]

In March 2013, a proposal was submitted to include the digit forms for ten and eleven propagated by the Dozenal Societies of Great Britain and America in the Unicode Standard.[27] Of these, the British forms were accepted for encoding as characters at code points U+218A turned digit two (↊) and U+218B turned digit three (↋) They have been included in the Unicode 8.0 release in June 2015.[2][17]

Unicode points U+218C and U+218D seem to be reserved for the Dwiggins digits (stylized X and E).[28]

Few fonts support these new characters, but Abibas, EB Garamond, Everson Mono, Squarish Sans CT, and Symbola do.

Also, the turned digits two and three are available in LaTeX as \textturntwo and \textturnthree.[29]

Duodecimal clock[edit]

Duodecimal metric systems[edit]

Systems of measurement proposed by dozenalists include:

  • Tom Pendlebury's TGM system[30]
  • Takashi Suga's Universal Unit System[31]

Comparison to other numeral systems[edit]

A duodecimal multiplication table

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 (base 20) 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, and so the digit set and the multiplication table are much larger. Binary has only two factors, 1 and 2, the latter being prime. Hexadecimal (base 16) has five factors, adding 4, 8 and 16 to those of 2, but no additional prime. Trigesimal (base 30) is the smallest system that has three different prime factors (all of the three smallest primes: 2, 3 and 5) and it has eight factors in total (1, 2, 3, 5, 6, 10, 15, and 30). Sexagesimal—which the ancient Sumerians and Babylonians among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors. The smallest system that has four different prime factors is base 210 and the pattern follows the primorials; in all base systems, there are similarities to the representation of multiples of numbers which are one less than the base.

Conversion tables to and from decimal[edit]

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 duodecimal number between 0.01 and ƐƐƐ,ƐƐƐ.ƐƐ to decimal, or any decimal number between 0.01 and 999,999.99 to duodecimal. To use them, the given number must first be decomposed 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 the digit conversion tables can be used to obtain the equivalent value in the target base for each digit. If the given number is in duodecimal and the target base is decimal, we get:

(duodecimal) 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, because the summands are already converted to base ten, the usual decimal arithmetic is used to perform the addition and recompose the number, arriving at the conversion result:

Duodecimal  ----->  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, (duodecimal) 123,456.78 equals (decimal) 296,130.638 ≈ 296,130.64

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

(decimal) 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = (duodecimal) 49,ᘔ54 + Ɛ,6ᘔ8 + 1,8ᘔ0 + 294 + 42 + 6 + 0.849724972497249724972497... + 0.0Ɛ62ᘔ68781Ɛ05915343ᘔ0Ɛ62...

However, in order to do this sum and recompose the number, now the addition tables for the duodecimal system have to be used, 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 duodecimal as well; in decimal, 6 + 6 equals 12, but in duodecimal it equals 10; so, if using decimal arithmetic with duodecimal numbers one would arrive at an incorrect result. Doing the arithmetic properly in duodecimal, one gets the result:

  Decimal  ----->  Duodecimal

  100,000     =     49,ᘔ54
   20,000     =      Ɛ,6ᘔ8
    3,000     =      1,8ᘔ0
      400     =        294
       50     =         42
 +      6     =   +      6
        0.7   =          0.849724972497249724972497...
        0.08  =          0.0Ɛ62ᘔ68781Ɛ05915343ᘔ0Ɛ62...
--------------------------------------------------------
  123,456.78  =     5Ɛ,540.943ᘔ0Ɛ62ᘔ68781Ɛ05915343ᘔ...

That is, (decimal) 123,456.78 equals (duodecimal) 5Ɛ,540.943ᘔ0Ɛ62ᘔ68781Ɛ059153... ≈ 5Ɛ,540.94

Duodecimal to decimal digit conversion[edit]

Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec.
1,000,000 2,985,984 100,000 248,832 10,000 20,736 1,000 1,728 100 144 10 12 1 1 0.1 0.083 0.01 0.00694
2,000,000 5,971,968 200,000 497,664 20,000 41,472 2,000 3,456 200 288 20 24 2 2 0.2 0.16 0.02 0.0138
3,000,000 8,957,952 300,000 746,496 30,000 62,208 3,000 5,184 300 432 30 36 3 3 0.3 0.25 0.03 0.02083
4,000,000 11,943,936 400,000 995,328 40,000 82,944 4,000 6,912 400 576 40 48 4 4 0.4 0.3 0.04 0.027
5,000,000 14,929,920 500,000 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
6,000,000 17,915,904 600,000 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
7,000,000 20,901,888 700,000 1,741,824 70,000 145,152 7,000 12,096 700 1,008 70 84 7 7 0.7 0.583 0.07 0.04861
8,000,000 23,887,872 800,000 1,990,656 80,000 165,888 8,000 13,824 800 1,152 80 96 8 8 0.8 0.6 0.08 0.05
9,000,000 26,873,856 900,000 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
ᘔ,000,000 29,859,840 ᘔ00,000 2,488,320 ᘔ0,000 207,360 ᘔ,000 17,280 ᘔ00 1,440 ᘔ0 120 10 0.ᘔ 0.83 0.0ᘔ 0.0694
Ɛ,000,000 32,845,824 Ɛ00,000 2,737,152 Ɛ0,000 228,096 Ɛ,000 19,008 Ɛ00 1,584 Ɛ0 132 Ɛ 11 0.Ɛ 0.916 0.0Ɛ 0.07638

Decimal to duodecimal digit conversion[edit]

Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod.
100,000 49,ᘔ54 10,000 5,954 1,000 6Ɛ4 100 84 10 1 1 0.1 0.12497 0.01 0.015343ᘔ0Ɛ62ᘔ68781Ɛ059
200,000 97,8ᘔ8 20,000 Ɛ,6ᘔ8 2,000 1,1ᘔ8 200 148 20 18 2 2 0.2 0.2497 0.02 0.02ᘔ68781Ɛ05915343ᘔ0Ɛ6
300,000 125,740 30,000 15,440 3,000 1,8ᘔ0 300 210 30 26 3 3 0.3 0.37249 0.03 0.043ᘔ0Ɛ62ᘔ68781Ɛ059153
400,000 173,594 40,000 1Ɛ,194 4,000 2,394 400 294 40 34 4 4 0.4 0.4972 0.04 0.05915343ᘔ0Ɛ62ᘔ68781Ɛ
500,000 201,428 50,000 24,Ɛ28 5,000 2,ᘔ88 500 358 50 42 5 5 0.5 0.6 0.05 0.07249
600,000 24Ɛ,280 60,000 2ᘔ,880 6,000 3,580 600 420 60 50 6 6 0.6 0.7249 0.06 0.08781Ɛ05915343ᘔ0Ɛ62ᘔ6
700,000 299,114 70,000 34,614 7,000 4,074 700 4ᘔ4 70 5ᘔ 7 7 0.7 0.84972 0.07 0.0ᘔ0Ɛ62ᘔ68781Ɛ05915343
800,000 326,Ɛ68 80,000 3ᘔ,368 8,000 4,768 800 568 80 68 8 8 0.8 0.9724 0.08 0.0Ɛ62ᘔ68781Ɛ05915343ᘔ
900,000 374,ᘔ00 90,000 44,100 9,000 5,260 900 630 90 76 9 9 0.9 0.ᘔ9724 0.09 0.10Ɛ62ᘔ68781Ɛ05915343ᘔ

Conversion of powers[edit]

Exponent b=2 b=3 b=4 b=5 b=6 b=7
Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod.
b6 64 54 729 509 4,096 2,454 15,625 9,061 46,656 23,000 117,649 58,101
b5 32 28 243 183 1,024 714 3,125 1,985 7,776 4,600 16,807 9,887
b4 16 14 81 69 256 194 625 441 1,296 900 2,401 1,481
b3 8 8 27 23 64 54 125 ᘔ5 216 160 343 247
b2 4 4 9 9 16 14 25 21 36 30 49 41
b1 2 2 3 3 4 4 5 5 6 6 7 7
b−1 0.5 0.6 0.3 0.4 0.25 0.3 0.2 0.2497 0.16 0.2 0.142857 0.186ᘔ35
b−2 0.25 0.3 0.1 0.14 0.0625 0.09 0.04 0.05915343ᘔ0
Ɛ62ᘔ68781Ɛ
0.027 0.04 0.0204081632653
06122448979591
836734693877551
0.02Ɛ322547ᘔ05ᘔ
644ᘔ9380Ɛ908996
741Ɛ615771283Ɛ
Exponent b=8 b=9 b=10 b=11 b=12
Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod. Dec. Duod.
b6 262,144 107,854 531,441 217,669 1,000,000 402,854 1,771,561 715,261 2,985,984 1,000,000
b5 32,768 16,Ɛ68 59,049 2ᘔ,209 100,000 49,ᘔ54 161,051 79,24Ɛ 248,832 100,000
b4 4,096 2,454 6,561 3,969 10,000 5,954 14,641 8,581 20,736 10,000
b3 512 368 729 509 1,000 6Ɛ4 1,331 92Ɛ 1,728 1,000
b2 64 54 81 69 100 84 121 ᘔ1 144 100
b1 8 8 9 9 10 11 Ɛ 12 10
b−1 0.125 0.16 0.1 0.14 0.1 0.12497 0.09 0.1 0.083 0.1
b−2 0.015625 0.023 0.012345679 0.0194 0.01 0.015343ᘔ0Ɛ6
2ᘔ68781Ɛ059
0.00826446280
99173553719
0.0123456789Ɛ 0.00694 0.01

Some properties[edit]

(In this section, all numbers are written with duodecimal, using ᘔ for ten and Ɛ for eleven)

All squares end with square digits (i.e. end with 0, 1, 4 or 9), if n is divisible by both 2 and 3, then n2 ends with 0, if n is not divisible by 2 or 3, then n2 ends with 1, if n is divisible by 2 but not by 3, then n2 ends with 4, if n is not divisible by 2 but by 3, then n2 ends with 9. If the unit digit of n2 is 0, then the dozens digit of n2 is either 0 or 3, if the unit digit of n2 is 1, then the dozens digit of n2 is even, if the unit digit of n2 is 4, then the dozen digit of n2 is 0, 1, 4, 5, 8 or 9, if the unit digit of n2 is 9, then the dozen digit of n2 is either 0 or 6. (More specially, all squares of (primes ≥ 5) end with 1)

The digital root of a square is 1, 3, 4, 5 or Ɛ.

No repdigits with more than one digit are squares, in fact, a square cannot end with three same digits except 000. (In contrast, in the decimal system squares may end in 444)

No four-digit palindromic numbers are squares.

A cube can end with all digits except 2, 6 and ᘔ (in fact, no perfect powers end with 2, 6 or ᘔ), if n is not congruent to 2 mod 4, then n3 ends with the same digit as n; if n is congruent to 2 mod 4, then n3 ends with the digit (the last digit of n +- 6).

The digital root of a cube can be any number.

If k≥2, then nk+2 ends with the same digit as nk.

Squares (and every powers) of 0, 1, 4, 9, 54, 69, 369, 854, 3854, 8369, Ɛ3854, 1Ɛ3854, ᘔ08369, ... end with the same digits as the number itself.

Except for 6 and 24, all even perfect numbers end with 54. Besides, except for 6, 24 and 354, all even perfect numbers end with 054 or 854.

The digital root of an even perfect number is 1, 4, 6 or ᘔ.

The unit digits of powers of 2 are 1, 2, 4, 8, 4, 8, 4, 8, 4, ... (periodic with period 2)

The unit digits of powers of 3 are 1, 3, 9, 3, 9, 3, 9, 3, 9, ... (periodic with period 2)

The unit digits of powers of 5 are 1, 5, 1, 5, 1, 5, 1, 5, 1, ... (periodic with period 2)

The unit digits of powers of 7 are 1, 7, 1, 7, 1, 7, 1, 7, 1, ... (periodic with period 2)

The unit digits of powers of Ɛ are 1, Ɛ, 1, Ɛ, 1, Ɛ, 1, Ɛ, 1, ... (periodic with period 2)

The period of the unit digits of powers of a number must be a divisor of 2 (= λ(10), where λ is the Carmichael function).

The final two digits of powers of 2 are 01, 02, 04, 08, 14, 28, 54, ᘔ8, 94, 68, 14, 28, 54, ᘔ8, 94, 68, 14, ... (periodic with period 6)

The final two digits of powers of 3 are 01, 03, 09, 23, 69, 83, 09, 23, 69, 83, 09, 23, 69, ... (periodic with period 4)

The final two digits of powers of 5 are 01, 05, 21, ᘔ5, 41, 85, 61, 65, 81, 45, ᘔ1, 25, 01, 05, 21, ᘔ5, 41, ... (periodic with period 10)

The final two digits of powers of 7 are 01, 07, 41, 47, 81, 87, 01, 07, 41, 47, 81, 87, 01, ... (periodic with period 6)

The final two digits of powers of Ɛ are 01, 0Ɛ, ᘔ1, 2Ɛ, 81, 4Ɛ, 61, 6Ɛ, 41, 8Ɛ, 21, ᘔƐ, 01, 0Ɛ, ᘔ1, 2Ɛ, 81, ... (periodic with period 10)

The period of the final two digits of powers of a number must be a divisor of 20 (= λ(100)).

More generally, for every n≥1, the period of the final n digits of powers of a number must be a divisor of 2×10n−1 (= λ(10n)).

The digital roots of powers of 2 are 1, 2, 4, 8, 5, ᘔ, 9, 7, 3, 6, 1, 2, 4, 8, 5, ᘔ, 9, 7, 3, 6, 1, ... (periodic with period ᘔ)

The digital roots of powers of 3 are 1, 3, 9, 5, 4, 1, 3, 9, 5, 4, 1, ... (periodic with period 5)

The digital roots of powers of 5 are 1, 5, 3, 4, 9, 1, 5, 3, 4, 9, 1, ... (periodic with period 5)

The digital roots of powers of 7 are 1, 7, 5, 2, 3, ᘔ, 4, 6, 9, 8, 1, 7, 5, 2, 3, ᘔ, 4, 6, 9, 8, 1, ... (periodic with period ᘔ)

The digital roots of powers of Ɛ are 1, Ɛ, Ɛ, Ɛ, Ɛ, Ɛ, ... (periodic with period 1)

The period of the digital roots of powers of a number must be a divisor of ᘔ (= λ(Ɛ)).

The unit digit of a Fibonacci number can be any digit except 6 (interestingly, if the unit digit of a Fibonacci number is 0, then the dozens digit of this number must also be 0, thus, all Fibonacci numbers divisible by 6 are also divisible by 100), and the unit digit of a Lucas number cannot be 0 or 9 (thus, no Lucas number is divisible by 10).

The unit digits of Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 1, 9, ᘔ, 7, 5, 0, 5, 5, ᘔ, 3, 1, 4, 5, 9, 2, Ɛ, 1, 0, 1, 1, 2, 3, 5, ... (periodic with period 20)

The unit digits of Lucas numbers are 2, 1, 3, 4, 7, Ɛ, 6, 5, Ɛ, 4, 3, 7, ᘔ, 5, 3, 8, Ɛ, 7, 6, 1, 7, 8, 3, Ɛ, 2, 1, 3, 4, 7, Ɛ, ... (periodic with period 20)

The final two digits of Fibonacci numbers are 00, 01, 01, 02, 03, 05, 08, 11, 19, 2ᘔ, 47, 75, 00, 75, 75, 2ᘔ, ᘔ3, 11, Ɛ4, 05, Ɛ9, 02, ƐƐ, 01, 00, 01, 01, 02, 03, 05, ... (periodic with period 20)

The final two digits of Lucas numbers are 02, 01, 03, 04, 07, 0Ɛ, 16, 25, 3Ɛ, 64, ᘔ3, 47, 2ᘔ, 75, ᘔ3, 58, 3Ɛ, 97, 16, Ɛ1, 07, Ɛ8, 03, ƐƐ, 02, 01, 03, 04, 07, 0Ɛ, ... (periodic with period 20)

The digital roots of Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 2, ᘔ, 1, Ɛ, 1, 1, 2, 3, 5, ... (periodic with period ᘔ)

The digital roots of Lucas numbers are 2, 1, 3, 4, 7, Ɛ, 7, 7, 3, ᘔ, 2, 1, 3, 4, 7, Ɛ, ... (periodic with period ᘔ)

The period of the unit digit of Fibonacci numbers is 20, the final two digits is also 20, the final three digits is 200, the final four digits is 2000, ..., the final n digits is 2×10n−1 (n≥2).

1001 is the first four-digit palindromic number, and it is also the smallest number expressible as the sum of two cubes in two different ways, i.e. 1001 = 1 + 1000 (=13 + 103) = 509 + 6Ɛ4 (=93 + ᘔ3) (see taxicab number for other numbers), and it is also the smallest absolute Euler pseudoprime, note that there is no absolute Euler-Jacobi pseudoprime and no absolute strong pseudoprime. Since 1001 = 7×11×17, we can use the divisibility rule of 1001 (i.e. form the alternating sum of blocks of three from right to left) for the divisibility rule of 7, 11 and 17. Besides, if 6k+1, 10k+1 and 16k+1 are all primes, then the product of them must be a Carmichael number (absolute Fermat pseudoprime), the smallest case is indeed 1001 (for k = 1), but 1001 is not the smallest Carmichael number (the smallest Carmichael number is 3ᘔ9).

1 + 1 + 1 + 1 + 1 + 1 + ... = −0.6

1 − 1 + 1 − 1 + 1 − 1 + ... = 0.6

1 + 2 + 3 + 4 + 5 + 6 + ... = −0.1

1 − 2 + 3 − 4 + 5 − 6 + ... = 0.3

1 + 2 + 4 + 8 + 14 + 28 + ... = −1

1 − 2 + 4 − 8 + 14 − 28 + ... = 0.4

1/1 + 1/2 + 1/4 + 1/8 + 1/14 + 1/28 + ... = 2

1/1 − 1/2 + 1/4 − 1/8 + 1/14 − 1/28 + ... = 0.8

1 + 3 + 9 + 23 + 69 + 183 + ... = −0.6

1 − 3 + 9 − 23 + 69 − 183 + ... = 0.3

1/1 + 1/3 + 1/9 + 1/23 + 1/69 + 1/183 + ... = 1.6

1/1 − 1/3 + 1/9 − 1/23 + 1/69 − 1/183 + ... = 0.9

1 + 4 + 14 + 54 + 194 + 714 + ... = −0.4

1/1 + 1/4 + 1/14 + 1/54 + 1/194 + 1/714 + ... = 1.4

(Note that all of these numbers either are integers or have only one digit after the duodecimal point)

All prime numbers end with prime digits or 1 (i.e. end with 1, 2, 3, 5, 7 or Ɛ), more generally, except for 2 and 3, all prime numbers end with 1, 5, 7 or Ɛ (1 and all prime digits that do not divide 10), since all prime numbers other than 2 and 3 are coprime to 10.

The density of primes end with 1 is a relatively low, but the density of primes end with 5, 7 and Ɛ are nearly equal. (since all prime squares except 4 and 9 end with 1, no prime squares end with 5, 7 or Ɛ)

Except (3, 5), all twin primes end with (5, 7) or (Ɛ, 1).

All palindromic primes except 11 has an odd number of digits, since all even-digit palindromic numbers are divisible by 11.

All lucky numbers end with digit 1, 3, 7 or 9.

Except for 3, all Fermat primes end with 5.

Except for 3, all Mersenne primes end with 7.

Except for 2 and 3, all Sophie Germain primes end with 5 or Ɛ.

Except for 5 and 7, all safe primes end with Ɛ.

A prime p is Gaussian prime if and only if p ends with 7 or Ɛ (or p=3).

A prime p is Eisenstein prime if and only if p ends with 5 or Ɛ (or p=2).

A prime p can be written as x2 + y2 if and only if p ends with 1 or 5 (or p=2).

A prime p can be written as x2 + 3y2 if and only if p ends with 1 or 7 (or p=3).

All full reptend primes end with 5 or 7. (in fact, for all primes p ≥ 5, (p-1)/(the period length of 1/p) is odd if and only if p is end with 5 or 7, since 10 is a quadratic nonresidue mod p if and only if p is end with 5 or 7, by quadratic reciprocity)

If p is a safe prime other than 5, 7 and Ɛ, then the period length of 1/p is (p-1)/2. (this is not true for all primes ends with Ɛ (other than Ɛ itself), the first counterexample is p = 2ƐƐ, where the period length of 1/p is only 37)

There are only 9 repunit primes below R1000: R2, R3, R5, R17, R81, R91, R225, R255 and R4ᘔ5 (Rn is the repunit with length n). If p is a Sophie Germain prime other than 2, 3 and 5, then Rp is divisible by 2p+1, thus Rp is not prime.

By Fermat's little theorem, if p is a prime other than 2, 3 and Ɛ, then p divides the repunit with length p−1. (The converse is not true, the first counterexample is 55, which is composite (equals 5×11) but divides the repunit with length 54, the counterexamples up to 1000 are 55, 77, Ɛ1, 101, 187, 275, 4ᘔ7, 777, 781, Ɛ55, they are exactly the Fermat pseudoprimes for base 10 which are not divisible by Ɛ) Thus, we can prove that every positive integer coprime to 10 has a repunit multiple, and every positive integer has a multiple uses only 0's and 1's. (the smallest multiple of n uses only 0's and 1's are 1, 10, 10, 10, 101, 10, 1001, 100, 100, 1010, 11111111111, 10, 11, 10010, 1010, 100, 10111, 100, 1001, 1010, 10010, 111111111110, 11101, 100, 110111, 110, 1000, 10010, 101, 1010, 101011, 1000, 111111111110, 101110, 101101, 100, ...)

n 1 5 7 Ɛ 11 15 17 21 25 27
smallest k such that k×n is a repunit 1 275 1ᘔ537 123456789Ɛ 1 92ᘔ79Ɛ43715865 8327 69Ɛ63848Ɛ 634ᘔ159788253ᘔ72Ɛ1 55 509867481Ɛ793ᘔᘔ5ᘔ1243628Ɛ317 45ᘔ3976ᘔ7Ɛ
the length of the repunit k×n 1 4 6 Ɛ 2 14 6 Ɛ 18 4 26 10

(this k is usually not prime, in fact, this k is not prime for all numbers n < 100 which are coprime to 10 except n = 55, and for n < 1000 which is coprime to 10, this k is prime only for n = 55, 101, 19Ɛ, 275 and 46Ɛ, and only 19Ɛ and 46Ɛ are itself prime, other 3 numbers are 5×11, 5×25 and 11×25, and this k for these n are successively 25, 11 and 5, which makes k×n = R4 = 1111 = 5×11×25, besides, this k for n = 46Ɛ is 2ᘔ3Ɛ, which makes k×n = R7 = 1111111, a repunit semiprime, and this k for n = 19Ɛ is a ᘔ8-digit prime number, with k×n = RᘔƐ, another repunit semiprime)

Write the recurring digits of 1/45 to 44/45, we get prime reciprocal magic square (its magic number is 1Ɛᘔ), it is conjectured that there are infinitely many such primes, but 45 is the only such prime below 1000.

The largest non-repunit permutable prime is 5ƐƐƐ, and the largest non-repunit circular prime is 115Ɛ77.

The largest left-truncatable prime is 471ᘔ34ᘔ164259Ɛᘔ16Ɛ324ᘔƐ8ᘔ32Ɛ7817, and the largest right-truncatable prime is 375ƐƐ5Ɛ515.

The only two base 10 Wieferich primes up to 1010 are 1685 and 5Ɛ685, note that both of the numbers end with 685, and it is conjectured that all base 10 Wieferich primes end with 685.

The only happy numbers below 1000 are 1, 10, 100, 222, 488, 848 and 884, almost all natural numbers are unhappy.

There are 1, 2, 3, 5 and 6-digit (but not 4-digit) narcissistic numbers, there are totally 73 narcissistic numbers, the largest of which has 43 digits.

Most numbers that end with 2 are nontotient (in fact, all nontotients < 58 except 2ᘔ end with 2), except 2 itself, the first counterexample is 92, which equals φ(ᘔ1) = φ(Ɛ2) and φ(182) = φ(2×Ɛ2), next counterexample is 362, which equals φ(381) = φ(1Ɛ2) and φ(742) = φ(2×1Ɛ2), there are only 9 such numbers ≤ 10000 (the number 2 itself is not counted), all such numbers (except the number 2 itself) are of the form φ(p2), where p is a prime ends with Ɛ.

All orders of non-cyclic simple group end with 0 (thus, all orders of unsolvable group end with 0), however, we can prove that no groups with order 10, 20, 30 or 40 are simple, thus 50 is the smallest order of non-cyclic simple group (thus, all groups with order < 50 are solvable), (50 is the order of the alternating group A5, which is a non-cyclic simple group, and thus an unsolvable group) next three orders of non-cyclic simple group are 120, 260 and 360. (Edit: I found that this is not completely true (although this is true for all orders ≤ 14000), the smallest counterexample is 14ᘔ28, however, all such orders are divisible by 4 and either 3 or 5 (i.e. divisible by either 10 or 18), and all such orders have at least 3 distinct prime factors, by these conditions, the smallest possible such order is indeed 50 = 22 × 3 × 5, next possible such order is 70 = 22 × 3 × 7, however, by Sylow theorems, the number of Sylow 7-subgroups of all groups with order 70 (i.e. the number of subgroups with order 7 of all groups with order 70) is congruent to 1 mod 7 and divides 70, hence must be 1, thus the subgroup with order 7 is a normal subgroup of the group with order 70, thus all groups with order 70 have a nontrivial normal subgroup and cannot be simple groups)

Some special numbers[edit]

(In this section, all numbers are written with duodecimal, using X for ten and E for eleven)

In duodecimal, as in decimal, there are special numbers that exhibit digit patterns when, for example, multiplied by small integers:

2497 * 1 = 2497
2497 * 2 = 4972
2497 * 3 = 7249
2497 * 4 = 9724
2497 * 5 = EEEE
186X35 * 1 = 186X35
186X35 * 2 = 35186X
186X35 * 3 = 5186X3
186X35 * 4 = 6X3518
186X35 * 5 = 86X351
186X35 * 6 = X35186
186X35 * 7 = EEEEEE
8579214E36429X7 * 1 = 8579214E36429X7
8579214E36429X7 * 2 = 14E36429X7085792
8579214E36429X7 * 3 = 214E36429X708579
8579214E36429X7 * 4 = 29X708579214E364
8579214E36429X7 * 5 = 36429X708579214E
8579214E36429X7 * 6 = 429X708579214E36
8579214E36429X7 * 7 = 4E36429X70857921
8579214E36429X7 * 8 = 579214E36429X708
8579214E36429X7 * 9 = 6429X708579214E3
8579214E36429X7 * X = 708579214E36429X
8579214E36429X7 * E = 79214E36429X7085
8579214E36429X7 * 10 = 8579214E36429X70
8579214E36429X7 * 11 = 9214E36429X70857
8579214E36429X7 * 12 = 9X708579214E3642
8579214E36429X7 * 13 = X708579214E36429
8579214E36429X7 * 14 = E36429X708579214
8579214E36429X7 * 15 = EEEEEEEEEEEEEEEE
123456789XE * 1 = 123456789XE
123456789XE * 2 = 2468E13579X
123456789XE * 3 = 36X147E2589
123456789XE * 4 = 4915X26E378
123456789XE * 5 = 5E4X3928167
123456789XE * 6 = 718293X4E56
123456789XE * 7 = 83E72X61945
123456789XE * 8 = 962E851X734
123456789XE * 9 = X8641E97523
123456789XE * X = EX987654312
123456789XE * E = 111111111101
123456789E * 1 = 123456789E (miss digit X)
123456789E * 2 = 2468E1357X (miss digit 9)
123456789E * 3 = 36X147E259 (miss digit 8)
123456789E * 4 = 4915X26E38 (miss digit 7)
123456789E * 5 = 5E4X392817 (miss digit 6)
123456789E * 6 = 718293X4E6 (miss digit 5)
123456789E * 7 = 83E72X6195 (miss digit 4)
123456789E * 8 = 962E851X74 (miss digit 3)
123456789E * 9 = X8641E9753 (miss digit 2)
123456789E * X = EX98765432 (miss digit 1)
123456789E * E = 11111111111
275 * 5 = 1111
275 * X = 2222
275 * 13 = 3333
275 * 18 = 4444
275 * 21 = 5555
275 * 26 = 6666
275 * 2E = 7777
275 * 34 = 8888
275 * 39 = 9999
275 * 42 = XXXX
275 * 47 = EEEE
1X537 * 7 = 111111
1X537 * 12 = 222222
1X537 * 19 = 333333
1X537 * 24 = 444444
1X537 * 2E = 555555
1X537 * 36 = 666666
1X537 * 41 = 777777
1X537 * 48 = 888888
1X537 * 53 = 999999
1X537 * 5X = XXXXXX
1X537 * 65 = EEEEEE
123456789E * E = 11111111111
123456789E * 1X = 22222222222
123456789E * 29 = 33333333333
123456789E * 38 = 44444444444
123456789E * 47 = 55555555555
123456789E * 56 = 66666666666
123456789E * 65 = 77777777777
123456789E * 74 = 88888888888
123456789E * 83 = 99999999999
123456789E * 92 = XXXXXXXXXXX
123456789E * X1 = EEEEEEEEEEE
92X79E43715865 * 15 = 1111111111111111
92X79E43715865 * 2X = 2222222222222222
92X79E43715865 * 43 = 3333333333333333
92X79E43715865 * 58 = 4444444444444444
92X79E43715865 * 71 = 5555555555555555
92X79E43715865 * 86 = 6666666666666666
92X79E43715865 * 9E = 7777777777777777
92X79E43715865 * E4 = 8888888888888888
92X79E43715865 * 109 = 9999999999999999
92X79E43715865 * 122 = XXXXXXXXXXXXXXXX
92X79E43715865 * 137 = EEEEEEEEEEEEEEEE
8327 * 17 = 111111
8327 * 32 = 222222
8327 * 49 = 333333
8327 * 64 = 444444
8327 * 7E = 555555
8327 * 96 = 666666
8327 * E1 = 777777
8327 * 108 = 888888
8327 * 123 = 999999
8327 * 13X = XXXXXX
8327 * 155 = EEEEEE
55 * 25 = 1111
55 * 4X = 2222
55 * 73 = 3333
55 * 98 = 4444
55 * 101 = 5555
55 * 126 = 6666
55 * 14E = 7777
55 * 174 = 8888
55 * 199 = 9999
55 * 202 = XXXX
55 * 227 = EEEE
1221 * E1 = 111111
1221 * 1X2 = 222222
1221 * 293 = 333333
1221 * 384 = 444444
1221 * 475 = 555555
1221 * 566 = 666666
1221 * 657 = 777777
1221 * 748 = 888888
1221 * 839 = 999999
1221 * 92X = XXXXXX
1221 * X1E = EEEEEE
112233445566778899XE * EE = 1111111111111111111111
112233445566778899XE * 1EX = 2222222222222222222222
112233445566778899XE * 2E9 = 3333333333333333333333
112233445566778899XE * 3E8 = 4444444444444444444444
112233445566778899XE * 4E7 = 5555555555555555555555
112233445566778899XE * 5E6 = 6666666666666666666666
112233445566778899XE * 6E5 = 7777777777777777777777
112233445566778899XE * 7E4 = 8888888888888888888888
112233445566778899XE * 8E3 = 9999999999999999999999
112233445566778899XE * 9E2 = XXXXXXXXXXXXXXXXXXXXXX
112233445566778899XE * XE1 = EEEEEEEEEEEEEEEEEEEEEE
E1X2938475665748392X1E1 * 121 = 11111111111111111111111111
E1X2938475665748392X1E1 * 242 = 22222222222222222222222222
E1X2938475665748392X1E1 * 363 = 33333333333333333333333333
E1X2938475665748392X1E1 * 484 = 44444444444444444444444444
E1X2938475665748392X1E1 * 5X5 = 55555555555555555555555555
E1X2938475665748392X1E1 * 706 = 66666666666666666666666666
E1X2938475665748392X1E1 * 827 = 77777777777777777777777777
E1X2938475665748392X1E1 * 948 = 88888888888888888888888888
E1X2938475665748392X1E1 * X69 = 99999999999999999999999999
E1X2938475665748392X1E1 * E8X = XXXXXXXXXXXXXXXXXXXXXXXXXX
E1X2938475665748392X1E1 * 10XE = EEEEEEEEEEEEEEEEEEEEEEEEEE
1^2 = 01, 0 + 1 = 1
E^2 = X1, X + 1 = E
56^2 = 2630, 26 + 30 = 56
66^2 = 3630, 36 + 30 = 66
EE^2 = EX01, EX + 01 = EE
444^2 = 170294, 170 + 294 = 444
778^2 = 4X4294, 4X4 + 294 = 778
EEE^2 = EEX001, EEX + 001 = EEE
12XX^2 = 1661144, 166 + 1144 = 12XX
1640^2 = 2401400, 240 + 1400 = 1640
2046^2 = 4161830, 416 + 1830 = 2046
2929^2 = 7802169, 780 + 2169 = 2929
3333^2 = X862469, X86 + 2469 = 3333
4973^2 = 1E062X69, 1E06 + 2X69 = 4973
5E60^2 = 2E603000, 2E60 + 3000 = 5E60
2^2 + 5^2 = 25
X^2 + 5^2 = X5
5^3 + 7^3 + 7^3 = 577
6^3 + 6^3 + 8^3 = 668
X^3 + 8^3 + 3^3 = X83
1^5 + 4^5 + 7^5 + 6^5 + 5^5 = 14765
9^5 + 3^5 + 8^5 + X^5 + 4^5 = 938X4
3^6 + 6^6 + 9^6 + 8^6 + 6^6 + 2^6 = 369862
X^6 + 2^6 + 3^6 + 9^6 + 4^6 + X^6 = X2394X
14 = (1 * 4)^2
20 = (2^2 + 0^2)!
21 = (2^2 + 1^2)^2
24 = 2^2 + 4!
30 = ((3 + 0)!)^2
43 = 4! + 3^3
48 = 4^3 - 8
84 = (8 + sqrt(4))^2
X1 = (X + 1)^2
X5 = (X - 5)^3
X8 = (X - 8)^7
XE = X + E^2
121 = (12 - 1)^2
160 = (1 * 6 + 0)^3
169 = (1 * (6 + 9))^2
183 = ((1 + 8) / 3)^5
194 = ((1 - 9) / 4)^8
230 = (30 / 2)^2
243 = - 2^2 + (4 + 3)^3
247 = (2 - sqrt(4) + 7)^3
294 = (2 * 9 + sqrt(4))^2
344 = (3! + 4 * 4)^2
381 = (3 * 8 - 1)^2
          1 * X + 1 = E
         12 * X + 2 = EX
        123 * X + 3 = EX9
       1234 * X + 4 = EX98
      12345 * X + 5 = EX987
     123456 * X + 6 = EX9876
    1234567 * X + 7 = EX98765
   12345678 * X + 8 = EX987654
  123456789 * X + 9 = EX9876543
 123456789X * X + X = EX98765432
123456789XE * X + E = EX987654321
          0 * E + 1 = 1
          1 * E + 2 = 11
         12 * E + 3 = 111
        123 * E + 4 = 1111
       1234 * E + 5 = 11111
      12345 * E + 6 = 111111
     123456 * E + 7 = 1111111
    1234567 * E + 8 = 11111111
   12345678 * E + 9 = 111111111
  123456789 * E + X = 1111111111
 123456789X * E + E = 11111111111
123456789XE * E + 10 = 111111111111
           1^2 = 1
          11^2 = 121
         111^2 = 12321
        1111^2 = 1234321
       11111^2 = 123454321
      111111^2 = 12345654321
     1111111^2 = 1234567654321
    11111111^2 = 123456787654321
   111111111^2 = 12345678987654321
  1111111111^2 = 123456789X987654321
 11111111111^2 = 123456789XEX987654321
111111111111^2 = 123456789E00EX987654321 (Rn^2 is not palindromic number if n >= 10)
           E^2 = X1
          EE^2 = EX01
         EEE^2 = EEX001
        EEEE^2 = EEEX0001
       EEEEE^2 = EEEEX00001
      EEEEEE^2 = EEEEEX000001
     EEEEEEE^2 = EEEEEEX0000001
    EEEEEEEE^2 = EEEEEEEX00000001
   EEEEEEEEE^2 = EEEEEEEEX000000001
  EEEEEEEEEE^2 = EEEEEEEEEX0000000001
 EEEEEEEEEEE^2 = EEEEEEEEEEX00000000001
EEEEEEEEEEEE^2 = EEEEEEEEEEEX000000000001
                      3 * 4 = 10
                    33 * 44 = 1210
                  333 * 444 = 123210
                3333 * 4444 = 12343210
              33333 * 44444 = 1234543210
            333333 * 444444 = 123456543210
          3333333 * 4444444 = 12345676543210
        33333333 * 44444444 = 1234567876543210
      333333333 * 444444444 = 123456789876543210
    3333333333 * 4444444444 = 123456789X9876543210
  33333333333 * 44444444444 = 123456789XEX9876543210
333333333333 * 444444444444 = 123456789E00EX9876543210
                      6 * E = 56
                    66 * EE = 6556
                  666 * EEE = 665556
                6666 * EEEE = 66655556
              66666 * EEEEE = 6666555556
            666666 * EEEEEE = 666665555556
          6666666 * EEEEEEE = 66666655555556
        66666666 * EEEEEEEE = 6666666555555556
      666666666 * EEEEEEEEE = 666666665555555556
    6666666666 * EEEEEEEEEE = 66666666655555555556
  66666666666 * EEEEEEEEEEE = 6666666666555555555556
666666666666 * EEEEEEEEEEEE = 666666666665555555555556
10XE * 1 = 10XE
10XE * 2 = 219X
10XE * 3 = 3289
10XE * 4 = 4378
10XE * 5 = 5467
10XE * 6 = 6556
10XE * 7 = 7645
10XE * 8 = 8734
10XE * 9 = 9823
10XE * X = X912
10XE * E = EX01
10EXE * 1 = 10EXE
10EXE * 2 = 21E9X
10EXE * 3 = 32E89
10EXE * 4 = 43E78
10EXE * 5 = 54E67
10EXE * 6 = 65E56
10EXE * 7 = 76E45
10EXE * 8 = 87E34
10EXE * 9 = 98E23
10EXE * X = X9E12
10EXE * E = EXE01
1 / E = 0.111111111111 ...
1 / 11 = 0.0E 0E 0E 0E 0E 0E ...
1 / EE = 0.01 01 01 01 01 01 ...
1 / 101 = 0.00EE 00EE 00EE ...
1 / E1 = 0.010EXE 010EXE ...
1 / 111 = 0.00E 00E 00E 00E ...
1 / X1 = 0.0123456789E 0123456789E ...
1 / 121 = 0.00X28466482X0 EE1937557391E ...
1 / XE1 = 0.00112233445566778899XE 00112233445566778899XE ...
1 / XEE1 = 0.000111222333444555666777888999XXE 000111222333444555666777888999XXE ...
1 / XEEE1 = 0.0000111122223333444455556666777788889999XXXE 0000111122223333444455556666777788889999XXXE ...
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
1 / EEX001 = 0.000 001 002 003 004 005 006 007 008 009 00X 00E 010 011 012 013 014 ...
1 / EEEX0001 = 0.0000 0001 0002 0003 0004 0005 0006 0007 0008 0009 000X 000E 0010 0011 0012 0013 0014 ...
1 / EXE01 = 0.0000101111212222323333434444545555656666767777878888989999X9XXXXEE 0000101111212222323333434444545555656666767777878888989999X9XXXXEE ...
1 / EXEE = 0.00 01 01 02 03 05 08 11 19 2X 47 ...
1 / EEXEEE = 0.000 001 001 002 003 005 008 011 019 02X 047 075 100 175 275 42X 6X3 ...
1 / EEEXEEEE = 0.0000 0001 0001 0002 0003 0005 0008 0011 0019 002X 0047 0075 0100 0175 0275 042X 06X3 0E11 15E4 ...
1 / EX = 0.01 02 04 08 14 28 54 X8
                                 194
                                   368
                                     714
                                      1228
                                        2454 ... (powers of 2)
1 / E9 = 0.01 03 09 23 69
                        183
                          509
                           1323
                             3969 ... (powers of 3)
1 / E8 = 0.01 04 14 54
                     194
                       714
                        2454 ... (powers of 4)
1 / XE = 0.0112358
                 11
                  19
                   2X
                    47
                     75
                     100
                      175
                       275
                        42X
                         6X3
                          E11
                          15E4 ... (Fibonacci numbers)
1 / 9E = 0.0125
              10
               25
                5X
                121
                 2X0
                  6X1
                  1462 ... (Pell numbers)
1 / 5E = 0.02 04 08 14 28 54 X8
                              194
                                368
                                  714
                                   1228
                                     2454 ... (powers of 2)
1 / 3E = 0.03 09 23 69
                     183
                       509
                        1323
                          3969 ... (powers of 3)
1 / 2E = 0.04 14 54
                  194
                    714
                     2454 ... (powers of 4)
                                     148421
                                    28
                                   54
                                  X8
                                194
                               368
                              714
                            1228
1 / 1E = (powers of 2) ... 2454
                                 23931
                                69
                              183
                             509
                           1323
1 / 2E = (power of 3) ... 3969
                                1441
                               54
                             194
                            714
1 / 3E = (power of 4) ... 2454
                                               11853211
                                              19
                                             2X
                                            47
                                           75
                                         100
                                        175
                                       275
                                      42X
                                     6X3
                                    E11
1 / 10E = (Fibonacci numbers) ... 15E4
                                     10521
                                    25
                                   5X
                                 121
                                2X0
                               6X1
1 / 11E = (Pell numbers) ... 1462
ab * 5 * 25 = abab
ab * 7 * 17 * 111 = ababab
ab * 5 * 25 * 75 * 175 = abababab
ab * E0E1 * 11111 = ababababab
ab * 5 * 7 * 17 * 25 * 111 * EE01 = abababababab
abc * 7 * 11 * 17 = abcabc
abc * 31 * 3X891 = abcabcabc
abc * 5 * 7 * 11 * 17 * 25 * EE01 = abcabcabcabc
abcd * 75 * 175 = abcdabcd
abcd * 7 * 17 * 111 * EE01 = abcdabcdabcd
abcde * 11 * E0E1 = abcdeabcde
abcdef * 5 * 25 * EE01 = abcdefabcdef

Prime numbers[edit]

(In this section, all numbers are written with duodecimal)

A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive divisors, 1 and the number itself. Natural numbers greater than 1 that are not prime are called composite.

The first 1ᘔ5 prime numbers (all the prime numbers less than 1000) are:

2, 3, 5, 7, Ɛ, 11, 15, 17, 1Ɛ, 25, 27, 31, 35, 37, 3Ɛ, 45, 4Ɛ, 51, 57, 5Ɛ, 61, 67, 6Ɛ, 75, 81, 85, 87, 8Ɛ, 91, 95, ᘔ7, ᘔƐ, Ɛ5, Ɛ7, 105, 107, 111, 117, 11Ɛ, 125, 12Ɛ, 131, 13Ɛ, 141, 145, 147, 157, 167, 16Ɛ, 171, 175, 17Ɛ, 181, 18Ɛ, 195, 19Ɛ, 1ᘔ5, 1ᘔ7, 1Ɛ1, 1Ɛ5, 1Ɛ7, 205, 217, 21Ɛ, 221, 225, 237, 241, 24Ɛ, 251, 255, 25Ɛ, 267, 271, 277, 27Ɛ, 285, 291, 295, 2ᘔ1, 2ᘔƐ, 2Ɛ1, 2ƐƐ, 301, 307, 30Ɛ, 315, 321, 325, 327, 32Ɛ, 33Ɛ, 347, 34Ɛ, 357, 35Ɛ, 365, 375, 377, 391, 397, 3ᘔ5, 3ᘔƐ, 3Ɛ5, 3Ɛ7, 401, 40Ɛ, 415, 41Ɛ, 421, 427, 431, 435, 437, 447, 455, 457, 45Ɛ, 465, 46Ɛ, 471, 481, 485, 48Ɛ, 497, 4ᘔ5, 4Ɛ1, 4ƐƐ, 507, 511, 517, 51Ɛ, 527, 531, 535, 541, 545, 557, 565, 575, 577, 585, 587, 58Ɛ, 591, 59Ɛ, 5Ɛ1, 5Ɛ5, 5Ɛ7, 5ƐƐ, 611, 615, 617, 61Ɛ, 637, 63Ɛ, 647, 655, 661, 665, 66Ɛ, 675, 687, 68Ɛ, 695, 69Ɛ, 6ᘔ7, 6Ɛ1, 701, 705, 70Ɛ, 711, 71Ɛ, 721, 727, 735, 737, 745, 747, 751, 767, 76Ɛ, 771, 775, 77Ɛ, 785, 791, 797, 7ᘔ1, 7ƐƐ, 801, 80Ɛ, 817, 825, 82Ɛ, 835, 841, 851, 855, 85Ɛ, 865, 867, 871, 881, 88Ɛ, 8ᘔ5, 8ᘔ7, 8ᘔƐ, 8Ɛ5, 8Ɛ7, 901, 905, 907, 90Ɛ, 91Ɛ, 921, 927, 955, 95Ɛ, 965, 971, 987, 995, 9ᘔ7, 9ᘔƐ, 9Ɛ1, 9Ɛ5, 9ƐƐ, ᘔ07, ᘔ0Ɛ, ᘔ11, ᘔ17, ᘔ27, ᘔ35, ᘔ37, ᘔ3Ɛ, ᘔ41, ᘔ45, ᘔ4Ɛ, ᘔ5Ɛ, ᘔ6Ɛ, ᘔ77, ᘔ87, ᘔ91, ᘔ95, ᘔ9Ɛ, ᘔᘔ7, ᘔᘔƐ, ᘔƐ7, ᘔƐƐ, Ɛ11, Ɛ15, Ɛ1Ɛ, Ɛ21, Ɛ25, Ɛ2Ɛ, Ɛ31, Ɛ37, Ɛ45, Ɛ61, Ɛ67, Ɛ6Ɛ, Ɛ71, Ɛ91, Ɛ95, Ɛ97, Ɛᘔ5, ƐƐ5, ƐƐ7

Except 2 and 3, all primes end with 1, 5, 7 or Ɛ, the first k such that all of 10k, 10k + 1, 10k + 2, ..., 10k + Ɛ are all composite is 38, i.e. all of 380, 381, 382, ..., 38Ɛ are composite.

The density of primes end with 1 is a relatively low, but the density of primes end with 5, 7 and Ɛ are nearly equal. (i.e. for a given natural number N, the number of primes end with 1 less than N is usually smaller than the number of primes end with 5 (or 7, or Ɛ) less than N) e.g. For all primes < 10000, there are 3Ɛ8 primes end with 1, 410 primes end with 5, 412 primes end with 7, 406 primes end with Ɛ. It is conjectured that for every natural number N ≥ 10, the number of primes end with 1 less than N is smaller than the number of primes end with 5 (or 7, or Ɛ) less than N.

All squares of primes (except 2 and 3) end with 1.

There are 2ᘔ primes between 1 and 100, 23 primes between 101 and 200, 1ᘔ primes between 201 and 300, 1ᘔ primes between 301 and 400, 1Ɛ primes between 401 and 500, 1ᘔ primes between 501 and 600, 16 primes between 601 and 700, 1ᘔ primes between 701 and 800, 18 primes between 801 and 900, 16 primes between 901 and ᘔ00, 1ᘔ primes between ᘔ01 and Ɛ00, 17 primes between Ɛ01 and 1000.

There are about N/ln(N) primes less than N, where ln is the natural logarithm, i.e. the logarithm with base e = 2.875236069821... (see prime number theorem)

N total numbers of primes ≤ N numbers of primes end with 1 ≤ N numbers of primes end with 5 ≤ N numbers of primes end with 7 ≤ N numbers of primes end with Ɛ ≤ N
10 5 0 1 1 1
40 13 2 4 4 3
100 2ᘔ 6 9 9 8
400 89 1ᘔ 23 23 23
1000 1ᘔ5 51 59 59 58
4000 621 157 16ᘔ 170 166
10000 1426 3Ɛ8 410 412 406
40000 4833 11ᘔ4 121Ɛ 1219 1211
100000 10852 31ᘔ4 3225 3225 321ᘔ
400000 3928Ɛ Ɛ333 Ɛ377 Ɛ3Ɛ9 Ɛ3ᘔ2
1000000 ᘔ4Ɛ20 27204 27295 2730ᘔ 27333

In the following table, numbers shaded in cyan are primes.

1 2 3 4 5 6 7 8 9 Ɛ 10
11 12 13 14 15 16 17 18 19 1ᘔ 20
21 22 23 24 25 26 27 28 29 2ᘔ 30
31 32 33 34 35 36 37 38 39 3ᘔ 40
41 42 43 44 45 46 47 48 49 4ᘔ 50
51 52 53 54 55 56 57 58 59 5ᘔ 60
61 62 63 64 65 66 67 68 69 6ᘔ 70
71 72 73 74 75 76 77 78 79 7ᘔ 80
81 82 83 84 85 86 87 88 89 8ᘔ 90
91 92 93 94 95 96 97 98 99 9ᘔ ᘔ0
ᘔ1 ᘔ2 ᘔ3 ᘔ4 ᘔ5 ᘔ6 ᘔ7 ᘔ8 ᘔ9 ᘔᘔ ᘔƐ Ɛ0
Ɛ1 Ɛ2 Ɛ3 Ɛ4 Ɛ5 Ɛ6 Ɛ7 Ɛ8 Ɛ9 Ɛᘔ ƐƐ 100

Divisibility rules[edit]

(In this section, all numbers are written with duodecimal)

1

Any integer is divisible by 1.

2

If a number is divisible by 2 then the unit digit of that number will be 0, 2, 4, 6, 8 or ᘔ.

3

If a number is divisible by 3 then the unit digit of that number will be 0, 3, 6 or 9.

4

If a number is divisible by 4 then the unit digit of that number will be 0, 4 or 8.

5

To test for divisibility by 5, double the units digit and subtract the result from the number formed by the rest of the digits. If the result is divisible by 5 then the given number is divisible by 5.

This rule comes from 21(5*5)

Examples:
13     rule => |1-2*3| = 5 which is divisible by 5.
2Ɛᘔ5   rule => |2Ɛᘔ-2*5| = 2Ɛ0(5*70) which is divisible by 5(or apply the rule on 2Ɛ0).

OR

To test for divisibility by 5, subtract the units digit and triple of the result to the number formed by the rest of the digits. If the result is divisible by 5 then the given number is divisible by 5.

This rule comes from 13(5*3)

Examples:
13     rule => |3-3*1| = 0 which is divisible by 5.
2Ɛᘔ5   rule => |5-3*2Ɛᘔ| = 8Ɛ1(5*195) which is divisible by 5(or apply the rule on 8Ɛ1).

OR

Form the alternating sum of blocks of two from right to left. If the result is divisible by 5 then the given number is divisible by 5.

This rule comes from 101, since 101 = 5*25, thus this rule can be also tested for the divisibility by 25.

Example:

97,374,627 => 27-46+37-97 = -7Ɛ which is divisible by 5.

6

If a number is divisible by 6 then the unit digit of that number will be 0 or 6.

7

To test for divisibility by 7, triple the units digit and add the result to the number formed by the rest of the digits. If the result is divisible by 7 then the given number is divisible by 7.

This rule comes from 2Ɛ(7*5)

Examples:
12     rule => |3*2+1| = 7 which is divisible by 7.
271Ɛ    rule => |3*Ɛ+271| = 29ᘔ(7*4ᘔ) which is divisible by 7(or apply the rule on 29ᘔ).

OR

To test for divisibility by 7, subtract the units digit and double the result from the number formed by the rest of the digits. If the result is divisible by 7 then the given number is divisible by 7.

This rule comes from 12(7*2)

Examples:
12     rule => |2-2*1| = 0 which is divisible by 7.
271Ɛ    rule => |Ɛ-2*271| = 513(7*89) which is divisible by 7(or apply the rule on 513).

OR

To test for divisibility by 7, 4 times the units digit and subtract the result from the number formed by the rest of the digits. If the result is divisible by 7 then the given number is divisible by 7.

This rule comes from 41(7*7)

Examples:
12     rule => |4*2-1| = 7 which is divisible by 7.
271Ɛ    rule => |4*Ɛ-271| = 235(7*3Ɛ) which is divisible by 7(or apply the rule on 235).

OR

Form the alternating sum of blocks of three from right to left. If the result is divisible by 7 then the given number is divisible by 7.

This rule comes from 1001, since 1001 = 7*11*17, thus this rule can be also tested for the divisibility by 11 and 17.

Example:

386,967,443 => 443-967+386 = -168 which is divisible by 7.

8

If the 2-digit number formed by the last 2 digits of the given number is divisible by 8 then the given number is divisible by 8.

Example: 1Ɛ48, 4120

     rule => since 48(8*7) divisible by 8, then 1Ɛ48 is divisible by 8.
     rule => since 20(8*3) divisible by 8, then 4120 is divisible by 8.
9

If the 2-digit number formed by the last 2 digits of the given number is divisible by 9 then the given number is divisible by 9.

Example: 7423, 8330

     rule => since 23(9*3) divisible by 9, then 7423 is divisible by 9.
     rule => since 30(9*4) divisible by 9, then 8330 is divisible by 9.

If the number is divisible by 2 and 5 then the number is divisible by .

Ɛ

If the sum of the digits of a number is divisible by Ɛ then the number is divisible by Ɛ (the equivalent of casting out nines in decimal).

Example: 29, 61Ɛ13

     rule => 2+9 = Ɛ which is divisible by Ɛ, then 29 is divisible by Ɛ.
     rule => 6+1+Ɛ+1+3 = 1ᘔ which is divisible by Ɛ, then 61Ɛ13 is divisible by Ɛ.
10

If a number is divisible by 10 then the unit digit of that number will be 0.

11

Sum the alternate digits and subtract the sums. If the result is divisible by 11 the number is divisible by 11 (the equivalent of divisibility by eleven in decimal).

Example: 66, 9427

     rule => |6-6| = 0 which is divisible by 11, then 66 is divisible by 11.
     rule => |(9+2)-(4+7)| = |ᘔ-ᘔ| = 0 which is divisible by 11, then 9427 is divisible by 11.
12

If the number is divisible by 2 and 7 then the number is divisible by 12.

13

If the number is divisible by 3 and 5 then the number is divisible by 13.

14

If the 2-digit number formed by the last 2 digits of the given number is divisible by 14 then the given number is divisible by 14.

Fractions and irrational numbers[edit]

Fractions[edit]

Duodecimal fractions 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
  • 1/10 = 0.1 (note that this is a twelfth, 1/ is a tenth)
  • 1/14 = 0.09 (note that this is a sixteenth, 1/12 is a fourteenth)

or complicated:

  • 1/5 = 0.249724972497... recurring (rounded to 0.24ᘔ)
  • 1/7 = 0.186ᘔ35186ᘔ35... recurring (rounded to 0.187)
  • 1/ = 0.1249724972497... recurring (rounded to 0.125)
  • 1/Ɛ = 0.111111111111... recurring (rounded to 0.111)
  • 1/11 = 0.0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ0Ɛ... recurring (rounded to 0.0Ɛ1)
  • 1/12 = 0.0ᘔ35186ᘔ35186... recurring (rounded to 0.0ᘔ3)
  • 1/13 = 0.0972497249724... recurring (rounded to 0.097)
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
1ᘔ.6 + 7.6 = 26 22.5 + 7.5 = 30

As explained in recurring decimals, whenever an irreducible fraction is written in radix point 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.

The number of denominators which give terminating fractions within a given number of digits, say n, in a base b is the number of factors (divisors) of bn, the nth power of the base b (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of bn is given using its prime factorization.

For decimal, 10n = 2n * 5n, the number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together. Factors of 10n = (n+1)(n+1) = (n+1)2.

For example, the number 8 is a factor of 103 (1000), so 1/8 and other fractions with a denominator of 8 can not require more than 3 fractional decimal digits to terminate. 5/8 = 0.625ten

For duodecimal, 12n = 22n * 3n. This has (2n+1)(n+1) divisors, the sample denominator of 8 is a factor of a gross (122 = 144), so eighths can not need more than two duodecimal fractional places to terminate. 5/8 = 0.76twelve

Because both ten and twelve have two unique prime factors, the number of divisors of bn for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of n2).

Recurring digits[edit]

The Dozenal Society of America argues that factors of 3 are more commonly encountered in real-life division problems than factors of 5.[32] Thus, in practical applications, the nuisance of repeating decimals is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.

However, when recurring fractions do occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because 12 (twelve) is between two prime numbers, 11 (eleven) and 13 (thirteen), whereas ten is adjacent to the 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, whereas 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, whereas 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 duodecimal than in decimal representation (e.g. 1/(22) = 0.25 ten = 0.3 twelve; 1/(23) = 0.125 ten = 0.16 twelve; 1/(24) = 0.0625 ten = 0.09 twelve; 1/(25) = 0.03125 ten = 0.046 twelve; etc.).

Values in bold indicate that value is exact.

Decimal base
Prime factors of the base: 2, 5
Prime factors of one below the base: 3
Prime factors of one above the base: 11
All other primes: 7, 13, 17, 19, 23, 29, 31
Duodecimal base
Prime factors of the base: 2, 3
Prime factors of one below the base: Ɛ
Prime factors of one above the base: 11
All other primes: 5, 7, 15, 17, , 25, 27
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 0.3 0.4 3 1/3
1/4 2 0.25 0.3 2 1/4
1/5 5 0.2 0.2497 5 1/5
1/6 2, 3 0.16 0.2 2, 3 1/6
1/7 7 0.142857 0.186ᘔ35 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/ᘔ
1/11 11 0.09 0.1 Ɛ 1/Ɛ
1/12 2, 3 0.083 0.1 2, 3 1/10
1/13 13 0.076923 0. 11 1/11
1/14 2, 7 0.0714285 0.0ᘔ35186 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 0.08579214Ɛ36429ᘔ7 15 1/15
1/18 2, 3 0.05 0.08 2, 3 1/16
1/19 19 0.052631578947368421 0.076Ɛ45 17 1/17
1/20 2, 5 0.05 0.07249 2, 5 1/18
1/21 3, 7 0.047619 0.06ᘔ3518 3, 7 1/19
1/22 2, 11 0.045 0.06 2, Ɛ 1/1ᘔ
1/23 23 0.0434782608695652173913 0.06316948421 1/1Ɛ
1/24 2, 3 0.0416 0.06 2, 3 1/20
1/25 5 0.04 0.05915343ᘔ0Ɛ62ᘔ68781Ɛ 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 0.05186ᘔ3 2, 7 1/24
1/29 29 0.0344827586206896551724137931 0.04Ɛ7 25 1/25
1/30 2, 3, 5 0.03 0.04972 2, 3, 5 1/26
1/31 31 0.032258064516129 0.0478ᘔᘔ093598166Ɛ74311Ɛ28623ᘔ55 27 1/27
1/32 2 0.03125 0.046 2 1/28
1/33 3, 11 0.03 0.04 3, Ɛ 1/29
1/34 2, 17 0.02941176470588235 0.0429ᘔ708579214Ɛ36 2, 15 1/2ᘔ
1/35 5, 7 0.0285714 0.0414559Ɛ3931 5, 7 1/2Ɛ
1/36 2, 3 0.027 0.04 2, 3 1/30

The duodecimal period length of 1/n are

0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, ... (sequence A246004 in the OEIS)

The duodecimal period length of 1/(nth prime) are

0, 0, 4, 6, 1, 2, 16, 6, 11, 4, 30, 9, 40, 42, 23, 52, 29, 15, 66, 35, 36, 26, 41, 8, 16, 100, 102, 53, 54, 112, 126, 65, 136, 138, 148, 150, 3, 162, 83, 172, 89, 90, 95, 24, 196, 66, 14, 222, 113, 114, 8, 119, 120, 125, 256, 131, 268, 54, 138, 280, ... (sequence A246489 in the OEIS)

Smallest prime with duodecimal period n are

11, 13, 157, 5, 22621, 7, 659, 89, 37, 19141, 23, 20593, 477517, 211, 61, 17, 2693651, 1657, 29043636306420266077, 85403261, 8177824843189, 57154490053, 47, 193, 303551, 79, 306829, 673, 59, 31, 373, 153953, 886381, 2551, 71, 73, ... (sequence A252170 in the OEIS)

Irrational numbers[edit]

As for irrational numbers, none of them have 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 × 102 + 2 × 101 + 3 × 100 + 4 × 1/101 + 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 does not exhibit a strictly repeating pattern; instead, the different digits often succeed in a seemingly random fashion. The following chart compares the first few digits of the decimal and duodecimal representation of several of the most important algebraic and transcendental irrational numbers, some of these numbers may be perceived as having fortuitous patterns, making them easier to memorize, when represented in one base or the other.

Algebraic irrational number In decimal In duodecimal
2 (the length of the diagonal of a unit square) 1.414213562373... (≈ 1.414) 1.4Ɛ79170ᘔ07Ɛ8... (≈ 1.4Ɛ7)
3 (the length of the diagonal of a unit cube, or twice the height of an equilateral triangle of unit side) 1.732050807568... (≈ 1.732) 1.894Ɛ97ƐƐ9687... (≈ 1.895)
5 (the length of the diagonal of a 1×2 rectangle) 2.236067977499... (≈ 2.236) 2.29ƐƐ13254058... (≈ 2.2ᘔ)
φ (phi, the golden ratio = ) 1.618033988749... (≈ 1.618) 1.74ƐƐ6772802ᘔ... (≈ 1.75)
Transcendental irrational number In decimal In duodecimal
π (pi, the ratio of circumference to diameter) 3.141592653589793238462643... (≈ 3.142) 3.184809493Ɛ918664573ᘔ6211... (≈ 3.185)
e (the base of the natural logarithm) 2.718281828459... (≈ 2.718) 2.875236069821... (≈ 2.875)

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

Number In decimal In duodecimal
γ (the limiting difference between the harmonic series and the natural logarithm) 0.577215664901... (≈ 0.577) 0.6Ɛ15188ᘔ6760... (≈ 0.6Ɛ1)

See also[edit]

References[edit]

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External links[edit]