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{{distinguish|Dewey Decimal Classification}}
{{thermodynamics|cTopic=Processes and Cycles}}
{{Numeral systems}}
<!-- BEFORE EDITING THIS PAGE:
The '''duodecimal''' system (also known as '''[[base (exponentiation)|base]]-12''' or '''dozenal''') is a [[positional notation]] [[numeral system]] using [[12 (number)|twelve]] as its [[radix|base]]. In this system, the number [[10 (number)|ten]] may be written as "A", "T" or "X", and the number [[11 (number)|eleven]] as "B" or "E" (another common notation, introduced by Sir [[Isaac Pitman]], is to use a rotated "2" for ten and a reversed "3" for eleven). The number twelve (that is, the number written as "12" in the [[decimal|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 (unit)|gross]]", "1000" means "1 [[great gross]]", and "0.1" means "1 twelfth" (instead of their decimal meanings "1 hundred", "1 thousand", and "1 tenth").
    Note that there is a separate page on the Diesel *engine*.
    If you are going to describe glow plugs, differences between fuels, use in
    trucks etc, that is the place for your contribution. -->
The '''Diesel cycle''' is a combustion process of a reciprocating [[internal combustion engine]]. In it, [[fuel]] is ignited by heat generated during the compression of air in the combustion chamber, into which fuel is then injected. This is in contrast to igniting the fuel-air mixture with a [[spark plug]] as in the [[Otto cycle]] (four-stroke/petrol) engine. Diesel engines ([[heat engine]]s using the Diesel cycle) are used in [[automobile]]s, [[power generation]], [[Diesel-electric transmission|diesel-electric]] [[locomotive]]s, and [[submarines]]. {{dubious|date=November 2014}}


The number twelve, a [[highly composite number]], is the smallest number with four non-trivial [[integer factorization|factor]]s (2, 3, 4, 6), and the smallest to include as factors all four numbers (1 to 4) within the [[subitizing and counting|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, with neither 3 nor 4), duodecimal representations fit more easily than decimal ones into many common patterns, as evidenced by the higher regularity observable in the duodecimal multiplication table. As a result, duodecimal is sometimes named the number system with the most optimal [[radix economy]].{{citation needed|date=May 2012}} Of its factors, 2 and 3 are [[prime number|prime]], which means the [[multiplicative inverse|reciprocals]] of all [[smooth number|3-smooth]] numbers (such as 2, 3, 4, 6, 8, 9...) have a [[terminating decimal|terminating]] representation in duodecimal. In particular, the five most elementary fractions (½, ⅓, ⅔, ¼ and ¾) all have a short terminating representation in duodecimal (0.6, 0.4, 0.8, 0.3 and 0.9, respectively), and twelve is the smallest radix with this feature (since it is the [[least common 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 numeral system|binary]], [[octal]] and [[hexadecimal]] systems, although the [[sexagesimal]] system (where the reciprocals of all [[regular number|5-smooth]] numbers terminate) does better in this respect (but at the cost of an unwieldy multiplication table).
The [[thermodynamic cycle]] which approximates the Diesel cycle [[pressure]] and [[Volume (thermodynamics)|volume]] of the [[combustion chamber]] of the [[diesel engine]], was invented by [[Rudolph Diesel]] in 1897. It is assumed to have constant pressure during the initial part of the "combustion" phase (<math>V_2</math> to <math>V_3</math> in the diagram, below). This is an idealized mathematical model: real physical diesels do have an increase in pressure during this period, but it is less pronounced than in the Otto cycle. In contrast, the idealized [[Otto cycle]] of a [[four-stroke cycle|gasoline engine]] approximates a constant volume process during that phase.


== Origin ==
== The Idealized Diesel Cycle ==
[[Image:DieselCycle PV.svg|thumb|300px|left|p-V Diagram for the ideal Diesel cycle. The cycle follows the numbers 1-4 in clockwise direction.]]


:''In this section, numerals are based on decimal [[Numerical digit|places]]. For example, 10 means [[10 (number)|ten]], 12 means [[12 (number)|twelve]].''
The image on the left shows a p-V diagram for the ideal Diesel cycle; where <math>p</math> is [[pressure]] and V the volume or <math>v</math> the [[specific volume]] if the process is placed on a unit mass basis. The ideal Diesel cycle follows the following four distinct processes:


Languages using duodecimal number systems are uncommon. Languages in the [[Nigeria]]n Middle Belt such as [[Janji]], [[Gbiri-Niragu]] (Kahugu), the Nimbia dialect of [[Gwandara]];<ref>{{citation
* Process 1 to 2 is [[isentropic]] compression of the fluid (blue)
| title=Decimal vs. Duodecimal: An interaction between two systems of numeration
* Process 2 to 3 is [[Reversible process (thermodynamics)|reversible]] constant pressure heating (red)
| last=Matsushita
* Process 3 to 4 is isentropic expansion (yellow)
| first=Shuji
* Process 4 to 1 is reversible constant volume cooling (green)<ref>Eastop & McConkey 1993, ''Applied Thermodynamics for Engineering Technologists'', Pearson Education Limited, Fifth Edition, p.137</ref>
| journal=2nd Meeting of the AFLANG, October 1998, Tokyo
| year=1998
| url=http://web.archive.org/web/20081005230737/http://www3.aa.tufs.ac.jp/~P_aflang/TEXTS/oct98/decimal.html
| accessdate=2011-05-29
}}</ref> the [[Chepang_language]] of [[Nepal]]<ref>{{citation
| contribution=Les principes de construction du nombre dans les langues tibéto-birmanes
| first=Martine
| last=Mazaudon
| title=La Pluralité
| editor-first=Jacques
| editor-last=François
| year=2002
| pages=91–119
| publisher=Peeters
| place=Leuven
| isbn=90-429-1295-2
| url=http://halshs.archives-ouvertes.fr/docs/00/16/68/91/PDF/numerationTB_SLP.pdf
}}</ref> and the [[Mahl language]] of [[Minicoy Island]] in [[India]] are known to use duodecimal numerals.  In fiction, [[J. R. R. Tolkien]]'s [[Elvish languages]] use a hybrid decimal-duodecimal system, primarily decimal but with special names for multiples of six.


[[Germanic languages]] have special words for 11 and 12, such as ''eleven'' and ''twelve'' in [[English language|English]], which are often misinterpreted as vestiges of a duodecimal system.{{citation needed|date=January 2012}}  However, they are considered to come from [[Proto-Germanic]] *''ainlif'' and *''twalif'' (respectively ''one left'' and ''two left''), both of which were decimal.
The Diesel engine is a heat engine: it converts [[heat]] into [[Work (thermodynamics)|work]].  During the bottom isentropic processes (blue), energy is transferred into the system in the form of work <math>W_{in}</math>, but by definition (isentropic) no energy is transferred into or out of the system in the form of heat. During the constant pressure (red, [[isobaric process |isobaric]]) process, energy enters the system as heat <math>Q_{in}</math>. During the top isentropic processes (yellow), energy is transferred out of the system in the form of <math>W_{out}</math>, but by definition (isentropic) no energy is transferred into or out of the system in the form of heat. During the constant volume (green, [[isochoric process|isochoric]]) process, some of energy flows out of the system as heat through the right depressurizing process <math>Q_{out}</math>. The work that leaves the system is equal to the work that enters the system plus the difference between the heat added to the system and the heat that leaves the system; in other words, net gain of work is equal to the difference between the heat added to the system and the heat that leaves the system.


Historically, [[Unit of measurement|unit]]s of [[time]] in many [[civilization]]s 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 calendar]]s, clocks, and compasses are based on the twelve [[Earthly Branches]].
* Work in (<math>W_{in}</math>) is done by the piston compressing the air (system)
* Heat in (<math>Q_{in}</math>) is done by the [[combustion]] of the fuel
* Work out (<math>W_{out}</math>) is done by the working fluid expanding and pushing a piston (this produces usable work)
* Heat out (<math>Q_{out}</math>) is done by venting the air
* Net work produced = <math>Q_{in}</math> - <math>Q_{out}</math>


Being a versatile denominator in fractions may explain why we have 12 inches in an imperial foot, 12 ounces in a [[troy weight|troy]] pound, 12 [[British One Penny coin (pre-decimal)|old British pence]] in a [[shilling]], 24 (12×2) hours in a day, and many other items counted by the [[dozen]], [[gross (unit)|gross]] ([[144 (number)|144]], [[square number|square]] of 12) or [[great gross]] ([[1728 (number)|1728]], [[cube (arithmetic)|cube]] of 12). The Romans used a fraction system based on 12, including the [[uncia (length)|uncia]] which became both the English words ''[[ounce]]'' and ''inch''.  Pre-[[Decimal Day|decimalisation]], the [[United Kingdom]] and [[Republic of Ireland]] 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.
The net work produced is also represented by the area enclosed by the cycle on the P-V diagram. The net work is produced per cycle and is also called the useful work, as it can be turned to other useful types of energy and propel a vehicle ([[kinetic energy]]) or produce electrical energy. The summation of many such cycles per unit of time is called the developed power. The <math>W_{out}</math> is also called the gross work, some of which is used in the next cycle of the engine to compress the next charge of air.


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 ([[Phalanx bone|phalanges]]) on one hand (three on each of four fingers).<ref>{{citation
| title=ヒマラヤの満月と十二進法 (The Full Moon in the Himalayas and the Duodecimal System)
| last=Nishikawa
| first=Yoshiaki
| year=2002
| url=http://www.kankyok.co.jp/nue/nue11/nue11_01.html
| accessdate=2008-03-24
}}</ref> It is possible to count to 12 with your 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.<ref name=Ifrah>{{Citation
  | last = Ifrah
  | first = Georges
  | author-link = Georges Ifrah
  | title = The Universal History of Numbers: From prehistory to the invention of the computer.
  | publisher = [[John Wiley and Sons]]
  | year= 2000
  | page =
  | isbn = 0-471-39340-1
}}. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk.</ref><ref name=Macey>{{Citation|last=Macey|first=Samuel L.|title=The Dynamics of Progress: Time, Method, and Measure|year=1989|publisher=University of Georgia Press|location=Atlanta, Georgia|isbn=978-0-8203-3796-8|pages=92|url=http://books.google.com/books?id=xlzCWmXguwsC&pg=PA92&lpg=PA92}}</ref>


== Places ==
{{clear}}
In a duodecimal place system, [[10 (number)|ten]] can be written as A, [[11 (number)|eleven]] can be written as B, and twelve is written as 10. For alternative symbols, see [[#Advocacy and "dozenalism"|below]].


According to this notation, duodecimal 50 expresses the same quantity as decimal [[60 (number)|60]] (= five times twelve), duodecimal 60 is equivalent to decimal [[72 (number)|72]] (= six times twelve = half a gross), duodecimal 100 has the same value as decimal [[144 (number)|144]] (= twelve times twelve = one gross), etc.
=== Maximum thermal efficiency ===
The maximum thermal efficiency of a Diesel cycle is dependent on the compression ratio and the cut-off ratio. It has the following formula under cold [[Standard state|air standard]] analysis:


== Comparison to other numeral systems ==
<math>\eta_{th}=1-\frac{1}{r^{\gamma-1}}\left ( \frac{\alpha^{\gamma}-1}{\gamma(\alpha-1)} \right )</math>


{| class="wikitable" style="float:right; text-align:center"
where
|+ A duodecimal [[multiplication table]]
:<math>\eta_{th} </math> is [[thermal efficiency]]
|-
:<math>\alpha</math> is the cut-off ratio <math>\frac{V_3}{V_2}</math> (ratio between the end and start volume for the combustion phase)
! style="width:1.4em" | 1
:{{math|r}} is the [[compression ratio]] <math>\frac{V_1}{V_2}</math>
! style="width:1.4em" | 2
:<math>\gamma </math> is ratio of [[specific heat capacity|specific heats]] (C<sub>p</sub>/C<sub>v</sub>)<ref>[http://230nsc1.phy-astr.gsu.edu/hbase/thermo/diesel.html The Diesel Engine]</ref>
! style="width:1.4em" | 3
! style="width:1.4em" | 4
! style="width:1.4em" | 5
! style="width:1.4em" | 6
! style="width:1.4em" | 7
! style="width:1.4em" | 8
! style="width:1.4em" | 9
! style="width:1.4em" | A
! style="width:1.4em" | B
! style="width:1.4em" | 10
|-
! 2
| 4 || 6|| 8 || A || 10 || 12 || 14 || 16 || 18 || 1A || 20
|-
! 3
| 6 || 9 || 10 || 13 || 16 || 19 || 20 || 23 || 26 || 29 || 30
|-
! 4
| 8 || 10 || 14 || 18 || 20 || 24 || 28 || 30 || 34 || 38 || 40
|-
! 5
| A || 13 || 18 || 21 || 26 || 2B || 34 || 39 || 42 || 47 || 50
|-
! 6
| 10 || 16 || 20 || 26 || 30 || 36 || 40 || 46 || 50 || 56 || 60
|-
! 7
| 12 || 19 || 24 || 2B || 36 || 41 || 48 || 53 || 5A || 65 || 70
|-
! 8
| 14 || 20 || 28 || 34 || 40 || 48 || 54 || 60 || 68 || 74 || 80
|-
! 9
| 16 || 23 || 30 || 39 || 46 || 53 || 60 || 69 || 76 || 83 || 90
|-
! A
| 18 || 26 || 34 || 42 || 50 || 5A || 68 || 76 || 84 || 92 || A0
|-
! B
| 1A || 29 || 38 || 47 || 56 || 65 || 74 || 83 || 92 || A1 || B0
|-
! 10
| 20 || 30 || 40 || 50 || 60 || 70 || 80 || 90 || A0 || B0 || 100
|}
The number 12 has six factors, which are [[1 (number)|1]], [[2 (number)|2]], [[3 (number)|3]], [[4 (number)|4]], [[6 (number)|6]], and [[12 (number)|12]], of which 2 and 3 are [[prime number|prime]].  The decimal system has only four factors, which are [[1 (number)|1]], [[2 (number)|2]], [[5 (number)|5]], and [[10 (number)|10]]; of which 2 and 5 are prime. Vigesimal adds two factors to those of ten, namely [[4 (number)|4]] and [[20 (number)|20]], but no additional prime factor. Although twenty has 6 factors, 2 of them prime, similarly to twelve, it is also a much larger base (i.e., the digit set and the multiplication table are much larger). Binary has only two factors, 1 and 2, the latter being prime. Hexadecimal has five factors, adding 4, [[8 (number)|8]] and [[16 (number)|16]] to those of 2, but no additional prime. [[Base 30|Trigesimal]] 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), the smallest system that has four different prime factors is Base 210 and the pattern follows the [[primorial]]s. [[Sexagesimal]]—which the ancient [[Sumerians]] and [[Babylonia]]ns among others actually used—adds the four convenient factors 4, 12, 20, and 60 to this but no new prime factors.{{-}}<!-- the {{-}} template keeps the multiplication table from squeezing the heading for the next section-->


== Conversion tables to and from decimal ==
The cut-off ratio can be expressed in terms of temperature as shown below:
:<math>\frac{T_2}{T_1} ={\left(\frac{V_1}{V_2}\right)^{\gamma-1}} = r^{\gamma-1}</math>


To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under [[Positional notation#Base conversion|positional notation]]). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any dozenal number between 0.01 and BBB,BBB.BB to decimal, or any decimal number between 0.01 and 999,999.99 to dozenal. To use them, we first decompose the given number into a sum of numbers with only one significant digit each. For example:
:<math> \displaystyle {T_2} ={T_1} r^{\gamma-1} </math>


123,456.78 = 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08
:<math>\frac{V_3}{V_2} = \frac{T_3}{T_2}</math>


This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 102,304.05), these are, of course, left out in the digit decomposition (102,304.05 = 100,000 + 2,000 + 300 + 4 + 0.05). Then we use the digit conversion tables to obtain the equivalent value in the target base for each digit. If the given number is in dozenal and the target base is decimal, we get:
:<math>\alpha = \left(\frac{T_3}{T_1}\right)\left(\frac{1}{r^{\gamma-1}}\right)</math>


<small>(dozenal)</small> 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = <small>(decimal)</small> 248,832 + 41,472 + 5,184 + 576 + 60 + 6 + 0.58{{overline|3}}333333333... + 0.0{{overline|5}}5555555555...
<math>T_3</math> can be approximated to the flame temperature of the fuel used. The flame temperature can be approximated to the [[adiabatic flame temperature]] of the fuel with corresponding air-to-fuel ratio and compression pressure, <math>p_3</math>.
<math>T_1</math> can be approximated to the inlet air temperature.


Now, since the summands are already converted to base ten, we use the usual decimal arithmetic to perform the addition and recompose the number, arriving at the conversion result:
This formula only gives the ideal thermal efficiency. The actual thermal efficiency will be significantly lower due to heat and friction losses. The formula is more complex than the Otto cycle (petrol/gasoline engine) relation that has the following formula;


  Dozenal  -----> Decimal
<math>\eta_{otto,th}=1-\frac{1}{r^{\gamma-1}}</math>
 
  100,000    =   248,832
    20,000    =    41,472
    3,000    =      5,184
      400    =        576
        50    =        60
  +      6    =  +      6
        0.7  =          0.58{{overline|3}}333333333...
        0.08  =          0.0{{overline|5}}5555555555...
--------------------------------------------
  123,456.78  =    296,130.63{{overline|8}}888888888...


That is, <small>(dozenal)</small> 123,456.78 equals <small>(decimal)</small> 296,130.63{{overline|8}}888888888... ≈ 296,130.64
The additional complexity for the Diesel formula comes around since the heat addition is at constant pressure and the heat rejection is at constant volume. The Otto cycle by comparison has both the heat addition and rejection at constant volume.


If the given number is in decimal and the target base is dozenal, the method is basically same. Using the digit conversion tables:
Comparing the two formulae it can be seen that for a given compression ratio ({{math|r}}), the ideal Otto cycle will be more efficient. However, a [[diesel engine]] will be more efficient overall since it will have the ability to operate at higher compression ratios.  If a petrol engine were to have the same compression ratio, then knocking (self-ignition) would occur and this would severely reduce the efficiency, whereas in a diesel engine, the self ignition is the desired behavior.  Additionally, both of these cycles are only idealizations, and the actual behavior does not divide as clearly or sharply. And the ideal Otto cycle formula stated above does not include throttling losses, which do not apply to diesel engines.


<small>(decimal)</small> 100,000 + 20,000 + 3,000 + 400 + 50 + 6 + 0.7 + 0.08 = <small>(dozenal)</small> 49,A54 + B,6A8 + 1,8A0 + 294 + 42 + 6 + 0.8{{overline|4972}}4972497249724972497... + 0.{{overline|0B62A68781B05915343A}}0B62...
== Applications ==
===Diesel engines ===
{{main|Diesel engine}}
Diesel engines have the lowest [[specific fuel consumption (shaft engine)|specific fuel consumption]] of any large internal combustion engine employing a single cycle, 0.26&nbsp;lb/hp·h (0.16&nbsp;kg/kWh) for very large marine engines (combined cycle power plants are more efficient, but employ two engines rather than one). Two-stroke diesels with high pressure forced induction, particularly [[turbocharging]], make up a large percentage of the very largest diesel engines.


However, in order to do this sum and recompose the number, we now have to use the addition tables for dozenal, instead of the addition tables for decimal most people are already familiar with, because the summands are now in base twelve and so the arithmetic with them has to be in dozenal as well. In decimal, 6 + 6 equals 12, but in dozenal it equals 10; so if we used decimal arithmetic with dozenal numbers we would arrive at an incorrect result. Doing the arithmetic properly in dozenal, we get the result:
In [[North America]], diesel engines are primarily used in large trucks, where the low-stress, high-efficiency cycle leads to much longer engine life and lower operational costs. These advantages also make the diesel engine ideal for use in the heavy-haul railroad environment.


  Decimal  ----->  Dozenal
=== Other internal combustion engines without spark plugs ===
 
Many [[model airplane]]s use very simple "glow" and "diesel" engines.  Glow engines use [[glow plug]]s.  "Diesel" model airplane engines have variable compression ratios.  Both types depend on special fuels.
  100,000    =     49,A54
    20,000    =     B,6A8
    3,000    =     1,8A0
      400    =       294
        50    =        42
  +      6    =   +      6
        0.7  =         0.8{{overline|4972}}4972497249724972497...
        0.08 =          0.{{overline|0B62A68781B05915343A}}0B62...
  --------------------------------------------------------
  123,456.78 =    5B,540.9{{overline|43A0B62A68781B059153}}43A...


That is, <small>(decimal)</small> 123,456.78 equals <small>(dozenal)</small> 5B,540.9{{overline|43A0B62A68781B059153}}43A... ≈ 5B,540.94
Some 19th-century or earlier experimental engines used external flames, exposed by valves, for ignition, but this becomes less attractive with increasing compression. (It was the research of [[Nicolas Léonard Sadi Carnot]] that established the thermodynamic value of compression.) A historical implication of this is that the diesel engine could have been invented without the aid of electricity.
<br /> See the development of the  [[hot bulb engine]] and [[indirect injection]] for historical significance.


=== Dozenal to decimal digit conversion ===
== References ==
 
<references/>
{|class="wikitable"
|-
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
| bgcolor=#c0c0c0 | '''''Doz.'''''
| ''Dec.''
|-
| bgcolor=#c0c0c0 | '''100,000'''
| 248,832
| bgcolor=#c0c0c0 | '''10,000'''
| 20,736
| bgcolor=#c0c0c0 | '''1,000'''
| 1,728
| bgcolor=#c0c0c0 | '''100'''
| 144
| bgcolor=#c0c0c0 | '''10'''
| 12
| bgcolor=#c0c0c0 | '''1'''
| 1
| bgcolor=#c0c0c0 | '''0.1'''
| 0.08{{overline|3}}
| bgcolor=#c0c0c0 | '''0.01'''
| 0.0069{{overline|4}}
|-
| bgcolor=#c0c0c0 | '''200,000'''
| 497,664
| bgcolor=#c0c0c0 | '''20,000'''
| 41,472
| bgcolor=#c0c0c0 | '''2,000'''
| 3,456
| bgcolor=#c0c0c0 | '''200'''
| 288
| bgcolor=#c0c0c0 | '''20'''
| 24
| bgcolor=#c0c0c0 | '''2'''
| 2
| bgcolor=#c0c0c0 | '''0.2'''
| 0.1{{overline|6}}
| bgcolor=#c0c0c0 | '''0.02'''
| 0.013{{overline|8}}
|-
| bgcolor=#c0c0c0 | '''300,000'''
| 746,496
| bgcolor=#c0c0c0 | '''30,000'''
| 62,208
| bgcolor=#c0c0c0 | '''3,000'''
| 5,184
| bgcolor=#c0c0c0 | '''300'''
| 432
| bgcolor=#c0c0c0 | '''30'''
| 36
| bgcolor=#c0c0c0 | '''3'''
| 3
| bgcolor=#c0c0c0 | '''0.3'''
| 0.25
| bgcolor=#c0c0c0 | '''0.03'''
| 0.0208{{overline|3}}
|-
| bgcolor=#c0c0c0 | '''400,000'''
| 995,328
| bgcolor=#c0c0c0 | '''40,000'''
| 82,944
| bgcolor=#c0c0c0 | '''4,000'''
| 6,912
| bgcolor=#c0c0c0 | '''400'''
| 576
| bgcolor=#c0c0c0 | '''40'''
| 48
| bgcolor=#c0c0c0 | '''4'''
| 4
| bgcolor=#c0c0c0 | '''0.4'''
| 0.{{overline|3}}
| bgcolor=#c0c0c0 | '''0.04'''
| 0.02{{overline|7}}
|-
| bgcolor=#c0c0c0 | '''500,000'''
| 1,244,160
| bgcolor=#c0c0c0 | '''50,000'''
| 103,680
| bgcolor=#c0c0c0 | '''5,000'''
| 8,640
| bgcolor=#c0c0c0 | '''500'''
| 720
| bgcolor=#c0c0c0 | '''50'''
| 60
| bgcolor=#c0c0c0 | '''5'''
| 5
| bgcolor=#c0c0c0 | '''0.5'''
| 0.41{{overline|6}}
| bgcolor=#c0c0c0 | '''0.05'''
| 0.0347{{overline|2}}
|-
| bgcolor=#c0c0c0 | '''600,000'''
| 1,492,992
| bgcolor=#c0c0c0 | '''60,000'''
| 124,416
| bgcolor=#c0c0c0 | '''6,000'''
| 10,368
| bgcolor=#c0c0c0 | '''600'''
| 864
| bgcolor=#c0c0c0 | '''60'''
| 72
| bgcolor=#c0c0c0 | '''6'''
| 6
| bgcolor=#c0c0c0 | '''0.6'''
| 0.5
| bgcolor=#c0c0c0 | '''0.06'''
| 0.041{{overline|6}}
|-
| bgcolor=#c0c0c0 | '''700,000'''
| 1,741,824
| bgcolor=#c0c0c0 | '''70,000'''
| 145,152
| bgcolor=#c0c0c0 | '''7,000'''
| 12,096
| bgcolor=#c0c0c0 | '''700'''
| 1008
| bgcolor=#c0c0c0 | '''70'''
| 84
| bgcolor=#c0c0c0 | '''7'''
| 7
| bgcolor=#c0c0c0 | '''0.7'''
| 0.58{{overline|3}}
| bgcolor=#c0c0c0 | '''0.07'''
| 0.0486{{overline|1}}
|-
| bgcolor=#c0c0c0 | '''800,000'''
| 1,990,656
| bgcolor=#c0c0c0 | '''80,000'''
| 165,888
| bgcolor=#c0c0c0 | '''8,000'''
| 13,824
| bgcolor=#c0c0c0 | '''800'''
| 1152
| bgcolor=#c0c0c0 | '''80'''
| 96
| bgcolor=#c0c0c0 | '''8'''
| 8
| bgcolor=#c0c0c0 | '''0.8'''
| 0.{{overline|6}}
| bgcolor=#c0c0c0 | '''0.08'''
| 0.0{{overline|5}}
|-
| bgcolor=#c0c0c0 | '''900,000'''
| 2,239,488
| bgcolor=#c0c0c0 | '''90,000'''
| 186,624
| bgcolor=#c0c0c0 | '''9,000'''
| 15,552
| bgcolor=#c0c0c0 | '''900'''
| 1,296
| bgcolor=#c0c0c0 | '''90'''
| 108
| bgcolor=#c0c0c0 | '''9'''
| 9
| bgcolor=#c0c0c0 | '''0.9'''
| 0.75
| bgcolor=#c0c0c0 | '''0.09'''
| 0.0625
|-
| bgcolor=#c0c0c0 | '''A00,000'''
| 2,488,320
| bgcolor=#c0c0c0 | '''A0,000'''
| 207,360
| bgcolor=#c0c0c0 | '''A,000'''
| 17,280
| bgcolor=#c0c0c0 | '''A00'''
| 1,440
| bgcolor=#c0c0c0 | '''A0'''
| 120
| bgcolor=#c0c0c0 | '''A'''
| 10
| bgcolor=#c0c0c0 | '''0.A'''
| 0.8{{overline|3}}
| bgcolor=#c0c0c0 | '''0.0A'''
| 0.069{{overline|4}}
|-
| bgcolor=#c0c0c0 | '''B00,000'''
| 2,737,152
| bgcolor=#c0c0c0 | '''B0,000'''
| 228,096
| bgcolor=#c0c0c0 | '''B,000'''
| 19,008
| bgcolor=#c0c0c0 | '''B00'''
| 1,584
| bgcolor=#c0c0c0 | '''B0'''
| 132
| bgcolor=#c0c0c0 | '''B'''
| 11
| bgcolor=#c0c0c0 | '''0.B'''
| 0.91{{overline|6}}
| bgcolor=#c0c0c0 | '''0.0B'''
| 0.0763{{overline|8}}
|}
 
=== Decimal to dozenal digit conversion ===
 
{|class="wikitable"
|-
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
|-
| bgcolor=#c0c0c0 | '''100,000'''
| 49,A54
| bgcolor=#c0c0c0 | '''10,000'''
| 5,954
| bgcolor=#c0c0c0 | '''1,000'''
| 6B4
| bgcolor=#c0c0c0 | '''100'''
| 84
| bgcolor=#c0c0c0 | '''10'''
| A
| bgcolor=#c0c0c0 | '''1'''
| 1
| bgcolor=#c0c0c0 | '''0.1'''
| 0.1{{overline|2497}}
| bgcolor=#c0c0c0 | '''0.01'''
| 0.0{{overline|15343A0B62A68781B059}}
|-
| bgcolor=#c0c0c0 | '''200,000'''
| 97,8A8
| bgcolor=#c0c0c0 | '''20,000'''
| B,6A8
| bgcolor=#c0c0c0 | '''2,000'''
| 1,1A8
| bgcolor=#c0c0c0 | '''200'''
| 148
| bgcolor=#c0c0c0 | '''20'''
| 18
| bgcolor=#c0c0c0 | '''2'''
| 2
| bgcolor=#c0c0c0 | '''0.2'''
| 0.{{overline|2497}}
| bgcolor=#c0c0c0 | '''0.02'''
| 0.0{{overline|2A68781B05915343A0B6}}
|-
| bgcolor=#c0c0c0 | '''300,000'''
| 125,740
| bgcolor=#c0c0c0 | '''30,000'''
| 15,440
| bgcolor=#c0c0c0 | '''3,000'''
| 1,8A0
| bgcolor=#c0c0c0 | '''300'''
| 210
| bgcolor=#c0c0c0 | '''30'''
| 26
| bgcolor=#c0c0c0 | '''3'''
| 3
| bgcolor=#c0c0c0 | '''0.3'''
| 0.3{{overline|7249}}
| bgcolor=#c0c0c0 | '''0.03'''
| 0.0{{overline|43A0B62A68781B059153}}
|-
| bgcolor=#c0c0c0 | '''400,000'''
| 173,594
| bgcolor=#c0c0c0 | '''40,000'''
| 1B,194
| bgcolor=#c0c0c0 | '''4,000'''
| 2,394
| bgcolor=#c0c0c0 | '''400'''
| 294
| bgcolor=#c0c0c0 | '''40'''
| 34
| bgcolor=#c0c0c0 | '''4'''
| 4
| bgcolor=#c0c0c0 | '''0.4'''
| 0.{{overline|4972}}
| bgcolor=#c0c0c0 | '''0.04'''
| 0.0{{overline|5915343A0B62A68781B0}}
|-
| bgcolor=#c0c0c0 | '''500,000'''
| 201,428
| bgcolor=#c0c0c0 | '''50,000'''
| 24,B28
| bgcolor=#c0c0c0 | '''5,000'''
| 2,A88
| bgcolor=#c0c0c0 | '''500'''
| 358
| bgcolor=#c0c0c0 | '''50'''
| 42
| bgcolor=#c0c0c0 | '''5'''
| 5
| bgcolor=#c0c0c0 | '''0.5'''
| 0.6
| bgcolor=#c0c0c0 | '''0.05'''
| 0.0{{overline|7249}}
|-
| bgcolor=#c0c0c0 | '''600,000'''
| 24B,280
| bgcolor=#c0c0c0 | '''60,000'''
| 2A,880
| bgcolor=#c0c0c0 | '''6,000'''
| 3,580
| bgcolor=#c0c0c0 | '''600'''
| 420
| bgcolor=#c0c0c0 | '''60'''
| 50
| bgcolor=#c0c0c0 | '''6'''
| 6
| bgcolor=#c0c0c0 | '''0.6'''
| 0.{{overline|7249}}
| bgcolor=#c0c0c0 | '''0.06'''
| 0.0{{overline|8781B05915343A0B62A6}}
|-
| bgcolor=#c0c0c0 | '''700,000'''
| 299,114
| bgcolor=#c0c0c0 | '''70,000'''
| 34,614
| bgcolor=#c0c0c0 | '''7,000'''
| 4,074
| bgcolor=#c0c0c0 | '''700'''
| 4A4
| bgcolor=#c0c0c0 | '''70'''
| 5A
| bgcolor=#c0c0c0 | '''7'''
| 7
| bgcolor=#c0c0c0 | '''0.7'''
| 0.8{{overline|4972}}
| bgcolor=#c0c0c0 | '''0.07'''
| 0.0{{overline|A0B62A68781B05915343}}
|-
| bgcolor=#c0c0c0 | '''800,000'''
| 326,B68
| bgcolor=#c0c0c0 | '''80,000'''
| 3A,368
| bgcolor=#c0c0c0 | '''8,000'''
| 4,768
| bgcolor=#c0c0c0 | '''800'''
| 568
| bgcolor=#c0c0c0 | '''80'''
| 68
| bgcolor=#c0c0c0 | '''8'''
| 8
| bgcolor=#c0c0c0 | '''0.8'''
| 0.{{overline|9724}}
| bgcolor=#c0c0c0 | '''0.08'''
| 0.{{overline|0B62A68781B05915343A}}
|-
| bgcolor=#c0c0c0 | '''900,000'''
| 374,A00
| bgcolor=#c0c0c0 | '''90,000'''
| 44,100
| bgcolor=#c0c0c0 | '''9,000'''
| 5,260
| bgcolor=#c0c0c0 | '''900'''
| 630
| bgcolor=#c0c0c0 | '''90'''
| 76
| bgcolor=#c0c0c0 | '''9'''
| 9
| bgcolor=#c0c0c0 | '''0.9'''
| 0.A{{overline|9724}}
| bgcolor=#c0c0c0 | '''0.09'''
| 0.1{{overline|0B62A68781B05915343A}}
|}
 
=== Conversion of powers ===
 
{|class="wikitable"
|-
| rowspan="2" | ''Exponent''
| colspan="2" | Powers of 2
| colspan="2" | Powers of 3
| colspan="2" | Powers of 4
| colspan="2" | Powers of 5
| colspan="2" | Powers of 6
| colspan="2" | Powers of 7
|-
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
|-
| ''^6''
| bgcolor=#c0c0c0 | '''64'''
| 54
| bgcolor=#c0c0c0 | '''729'''
| 509
| bgcolor=#c0c0c0 | '''4,096'''
| 2454
| bgcolor=#c0c0c0 | '''15,625'''
| 9,061
| bgcolor=#c0c0c0 | '''46,656'''
| 23,000
| bgcolor=#c0c0c0 | '''117,649'''
| 58,101
|-
| ''^5''
| bgcolor=#c0c0c0 | '''32'''
| 28
| bgcolor=#c0c0c0 | '''243'''
| 183
| bgcolor=#c0c0c0 | '''1,024'''
| 714
| bgcolor=#c0c0c0 | '''3,125'''
| 1,985
| bgcolor=#c0c0c0 | '''7,776'''
| 4,600
| bgcolor=#c0c0c0 | '''16,807'''
| 9,887
|-
| ''^4''
| bgcolor=#c0c0c0 | '''16'''
| 14
| bgcolor=#c0c0c0 | '''81'''
| 69
| bgcolor=#c0c0c0 | '''256'''
| 194
| bgcolor=#c0c0c0 | '''625'''
| 441
| bgcolor=#c0c0c0 | '''1,296'''
| 900
| bgcolor=#c0c0c0 | '''2,401'''
| 1,481
|-
| ''^3''
| bgcolor=#c0c0c0 | '''8'''
| 8
| bgcolor=#c0c0c0 | '''27'''
| 23
| bgcolor=#c0c0c0 | '''64'''
| 54
| bgcolor=#c0c0c0 | '''125'''
| A5
| bgcolor=#c0c0c0 | '''216'''
| 160
| bgcolor=#c0c0c0 | '''343'''
| 247
|-
| ''^2''
| bgcolor=#c0c0c0 | '''4'''
| 4
| bgcolor=#c0c0c0 | '''9'''
| 9
| bgcolor=#c0c0c0 | '''16'''
| 14
| bgcolor=#c0c0c0 | '''25'''
| 21
| bgcolor=#c0c0c0 | '''36'''
| 30
| bgcolor=#c0c0c0 | '''49'''
| 41
|-
| ''^1''
| bgcolor=#c0c0c0 | '''2'''
| 2
| bgcolor=#c0c0c0 | '''3'''
| 3
| bgcolor=#c0c0c0 | '''4'''
| 4
| bgcolor=#c0c0c0 | '''5'''
| 5
| bgcolor=#c0c0c0 | '''6'''
| 6
| bgcolor=#c0c0c0 | '''7'''
| 7
|-
| ''^−1''
| bgcolor=#c0c0c0 | '''0.5'''
| 0.6
| bgcolor=#c0c0c0 | '''0.{{overline|3}}'''
| 0.4
| bgcolor=#c0c0c0 | '''0.25'''
| 0.3
| bgcolor=#c0c0c0 | '''0.2'''
| 0.{{overline|2497}}
| bgcolor=#c0c0c0 | '''0.1{{overline|6}}'''
| 0.2
| bgcolor=#c0c0c0 | '''0.{{overline|142857}}'''
| 0.{{overline|186A35}}
|-
| ''^−2''
| bgcolor=#c0c0c0 | '''0.25'''
| 0.3
| bgcolor=#c0c0c0 | '''0.{{overline|1}}'''
| 0.14
| bgcolor=#c0c0c0 | '''0.0625'''
| 0.09
| bgcolor=#c0c0c0 | '''0.04'''
| 0.{{overline|05915343A0<br/>B62A68781B}}
| bgcolor=#c0c0c0 | '''0.02{{overline|7}}'''
| 0.04
| bgcolor=#c0c0c0 | '''0.{{overline|0204081632653<br/>06122448979591<br/>836734693877551}}'''
| 0.{{overline|02B322547A05A<br/>644A9380B908996<br/>741B615771283B}}
|}
 
{|class="wikitable"
|-
| rowspan="2" | ''Exponent''
| colspan="2" | Powers of 8
| colspan="2" | Powers of 9
| colspan="2" | '''Powers of 10'''
| colspan="2" | Powers of 11
| colspan="2" | '''Powers of 12'''
|-
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
| bgcolor=#c0c0c0 | '''''Dec.'''''
| ''Doz.''
|-
| ''^6''
| bgcolor=#c0c0c0 | '''262,144'''
| 107,854
| bgcolor=#c0c0c0 | '''531,441'''
| 217,669
| bgcolor=#c0c0c0 | '''1,000,000'''
| 402,854
| bgcolor=#c0c0c0 | '''1,771,561'''
| 715,261
| bgcolor=#c0c0c0 | '''2,985,984'''
| 1,000,000
|-
| ''^5''
| bgcolor=#c0c0c0 | '''32,768'''
| 16,B68
| bgcolor=#c0c0c0 | '''59,049'''
| 2A,209
| bgcolor=#c0c0c0 | '''100,000'''
| 49,A54
| bgcolor=#c0c0c0 | '''161,051'''
| 79,24B
| bgcolor=#c0c0c0 | '''248,832'''
| 100,000
|-
| ''^4''
| bgcolor=#c0c0c0 | '''4,096'''
| 2,454
| bgcolor=#c0c0c0 | '''6,561'''
| 3,969
| bgcolor=#c0c0c0 | '''10,000'''
| 5,954
| bgcolor=#c0c0c0 | '''14,641'''
| 8,581
| bgcolor=#c0c0c0 | '''20,736'''
| 10,000
|-
| ''^3''
| bgcolor=#c0c0c0 | '''512'''
| 368
| bgcolor=#c0c0c0 | '''729'''
| 509
| bgcolor=#c0c0c0 | '''1,000'''
| 6B4
| bgcolor=#c0c0c0 | '''1,331'''
| 92B
| bgcolor=#c0c0c0 | '''1,728'''
| 1,000
|-
| ''^2''
| bgcolor=#c0c0c0 | '''64'''
| 54
| bgcolor=#c0c0c0 | '''81'''
| 69
| bgcolor=#c0c0c0 | '''100'''
| 84
| bgcolor=#c0c0c0 | '''121'''
| A1
| bgcolor=#c0c0c0 | '''144'''
| 100
|-
| ''^1''
| bgcolor=#c0c0c0 | '''8'''
| 8
| bgcolor=#c0c0c0 | '''9'''
| 9
| bgcolor=#c0c0c0 | '''10'''
| A
| bgcolor=#c0c0c0 | '''11'''
| B
| bgcolor=#c0c0c0 | '''12'''
| 10
|-
| ''^−1''
| bgcolor=#c0c0c0 | '''0.125'''
| 0.16
| bgcolor=#c0c0c0 | '''0.{{overline|1}}'''
| 0.14
| bgcolor=#c0c0c0 | '''0.1'''
| 0.1{{overline|2497}}
| bgcolor=#c0c0c0 | '''0.{{overline|09}}'''
| 0.{{overline|1}}
| bgcolor=#c0c0c0 | '''0.08{{overline|3}}'''
| 0.1
|-
| ''^−2''
| bgcolor=#c0c0c0 | '''0.015625'''
| 0.023
| bgcolor=#c0c0c0 | '''0.{{overline|012345679}}'''
| 0.0194
| bgcolor=#c0c0c0 | '''0.01'''
| 0.0{{overline|15343A0B6<br/>2A68781B059}}
| bgcolor=#c0c0c0 | '''0.{{overline|00826446280<br/>99173553719}}'''
| 0.{{overline|0123456789B}}
| bgcolor=#c0c0c0 | '''0.0069{{overline|4}}'''
| 0.01
|}
 
==Fractions and irrational numbers==
===Fractions===
Duodecimal [[Fraction (mathematics)|fraction]]s may be simple:
 
* {{frac|2}} = 0.6
* {{frac|3}} = 0.4
* {{frac|4}} = 0.3
* {{frac|6}} = 0.2
* {{frac|8}} = 0.16
* {{frac|9}} = 0.14
 
or complicated
 
* {{frac|5}}  = 0.24972497... recurring (easily rounded to 0.25)
* {{frac|7}}  = 0.186A35186A35... recurring (easily rounded to 0.187)
* {{frac|A}}  = 0.124972497... recurring (rounded to 0.125)
* {{frac|B}}  = 0.11111... recurring (rounded to 0.11)
* {{frac|11}} = 0.0B0B... recurring (rounded to 0.0B)
 
{|class="wikitable"
|-
| ''Examples in duodecimal''
| ''Decimal equivalent''
|-
| 1 × ({{frac|5|8}}) = 0.76
| 1 × ({{frac|5|8}}) = 0.625
|-
| 100 × ({{frac|5|8}}) = 76
| 144 × ({{frac|5|8}}) = 90
|-
| {{frac|576|9}} = 76
| {{frac|810|9}} = 90
|-
| {{frac|400|9}} = 54
| {{frac|576|9}} = 64
|-
| 1A.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. Thus, in base-ten (=&nbsp;2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate:  {{frac|8}}&nbsp;=&nbsp;{{frac|(2×2×2)}}, {{frac|20}}&nbsp;=&nbsp;{{frac|(2×2×5)}} and {{frac|500}}&nbsp;=&nbsp;{{frac|(2×2×5×5×5)}} can be expressed exactly as 0.125, 0.05 and 0.002 respectively. {{frac|3}} and {{frac|7}}, however, recur (0.333... and 0.142857142857...). In the duodecimal (=&nbsp;2×2×3) system, {{frac|8}} is exact; {{frac|20}} and {{frac|500}} recur because they include 5 as a factor; {{frac|3}} is exact; and {{frac|7}} recurs, just as it does in decimal.
 
=== Recurring digits ===
 
Arguably, factors of 3 are more commonly encountered in real-life [[division (mathematics)|division]] problems than factors of 5 (or would be, were it not for the decimal system having influenced most cultures). Thus, in practical applications, the nuisance of [[recurring decimals]] is encountered less often when duodecimal notation is used. Advocates of duodecimal systems argue that this is particularly true of financial calculations, in which the twelve months of the year often enter into calculations.
 
However, when recurring fractions <i>do</i> occur in duodecimal notation, they are less likely to have a very short period than in decimal notation, because [[12 (number)|12]] (twelve) is between two [[prime number]]s, [[11 (number)|11]] (eleven) and [[13 (number)|13]] (thirteen), whereas ten is adjacent to [[composite number]] [[9 (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 (number)|3]] in its factorization, while only one out of every five contains the prime factor [[5 (number)|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 (number)|2]] appears twice in the factorization of twelve, while 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 dozenal than in decimal (e.g., 1/(2<sup>2</sup>) = 0.25 <SMALL>dec</SMALL> = 0.3 <SMALL>doz</SMALL>; 1/(2<sup>3</sup>) = 0.125 <SMALL>dec</SMALL> = 0.16 <SMALL>doz</SMALL>; 1/(2<sup>4</sup>) = 0.0625 <SMALL>dec</SMALL> = 0.09 <SMALL>doz</SMALL>; 1/(2<sup>5</sup>) = 0.03125 <SMALL>dec</SMALL> = 0.046 <SMALL>doz</SMALL>; etc.).
 
{|class="wikitable"
| colspan="3" align="center" | Decimal base<br><SMALL>Prime factors of the base: <font style="color:Green">'''2'''</font>, <font style="color:Green">'''5'''</font></SMALL>
| colspan="3" align="center" | '''Duodecimal / Dozenal base'''<br><SMALL>Prime factors of the base: <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font></SMALL>
|-
| align="center" | Fraction
| align="center" | <SMALL>Prime factors<br>of the denominator<SMALL>
| align="center" | Positional representation
| align="center" | Positional representation
| align="center" | <SMALL>Prime factors<br>of the denominator<SMALL>
| align="center" | Fraction
|-
| align="center" | 1/2
| align="center" | <font style="color:Green">'''2'''</font>
| '''0.5'''
| '''0.6'''
| align="center" | <font style="color:Green">'''2'''</font>
| align="center" | 1/2
|-
| align="center" | 1/3
| align="center" | <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.'''3333... = '''0.'''{{overline|3}}
| '''0.4'''
| align="center" | <font style="color:Green">'''3'''</font>
| align="center" | 1/3
|-
| align="center" | 1/4
| align="center" | <font style="color:Green">'''2'''</font>
| '''0.25'''
| '''0.3'''
| align="center" | <font style="color:Green">'''2'''</font>
| align="center" | 1/4
|-
| align="center" | 1/5
| align="center" | <font style="color:Green">'''5'''</font>
| '''0.2'''
| bgcolor=#c0c0c0 | '''0.'''24972497... = '''0.'''{{overline|2497}}
| align="center" | <font style="color:Red">'''5'''</font>
| align="center" | 1/5
|-
| align="center" | 1/6
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.1'''{{overline|6}}
| '''0.2'''
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>
| align="center" | 1/6
|-
| align="center" | 1/7
| align="center" | <font style="color:Red">'''7'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|142857}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|186A35}}
| align="center" | <font style="color:Red">'''7'''</font>
| align="center" | 1/7
|-
| align="center" | 1/8
| align="center" | <font style="color:Green">'''2'''</font>
| '''0.125'''
| '''0.16'''
| align="center" | <font style="color:Green">'''2'''</font>
| align="center" | 1/8
|-
| align="center" | 1/9
| align="center" | <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| '''0.14'''
| align="center" | <font style="color:Green">'''3'''</font>
| align="center" | 1/9
|-
| align="center" | 1/10
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''5'''</font>
| '''0.1'''
| bgcolor=#c0c0c0 | '''0.1'''{{overline|2497}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''5'''</font>
| align="center" | 1/A
|-
| align="center" | 1/11
| align="center" | <font style="color:Red">'''11'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|09}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| align="center" | <font style="color:Red">'''B'''</font>
| align="center" | 1/B
|-
| align="center" | 1/12
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.08'''{{overline|3}}
| '''0.1'''
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>
| align="center" | 1/10
|-
| align="center" | 1/13
| align="center" | <font style="color:Red">'''13'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|076923}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0B}}
| align="center" | <font style="color:Red">'''11'''</font>
| align="center" | 1/11
|-
| align="center" | 1/14
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''7'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|714285}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|A35186}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''7'''</font>
| align="center" | 1/12
|-
| align="center" | 1/15
| align="center" | <font style="color:Red">'''3'''</font>, <font style="color:Green">'''5'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|6}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|9724}}
| align="center" | <font style="color:Green">'''3'''</font>, <font style="color:Red">'''5'''</font>
| align="center" | 1/13
|-
| align="center" | 1/16
| align="center" | <font style="color:Green">'''2'''</font>
| '''0.0625'''
| '''0.09'''
| align="center" | <font style="color:Green">'''2'''</font>
| align="center" | 1/14
|-
| align="center" | 1/17
| align="center" | <font style="color:Red">'''17'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0588235294117647}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|08579214B36429A7}}
| align="center" | <font style="color:Red">'''15'''</font>
| align="center" | 1/15
|-
| align="center" | 1/18
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|5}}
| '''0.08'''
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>
| align="center" | 1/16
|-
| align="center" | 1/19
| align="center" | <font style="color:Red">'''19'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|052631578947368421}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|076B45}}
| align="center" | <font style="color:Red">'''17'''</font>
| align="center" | 1/17
|-
| align="center" | 1/20
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''5'''</font>
| '''0.05'''
| bgcolor=#c0c0c0 | '''0.0'''{{overline|7249}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''5'''</font>
| align="center" | 1/18
|-
| align="center" | 1/21
| align="center" | <font style="color:Red">'''3'''</font>, <font style="color:Red">'''7'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|047619}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|6A3518}}
| align="center" | <font style="color:Green">'''3'''</font>, <font style="color:Red">'''7'''</font>
| align="center" | 1/19
|-
| align="center" | 1/22
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''11'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|45}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|6}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''B'''</font>
| align="center" | 1/1A
|-
| align="center" | 1/23
| align="center" | <font style="color:Red">'''23'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0434782608695652173913}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|06316948421}}
| align="center" | <font style="color:Red">'''1B'''</font>
| align="center" | 1/1B
|-
| align="center" | 1/24
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.041'''{{overline|6}}
| '''0.06'''
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>
| align="center" | 1/20
|-
| align="center" | 1/25
| align="center" | <font style="color:Green">'''5'''</font>
| '''0.04'''
| bgcolor=#c0c0c0 | '''0.'''{{overline|05915343A0B62A68781B}}
| align="center" | <font style="color:Red">'''5'''</font>
| align="center" | 1/21
|-
| align="center" | 1/26
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''13'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|384615}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|56}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''11'''</font>
| align="center" | 1/22
|-
| align="center" | 1/27
| align="center" | <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|037}}
| '''0.054'''
| align="center" | <font style="color:Green">'''3'''</font>
| align="center" | 1/23
|-
| align="center" | 1/28
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''7'''</font>
| bgcolor=#c0c0c0 | '''0.03'''{{overline|571428}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|5186A3}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''7'''</font>
| align="center" | 1/24
|-
| align="center" | 1/29
| align="center" | <font style="color:Red">'''29'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0344827586206896551724137931}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|04B7}}
| align="center" | <font style="color:Red">'''25'''</font>
| align="center" | 1/25
|-
| align="center" | 1/30
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>, <font style="color:Green">'''5'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|3}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|4972}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>, <font style="color:Red">'''5'''</font>
| align="center" | 1/26
|-
| align="center" | 1/31
| align="center" | <font style="color:Red">'''31'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|032258064516129}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0478AA093598166B74311B28623A55}}
| align="center" | <font style="color:Red">'''27'''</font>
| align="center" | 1/27
|-
| align="center" | 1/32
| align="center" | <font style="color:Green">'''2'''</font>
| '''0.03125'''
| '''0.046'''
| align="center" | <font style="color:Green">'''2'''</font>
| align="center" | 1/28
|-
| align="center" | 1/33
| align="center" | <font style="color:Red">'''3'''</font>, <font style="color:Red">'''11'''</font>
| bgcolor=#c0c0c0 | '''0.'''{{overline|03}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|4}}
| align="center" | <font style="color:Green">'''3'''</font>, <font style="color:Red">'''B'''</font>
| align="center" | 1/29
|-
| align="center" | 1/34
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''17'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|2941176470588235}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|429A708579214B36}}
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''15'''</font>
| align="center" | 1/2A
|-
| align="center" | 1/35
| align="center" | <font style="color:Green">'''5'''</font>, <font style="color:Red">'''7'''</font>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|285714}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0414559B3931}}
| align="center" | <font style="color:Red">'''5'''</font>, <font style="color:Red">'''7'''</font>
| align="center" | 1/2B
|-
| align="center" | 1/36
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Red">'''3'''</font>
| bgcolor=#c0c0c0 | '''0.02'''{{overline|7}}
| '''0.04'''
| align="center" | <font style="color:Green">'''2'''</font>, <font style="color:Green">'''3'''</font>
| align="center" | 1/30
|}
 
=== Irrational numbers ===
 
As for [[irrational number]]s, none of them has a finite representation in ''any'' of the [[rational number|rational]]-based positional number systems (such as the decimal and duodecimal ones); this is because a rational-based positional number system is essentially nothing but a way of expressing quantities as a sum of fractions whose denominators are powers of the base, and by definition no ''finite'' sum of rational numbers can ever result in an irrational number. For example, 123.456 = 1 × 1/10<sup>−2</sup> + 2 × 1/10<sup>−1</sup> + 3 × 1/10<sup>0</sup> + 4 × 1/10<sup>1</sup> + 5 × 1/10<sup>2</sup> + 6 × 1/10<sup>3</sup> (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 pattern of repetition; instead, the different digits succeed in a seemingly random fashion. The following chart compares the first few digits of the decimal and duodecimal representation of several of the most important [[algebraic number|algebraic]] and [[transcendental number|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.
 
{|class="wikitable"
| align="center" | ''Algebraic irrational number''
| align="center" | In decimal
| align="center" | '''In duodecimal / dozenal'''
|-
| align="center" | [[Square root of 2|√2]] <SMALL>(the length of the [[diagonal]] of a unit [[Square (geometry)|square]])</SMALL>
| 1.41421356237309... (≈ 1.4142)
| 1.4B79170A07B857... (≈ 1.5)
|-
| align="center" | [[Square root of 3|√3]] <SMALL>(the length of the diagonal of a unit [[cube]], or twice the [[height]] of an [[equilateral triangle]] of unit side)</SMALL>
| 1.73205080756887... (≈ 1.732)
| 1.894B97BB968704... (≈ 1.895)
|-
| align="center" | [[Square root of 5|√5]] <SMALL>(the length of the [[diagonal]] of a 1×2 [[rectangle]])</SMALL>
| 2.2360679774997... (≈ 2.236)
| 2.29BB132540589... (≈ 2.2A)
|-
| align="center" | [[Golden ratio|φ]] <SMALL>(phi, the golden ratio = <math>\scriptstyle \frac{1+\sqrt{5}}{2}</math>)</SMALL>
| 1.6180339887498... (≈ 1.618)
| 1.74BB6772802A4... (≈ 1.75)
|-
| align="center" | ''Transcendental irrational number''
| align="center" | In decimal
| align="center" | '''In duodecimal / dozenal'''
|-
| align="center" | ''[[Pi|π]]'' <SMALL>(pi, the ratio of [[circumference]] to [[diameter]])<SMALL>
| 3.1415926535897932384626433<br/>8327950288419716939937510...<br/>(≈ 3.1416)
| 3.184809493B918664573A6211B<br/>B151551A05729290A7809A492...<br/>(≈ 3.1848)
|-
| align="center" | [[E (mathematical constant)|e]] <SMALL>(the base of the [[natural logarithm]])</SMALL>
| 2.718281828459045... (≈ 2.718)
| 2.8752360698219B8... (≈ 2.875)
|}
 
The first few digits of the decimal and dozenal representation of another important number, the [[Euler-Mascheroni constant]] (the status of which as a rational or irrational number is not yet known), are:
 
{|class="wikitable"
| align="center" | ''Number''
| align="center" | In decimal
| align="center" | '''In duodecimal / dozenal'''
|-
| align="center" | [[Euler-Mascheroni constant|γ]] <SMALL>(the limiting difference between the [[harmonic series (mathematics)|harmonic series]] and the natural logarithm)</SMALL>
| 0.57721566490153... (~ 0.577)
| 0.6B15188A6760B3... (~ 0.7)
|}
 
==Advocacy and "dozenalism"==
The case for the duodecimal system was put forth at length in F. Emerson Andrews' 1935 book ''New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics''. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized ''either'' by the adoption of ten-based weights and measure ''or'' by the adoption of the duodecimal number system.
 
Rather than the symbols "A" for ten and "B" for eleven as used in [[hexadecimal]] notation and [[vigesimal]] notation (or "T" and "E" for ten and eleven), he suggested in his book and used a script X and a script E, <math>x\!</math> ([[Unicode|U+]]1D4B3) and [[Image:Scripte.png]] (U+2130), to represent the digits ten and eleven respectively, because, at least on a page of Roman script, these characters were distinct from any existing letters or numerals, yet were readily available in printers' fonts. He chose <math>x\!</math> for its resemblance to the Roman numeral X, and [[Image:Scripte.png]] as the first letter of the word "eleven".
 
Another popular notation, introduced by Sir [[Isaac Pitman]], is to use a rotated 2 (resembling a script ''τ'' for "ten") to represent ten and a rotated or horizontally flipped 3 (which again resembles ''ε'') to represent eleven. This is the convention commonly employed by the Dozenal Society of Great Britain and has the advantage of being easily recognizable as digits because of their resemblance in shape to existing digits. On the other hand, the Dozenal Society of America adopted for some years the convention of using an [[asterisk]] * for ten and a [[Number sign|hash]] # for eleven. The reason was the symbol * resembles a struck-through X while # resembles a doubly-struck-through 11, and both symbols are already present in [[telephone]] [[Rotary dial|dial]]s. However, critics pointed out these symbols do not look anything like digits. Some other systems write 10 as Φ (a combination of 1 and 0) and eleven as a cross of two lines (+, x, or † for example)<!-- "+" is similar to the Chinese character for ten & "X" is the Roman numeral for ten. -->.  Problems with these symbols are evident, most notably that most of them can not be represented in the [[seven-segment display]] of most [[calculator]] displays ([[Image:Scripte.png]] being an exception, although "E" is used on calculators to indicate an [[error message]]). However, 10 and 11 do fit, both within a single digit (11 fits as is, while the 10 has to be tilted sideways, resulting in a character that resembles an O with a [[macron]], ō or <u>o</u>). A and B also fit (although B must be represented as lowercase "b" and as such, 6 must have a bar over it to distinguish the two figures) and are used on calculators for bases higher than ten.
 
In "Little Twelvetoes", American television series ''[[Schoolhouse Rock!]]'' portrayed an alien child using base-twelve arithmetic, using "dek", "el" and "doh" as names for ten, eleven and twelve, and Andrews' script-X and script-E for the digit symbols. ("Dek" is from the prefix "deca", "el" being short for "eleven" and "doh" an apparent shortening of "dozen".)<ref>[http://www.schoolhouserock.tv/Little.html "Little Twelvetoes"]</ref>
 
The Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the base-twelve system. They use the word '''dozenal''' instead of "duodecimal" because the latter comes from Latin roots that express twelve in base-ten terminology.
 
The renowned mathematician and mental calculator [[Alexander Aitken|Alexander Craig Aitken]] was an outspoken advocate of the advantages and superiority of duodecimal over decimal:
{{quote|The duodecimal tables are easy to master, easier than the decimal ones; and in elementary teaching they would be so much more interesting, since young children would find more fascinating things to do with twelve rods or blocks than with ten. Anyone having these tables at command will do these calculations more than one-and-a-half times as fast in the duodecimal scale as in the decimal. This is my experience; I am certain that even more so it would be the experience of others.|A. C. Aitken|in ''The Listener'', January 25, 1962<ref>[http://www.dozenalsociety.org.uk/leafletsetc/aitken.html Basic Stuff<!-- Bot generated title -->]</ref>}}
 
{{quote|But the final quantitative advantage, in my own experience, is this: in varied and extensive calculations of an ordinary and not unduly complicated kind, carried out over many years, I come to the conclusion that the efficiency of the decimal system might be rated at about 65 or less, if we assign 100 to the duodecimal.|A. C. Aitken|''The Case Against Decimalisation'' (Edinburgh / London: Oliver & Boyd, 1962)<ref>[http://www.dozenalsociety.org.uk/pdfs/aitken.pdf The Case against Decimalisation<!-- Bot generated title -->]</ref>}}
 
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.
 
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]] (one of the widely popular [[New Age]] channeled entities) 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.<ref>''[https://www.kryon.com/k_13.html Kryon—Alchemy of the Human Spirit]'', ISBN 0-9636304-8-2</ref>
 
=== Dozenal clock ===
 
[http://www.dozenalsociety.org.uk/apps/dozenalclock.html Dozenal Clock by Joshua Harkey]
 
=== Dozenal metric systems ===
 
[[Systems of measurement]] proposed by dozenalists include:
 
* Tom Pendlebury's [[TGM (measurement system)|TGM]] system<ref>{{cite web|last=Pendlebury|first=Tom|title=TGM|url=http://www.dozenalsociety.org.uk/pdfs/TGMbooklet.pdf}}</ref>
* Takashi Suga's [[Universal Unit System]]<ref>{{cite web|last=Suga|first=Takashi|title=Universal Unit System|url=http://www.asahi-net.or.jp/~dd6t-sg/univunit-e/}}</ref>


== See also ==
== See also ==
* [[Diesel engine]]
* [[Hot bulb engine]]
* [[Mixed/dual cycle]]
{{Thermodynamic cycles|state=uncollapsed}}


*[[Senary]] (base 6)
{{DEFAULTSORT:Diesel Cycle}}
*[[base 24|Quadrovigesimal]] (base 24)
[[Category:Thermodynamic cycles]]
*[[base 36|Hexatridecimal]] (base 36)
*[[Sexagesimal]] (base 60)
*[[Babylonian numerals]]
 
== References ==
{{reflist}}
 
== External links ==
*[http://www.dozenal.org/ Dozenal Society of America]
*[http://www.dozenalsociety.org.uk/ Dozenal Society of Great Britain website]
*[http://flud.org/dozenal-calc.html Online Decimal-Dozenal Converter]
*[http://7r4n5.com/papers/bases-of-counting/ The Bases of Counting]
 
[[Category:Positional numeral systems]]
 
[[ar:نظام عد ثنائي عشر]]
[[ca:Sistema duodecimal]]
[[da:Duodecimal]]
[[de:Duodezimalsystem]]
[[es:Sistema duodecimal]]
[[eo:Dekduuma sistemo]]
[[eu:Zenbaki-sistema hamabitar]]
[[fa:دستگاه اعداد پایه ۱۲]]
[[fr:Système duodécimal]]
[[ko:십이진법]]
[[is:Tylftakerfi]]
[[he:בסיס דואודצימלי]]
[[hu:Tizenkettes számrendszer]]
[[nl:Twaalftallig stelsel]]
[[ja:十二進法]]
[[no:Tolvtallsystemet]]
[[nn:Tolvtalsystemet]]
[[pl:Dwunastkowy system liczbowy]]
[[pt:Sistema de numeração duodecimal]]
[[ru:Двенадцатеричная система счисления]]
[[sl:Dvanajstiški številski sistem]]
[[fi:Duodesimaalijärjestelmä]]
[[sv:Duodecimalsystemet]]
[[th:เลขฐานสิบสอง]]
[[tr:On ikili sayı sistemi]]
[[uk:Дванадцяткова система числення]]
[[zh:十二进制]]

Revision as of 11:30, 8 August 2014

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The thermodynamic cycle which approximates the Diesel cycle pressure and volume of the combustion chamber of the diesel engine, was invented by Rudolph Diesel in 1897. It is assumed to have constant pressure during the initial part of the "combustion" phase ( to in the diagram, below). This is an idealized mathematical model: real physical diesels do have an increase in pressure during this period, but it is less pronounced than in the Otto cycle. In contrast, the idealized Otto cycle of a gasoline engine approximates a constant volume process during that phase.

The Idealized Diesel Cycle

p-V Diagram for the ideal Diesel cycle. The cycle follows the numbers 1-4 in clockwise direction.

The image on the left shows a p-V diagram for the ideal Diesel cycle; where is pressure and V the volume or the specific volume if the process is placed on a unit mass basis. The ideal Diesel cycle follows the following four distinct processes:

  • Process 1 to 2 is isentropic compression of the fluid (blue)
  • Process 2 to 3 is reversible constant pressure heating (red)
  • Process 3 to 4 is isentropic expansion (yellow)
  • Process 4 to 1 is reversible constant volume cooling (green)[1]

The Diesel engine is a heat engine: it converts heat into work. During the bottom isentropic processes (blue), energy is transferred into the system in the form of work , but by definition (isentropic) no energy is transferred into or out of the system in the form of heat. During the constant pressure (red, isobaric) process, energy enters the system as heat . During the top isentropic processes (yellow), energy is transferred out of the system in the form of , but by definition (isentropic) no energy is transferred into or out of the system in the form of heat. During the constant volume (green, isochoric) process, some of energy flows out of the system as heat through the right depressurizing process . The work that leaves the system is equal to the work that enters the system plus the difference between the heat added to the system and the heat that leaves the system; in other words, net gain of work is equal to the difference between the heat added to the system and the heat that leaves the system.

The net work produced is also represented by the area enclosed by the cycle on the P-V diagram. The net work is produced per cycle and is also called the useful work, as it can be turned to other useful types of energy and propel a vehicle (kinetic energy) or produce electrical energy. The summation of many such cycles per unit of time is called the developed power. The is also called the gross work, some of which is used in the next cycle of the engine to compress the next charge of air.


50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .

Maximum thermal efficiency

The maximum thermal efficiency of a Diesel cycle is dependent on the compression ratio and the cut-off ratio. It has the following formula under cold air standard analysis:

where

is thermal efficiency
is the cut-off ratio (ratio between the end and start volume for the combustion phase)
Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. is the compression ratio
is ratio of specific heats (Cp/Cv)[2]

The cut-off ratio can be expressed in terms of temperature as shown below:

can be approximated to the flame temperature of the fuel used. The flame temperature can be approximated to the adiabatic flame temperature of the fuel with corresponding air-to-fuel ratio and compression pressure, . can be approximated to the inlet air temperature.

This formula only gives the ideal thermal efficiency. The actual thermal efficiency will be significantly lower due to heat and friction losses. The formula is more complex than the Otto cycle (petrol/gasoline engine) relation that has the following formula;

The additional complexity for the Diesel formula comes around since the heat addition is at constant pressure and the heat rejection is at constant volume. The Otto cycle by comparison has both the heat addition and rejection at constant volume.

Comparing the two formulae it can be seen that for a given compression ratio (Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.), the ideal Otto cycle will be more efficient. However, a diesel engine will be more efficient overall since it will have the ability to operate at higher compression ratios. If a petrol engine were to have the same compression ratio, then knocking (self-ignition) would occur and this would severely reduce the efficiency, whereas in a diesel engine, the self ignition is the desired behavior. Additionally, both of these cycles are only idealizations, and the actual behavior does not divide as clearly or sharply. And the ideal Otto cycle formula stated above does not include throttling losses, which do not apply to diesel engines.

Applications

Diesel engines

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Diesel engines have the lowest specific fuel consumption of any large internal combustion engine employing a single cycle, 0.26 lb/hp·h (0.16 kg/kWh) for very large marine engines (combined cycle power plants are more efficient, but employ two engines rather than one). Two-stroke diesels with high pressure forced induction, particularly turbocharging, make up a large percentage of the very largest diesel engines.

In North America, diesel engines are primarily used in large trucks, where the low-stress, high-efficiency cycle leads to much longer engine life and lower operational costs. These advantages also make the diesel engine ideal for use in the heavy-haul railroad environment.

Other internal combustion engines without spark plugs

Many model airplanes use very simple "glow" and "diesel" engines. Glow engines use glow plugs. "Diesel" model airplane engines have variable compression ratios. Both types depend on special fuels.

Some 19th-century or earlier experimental engines used external flames, exposed by valves, for ignition, but this becomes less attractive with increasing compression. (It was the research of Nicolas Léonard Sadi Carnot that established the thermodynamic value of compression.) A historical implication of this is that the diesel engine could have been invented without the aid of electricity.
See the development of the hot bulb engine and indirect injection for historical significance.

References

  1. ↑ Eastop & McConkey 1993, Applied Thermodynamics for Engineering Technologists, Pearson Education Limited, Fifth Edition, p.137
  2. ↑ The Diesel Engine

See also

Template:Thermodynamic cycles

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