Editing Duodecimal (section)

== Fractions and irrational numbers ==
=== Fractions ===
Duodecimal [[Fraction (mathematics)|fractions]] for rational numbers with [[Smooth number|3-smooth]] denominators terminate:
* {{sfrac|2}} = 0.6
* {{sfrac|3}} = 0.4
* {{sfrac|4}} = 0.3
* {{sfrac|6}} = 0.2
* {{sfrac|8}} = 0.16
* {{sfrac|9}} = 0.14
* {{sfrac|10}} = 0.1 (this is one twelfth, {{sfrac|{{d2}}}} is one tenth)
* {{sfrac|14}} = 0.09 (this is one sixteenth, {{sfrac|12}} is one fourteenth)

while other rational numbers have [[recurring decimal|recurring]] duodecimal fractions:
* {{sfrac|5}} = 0.{{Overline|2497}}
* {{sfrac|7}} = 0.{{Overline|186{{D2}}35}}
* {{sfrac|{{d2}}}} = 0.1{{Overline|2497}} (one tenth)
* {{sfrac|{{d3}}}} = 0.{{Overline|1}} (one eleventh)
* {{sfrac|11}} = 0.{{Overline|0{{D3}}}} (one thirteenth)
* {{sfrac|12}} = 0.0{{Overline|{{D2}}35186}} (one fourteenth)
* {{sfrac|13}} = 0.0{{Overline|9724}} (one fifteenth)

{| class="wikitable"
! Examples in duodecimal
! Decimal equivalent
|-
| 1 × {{sfrac|5|8}} = 0.76
| 1 × {{sfrac|5|8}} = 0.625
|-
| 100 × {{sfrac|5|8}} = 76
| 144 × {{sfrac|5|8}} = 90
|-
| {{sfrac|576|9}} = 76
| {{sfrac|810|9}} = 90
|-
| {{sfrac|400|9}} = 54
| {{sfrac|576|9}} = 64
|-
| 1{{d2}}.6 + 7.6 = 26
| 22.5 + 7.5 = 30
|}

As explained in [[recurring decimal]]s, 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 factor]]s of its denominator are also prime factors of the base.

Because <math>2\times5=10</math> in the decimal system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: {{sfrac|8}}&nbsp;=&nbsp;{{sfrac|(2×2×2)}}, {{sfrac|20}}&nbsp;=&nbsp;{{sfrac|(2×2×5)}}, and {{sfrac|500}}&nbsp;=&nbsp;{{sfrac|(2×2×5×5×5)}} can be expressed exactly as 0.125, 0.05, and 0.002 respectively. {{sfrac|3}} and {{sfrac|7}}, however, recur (0.333... and 0.142857142857...).

Because <math>2\times2\times3=12</math> in the duodecimal system, {{sfrac|8}} is exact; {{sfrac|20}} and {{sfrac|500}} recur because they include 5 as a factor; {{sfrac|3}} is exact, and {{sfrac|7}} recurs, just as it does in decimal.

The number of denominators that give terminating fractions within a given number of digits, {{math|''n''}}, in a base {{math|''b''}} is the number of factors (divisors) of <math>b^n</math>, the {{math|''n''}}th power of the base {{math|''b''}} (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of ''<math>b^n</math>'' is given using its prime factorization.

For decimal, <math>10^n=2^n\times 5^n</math>. The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together, so the number of factors of ''<math>10^n</math>'' is <math>(n+1)(n+1)=(n+1)^2</math>.

For example, the number 8 is a factor of 10<sup>3</sup> (1000), so <math display="inline">\frac{1}{8}</math> and other fractions with a denominator of 8 cannot require more than three fractional decimal digits to terminate. <math display="inline">\frac{5}{8}=0.625_{10}.</math>

For duodecimal, <math>10^n=2^{2n}\times 3^n</math>. This has <math>(2n+1)(n+1)</math> divisors. The sample denominator of 8 is a factor of a gross <math display="inline">12^2=144</math> (in decimal), so eighths cannot need more than two duodecimal fractional places to terminate. <math display="inline">\frac{5}{8}=0.76_{12}.</math>

Because both ten and twelve have two unique prime factors, the number of divisors of ''<math>b^n</math>'' for {{math|''b'' {{=}} 10 or 12}} grows quadratically with the exponent {{math|''n''}} (in other words, of the order of <math>n^2</math>).

=== Recurring digits ===
The Dozenal Society of America argues that factors of 3 are more commonly encountered in real-life [[division (mathematics)|division]] problems than factors of 5.<ref name="dsafaq">{{cite web |author=De Vlieger |first=Michael Thomas |date=30 November 2011 |title=Dozenal FAQs |url=https://dozenal.org/articles/DSA-DozenalFAQs.pdf |access-date=November 20, 2022 |website=dozenal.org |publisher=The Dozenal Society of America }}</ref> 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 number]]s, 11 (eleven) and 13 (thirteen), whereas ten is adjacent to the [[composite number]] 9. Nonetheless, having a shorter or longer period does not 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 [[power of two|powers of two]] will have a shorter, more convenient terminating representation in duodecimal than in decimal:

* 1/(2<sup>2</sup>) = {{base|0.25|10}} = {{base|0.3|12}}
* 1/(2<sup>3</sup>) = {{base|0.125|10}} = {{base|0.16|12}}
* 1/(2<sup>4</sup>) = {{base|0.0625|10}} = {{base|0.09|12}}
* 1/(2<sup>5</sup>) = {{base|0.03125|10}} = {{base|0.046|12}}

{| class="wikitable"
|- style="text-align:center;"
| colspan="3"| '''Decimal base'''<br><SMALL>Prime factors of the base: <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span></SMALL><br><SMALL>Prime factors of one below the base: <span style="color:Blue">'''3'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Magenta">'''11'''</span></SMALL><br><SMALL>All other primes: <span style="color:Red">'''7'''</span>, <span style="color:Red">'''13'''</span>, <span style="color:Red">'''17'''</span>, <span style="color:Red">'''19'''</span>, <span style="color:Red">'''23'''</span>, <span style="color:Red">'''29'''</span>, <span style="color:Red">'''31'''</span></SMALL>
| colspan="3"| '''Duodecimal base'''<br><SMALL>Prime factors of the base: <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span></SMALL><br><SMALL>Prime factors of one below the base: <span style="color:Blue">'''{{d3}}'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Magenta">'''11 (={{base|13|10}})'''</span></SMALL><br><SMALL>All other primes: <span style="color:Red">'''5'''</span>, <span style="color:Red">'''7'''</span>, <span style="color:Red">'''15 (={{base|17|10}})'''</span>, <span style="color:Red">'''17 (={{base|19|10}})'''</span>, <span style="color:Red">'''1{{d3}} (={{base|23|10}})'''</span>, <span style="color:Red">'''25 (={{base|29|10}})'''</span>, <span style="color:Red">'''27 (={{base|31|10}})'''</span></SMALL>
|-
! Fraction
! <SMALL>Prime factors<br>of the denominator</SMALL>
! Positional representation
! Positional representation
! <SMALL>Prime factors<br>of the denominator</SMALL>
! Fraction
|-
| style="text-align:center;"| 1/2
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| 0.5
| 0.6
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| style="text-align:center;"| 1/2
|-
| style="text-align:center;"| 1/3
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>
| 0.{{overline|3}}
| 0.4
| style="text-align:center;"| <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/3
|-
| style="text-align:center;"| 1/4
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| 0.25
| 0.3
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| style="text-align:center;"| 1/4
|-
| style="text-align:center;"| 1/5
| style="text-align:center;"| <span style="color:Green">'''5'''</span>
| 0.2
| 0.{{overline|2497}}
| style="text-align:center;"| <span style="color:Red">'''5'''</span>
| style="text-align:center;"| 1/5
|-
| style="text-align:center;"| 1/6
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| 0.1{{overline|6}}
| 0.2
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/6
|-
| style="text-align:center;"| 1/7
| style="text-align:center;"| <span style="color:Red">'''7'''</span>
| 0.{{overline|142857}}
| 0.{{overline|186{{d2}}35}}
| style="text-align:center;"| <span style="color:Red">'''7'''</span>
| style="text-align:center;"| 1/7
|-
| style="text-align:center;"| 1/8
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| 0.125
| 0.16
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| style="text-align:center;"| 1/8
|-
| style="text-align:center;"| 1/9
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>
| 0.{{overline|1}}
| 0.14
| style="text-align:center;"| <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/9
|-
| style="text-align:center;"| 1/10
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
| 0.1
| 0.1{{overline|2497}}
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''5'''</span>
| style="text-align:center;"| 1/{{d2}}
|-
| style="text-align:center;"| 1/11
| style="text-align:center;"| <span style="color:Magenta">'''11'''</span>
| 0.{{overline|09}}
| 0.{{overline|1}}
| style="text-align:center;"| <span style="color:Blue">'''{{d3}}'''</span>
| style="text-align:center;"| 1/{{d3}}
|-
| style="text-align:center;"| 1/12
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| 0.08{{overline|3}}
| 0.1
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/10
|-
| style="text-align:center;"| 1/13
| style="text-align:center;"| <span style="color:Red">'''13'''</span>
| 0.{{overline|076923}}
| 0.{{overline|0{{d3}}}}
| style="text-align:center;"| <span style="color:Magenta">'''11'''</span>
| style="text-align:center;"| 1/11
|-
| style="text-align:center;"| 1/14
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| 0.0{{overline|714285}}
| 0.0{{overline|{{d2}}35186}}
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| style="text-align:center;"| 1/12
|-
| style="text-align:center;"| 1/15
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>
| 0.0{{overline|6}}
| 0.0{{overline|9724}}
| style="text-align:center;"| <span style="color:Green">'''3'''</span>, <span style="color:Red">'''5'''</span>
| style="text-align:center;"| 1/13
|-
| style="text-align:center;"| 1/16
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| 0.0625
| 0.09
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| style="text-align:center;"| 1/14
|-
| style="text-align:center;"| 1/17
| style="text-align:center;"| <span style="color:Red">'''17'''</span>
| 0.{{overline|0588235294117647}}
| 0.{{overline|08579214{{d3}}36429{{d2}}7}}
| style="text-align:center;"| <span style="color:Red">'''15'''</span>
| style="text-align:center;"| 1/15
|-
| style="text-align:center;"| 1/18
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| 0.0{{overline|5}}
| 0.08
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/16
|-
| style="text-align:center;"| 1/19
| style="text-align:center;"| <span style="color:Red">'''19'''</span>
| 0.{{overline|052631578947368421}}
| 0.{{overline|076{{d3}}45}}
| style="text-align:center;"| <span style="color:Red">'''17'''</span>
| style="text-align:center;"| 1/17
|-
| style="text-align:center;"| 1/20
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
| 0.05
| 0.0{{overline|7249}}
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''5'''</span>
| style="text-align:center;"| 1/18
|-
| style="text-align:center;"| 1/21
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''7'''</span>
| 0.{{overline|047619}}
| 0.0{{overline|6{{d2}}3518}}
| style="text-align:center;"| <span style="color:Green">'''3'''</span>, <span style="color:Red">'''7'''</span>
| style="text-align:center;"| 1/19
|-
| style="text-align:center;"| 1/22
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span>
| 0.0{{overline|45}}
| 0.0{{overline|6}}
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''{{d3}}'''</span>
| style="text-align:center;"| 1/1{{d2}}
|-
| style="text-align:center;"| 1/23
| style="text-align:center;"| <span style="color:Red">'''23'''</span>
| 0.{{overline|0434782608695652173913}}
| 0.{{overline|06316948421}}
| style="text-align:center;"| <span style="color:Red">'''1{{d3}}'''</span>
| style="text-align:center;"| 1/1{{d3}}
|-
| style="text-align:center;"| 1/24
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| 0.041{{overline|6}}
| 0.06
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/20
|-
| style="text-align:center;"| 1/25
| style="text-align:center;"| <span style="color:Green">'''5'''</span>
| 0.04
| 0.{{overline|05915343{{d2}}0{{d3}}62{{d2}}68781{{d3}}}}
| style="text-align:center;"| <span style="color:Red">'''5'''</span>
| style="text-align:center;"| 1/21
|-
| style="text-align:center;"| 1/26
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span>
| 0.0{{overline|384615}}
| 0.0{{overline|56}}
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span>
| style="text-align:center;"| 1/22
|-
| style="text-align:center;"| 1/27
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>
| 0.{{overline|037}}
| 0.054
| style="text-align:center;"| <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/23
|-
| style="text-align:center;"| 1/28
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| 0.03{{overline|571428}}
| 0.0{{overline|5186{{d2}}3}}
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| style="text-align:center;"| 1/24
|-
| style="text-align:center;"| 1/29
| style="text-align:center;"| <span style="color:Red">'''29'''</span>
| 0.{{overline|0344827586206896551724137931}}
| 0.{{overline|04{{d3}}7}}
| style="text-align:center;"| <span style="color:Red">'''25'''</span>
| style="text-align:center;"| 1/25
|-
| style="text-align:center;"| 1/30
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>
| 0.0{{overline|3}}
| 0.0{{overline|4972}}
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>, <span style="color:Red">'''5'''</span>
| style="text-align:center;"| 1/26
|-
| style="text-align:center;"| 1/31
| style="text-align:center;"| <span style="color:Red">'''31'''</span>
| 0.{{overline|032258064516129}}
| 0.{{overline|0478{{d2}}{{d2}}093598166{{d3}}74311{{d3}}28623{{d2}}55}}
| style="text-align:center;"| <span style="color:Red">'''27'''</span>
| style="text-align:center;"| 1/27
|-
| style="text-align:center;"| 1/32
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| 0.03125
| 0.046
| style="text-align:center;"| <span style="color:Green">'''2'''</span>
| style="text-align:center;"| 1/28
|-
| style="text-align:center;"| 1/33
| style="text-align:center;"| <span style="color:Blue">'''3'''</span>, <span style="color:Magenta">'''11'''</span>
| 0.{{overline|03}}
| 0.0{{overline|4}}
| style="text-align:center;"| <span style="color:Green">'''3'''</span>, <span style="color:Blue">'''{{d3}}'''</span>
| style="text-align:center;"| 1/29
|-
| style="text-align:center;"| 1/34
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''17'''</span>
| 0.0{{overline|2941176470588235}}
| 0.0{{overline|429{{d2}}708579214{{d3}}36}}
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Red">'''15'''</span>
| style="text-align:center;"| 1/2{{d2}}
|-
| style="text-align:center;"| 1/35
| style="text-align:center;"| <span style="color:Green">'''5'''</span>, <span style="color:Red">'''7'''</span>
| 0.0{{overline|285714}}
| 0.{{overline|0414559{{d3}}3931}}
| style="text-align:center;"| <span style="color:Red">'''5'''</span>, <span style="color:Red">'''7'''</span>
| style="text-align:center;"| 1/2{{d3}}
|-
| style="text-align:center;"| 1/36
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| 0.02{{overline|7}}
| 0.04
| style="text-align:center;"| <span style="color:Green">'''2'''</span>, <span style="color:Green">'''3'''</span>
| style="text-align:center;"| 1/30
|}

The duodecimal period length of 1/''n'' are (in decimal)
: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, ... {{OEIS|id=A246004}}

The duodecimal period length of 1/(''n''th prime) are (in decimal)
: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, ... {{OEIS|id=A246489}}

Smallest prime with duodecimal period ''n'' are (in decimal)
: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, ... {{OEIS|id=A252170}}

=== Irrational numbers ===
The representations of [[irrational number]]s in any positional number system (including decimal and duodecimal) neither terminate nor [[Repeating decimal|repeat]]. The following table gives the first digits for some important [[algebraic number|algebraic]] and [[transcendental number|transcendental]] numbers in both decimal and duodecimal.

{| class="wikitable"
! Algebraic irrational number
! In decimal
! In duodecimal
|-
| style="text-align:center;"| [[Square root of 2|{{sqrt|2}}]], the square root of 2
| 1.414213562373...
| 1.4{{d3}}79170{{d2}}07{{d3}}8...
|-
| style="text-align:center;"| {{mvar|[[Golden ratio|φ]]}} (phi), the golden ratio = <math>\tfrac{1+\sqrt{5}}{2}</math>
| 1.618033988749...
| 1.74{{d3}}{{d3}}6772802{{d2}}...
|-
! Transcendental number
! In decimal
! In duodecimal
|-
| style="text-align:center;"| {{mvar|[[Pi|π]]}} (pi), the ratio of a circle's [[circumference]] to its [[diameter]]
| 3.141592653589...
| 3.184809493{{d3}}91...
|-
| style="text-align:center;"| {{mvar|[[E (mathematical constant)|e]]}}, the base of the [[natural logarithm]]
| 2.718281828459...
| 2.875236069821...
|}