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=== 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}}
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