A Comprehensive Dozenal Counting System:  How to count in duodecimal – number names, scientific notation, prefixes, and abbreviations

Originated: 25 November 2004

Revised: 10 March 2006

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Counting in Dozenal:

For ordinary purposes, counting in dozenal is not difficult at all.  The English language accommodates this quite easily.  The terms ‘dozen’ and ‘gross’ represent number sets of 12 and 144 respectively.  This makes counting in numbers less than 1 728 (which we do most often) very convenient.  Beyond that, however, it gets a little troublesome. For example, 1 728 – the dozenal notation equivalent of 1 000 – was traditionally called a ‘great gross’ – in my opinion, a cumbersome name.  However, since very few people have even heard of a ‘great gross’, I feel comfortable renaming that particular unit to make conform to the pattern dozenal counting system outlined below.  Most dozenists I have conversed with advocate for a system of number names with the basic numbers having no more than one syllable  (thus, ‘twelve’ is preferred to ‘one dozen’ in most dozenal systems I have seen proposed).  I, however, prefer the term ‘dozen’ to be retained.  The reason is that I want to make counting as easy as possible for people used to thinking in base ten.  My thinking is that, though more cumbersome than ‘thirteen, fourteen, fifteen…’, ‘one dozen one, one dozen two, one dozen three…’ is not unlike the hundred sets that people are used to counting in.  Familiarity is important.  If you are going to try to get people to use something new for an important thing like counting, you want them to be able to conceptualize it as quickly and easily as possible.  Despite my preference for the ‘dozen’ I would like to see ‘twelve’ also kept as it is a useful term in some circumstances – for example, one dozen o’ clock is just a tad more cumbersome than twelve o’ clock.

The chart below lists the numbers of the dozenal counting system I am proposing.  Many of them look similar, but many of them look different.  The far left column shows what the number would look like in dozenal notation (I have substituted the characters X and E for ’10’ and ‘11’ since the standard keyboard does not accommodate my preferred characters for these).  The far right column shows what the number expressed looks like in our current decimal notation.  The center column gives the proposed names for the numbers in this new dozenal system.  Before we get to the chart, allow me to explain how I decided on these particular names.

I have renamed ‘zero’ for the sake of efficiency.  The word I use, ‘neen’, is a transliteration of an Old English word meaning ‘none’.  My reasoning should be self-explanatory: try saying ‘point zero, zero, zero, zero, zero, one’.  Not that easy, is it?  Americans get around this trouble by substituting ‘oh’ for ‘zero’, and the British do it by substituting ‘naught’ – notice that both substitutions are one-syllable words.  In my system, ‘zero’ would still be an acceptable name, but ‘neen’ would be preferred.

The next oddity is ‘seof’ instead of ‘seven’.  As with most other dozenists, I prefer a basic set of one-syllable numbers – it is more efficient.  ‘Seof’ is derived from an Old English word for ‘seven’.

For the same reason as ‘seven’, I have replaced ‘eleven’ with an Old English derivative: ‘eolf’.  This term is actually only loosely based on its Old English counterpart.

I have already explained ‘dozen’ in the first paragraph; however, I did not go into great detail concerning the proper form.  You probably have noticed in certain formal documents that the term ‘and’ is used to describe numbers between sets of one hundred (for example, the Constitution of the United States of America describes the date as “the Year of our Lord one thousand seven hundred and Eighty seven”).  I would like to preserve this construction for formal speech and in certain other circumstances (where it just sounds right – for example: ‘I am two dozen and two years old’ – as opposed to being asked ‘How old are you?’ and responding with  ‘two dozen two.’). For ordinary counting, the ‘and’ is just an extra syllable.  Therefore constructions like ‘three dozen five, three dozen six…’ are to be preferred.

Twelve times twelve in one gross.  While retaining the term ‘gross’, I have changed the official term to ‘grosan’.  The reason being relates to the way it sounds.  In English, I have heard ‘I’ll take a gross of these’ and, in response to ‘How many?’, ‘a gross’.  Try saying ‘I drove 144 miles’ in that terminology.  ‘I drove a gross miles’ doesn’t sound like a proper use of the term ‘gross’.  Likewise, ‘I drove a gross of miles’ doesn’t sound quite right, either.  To solve this usage problem, I have added the term ‘grosan’.  ‘I drove a grosan miles’ sounds more natural (not to mention that it matches my other set names in sound).

Now, the next pattern of set names requires a little bit of explanation.  In our current decimal system, one hundred hundreds is called ‘one thousand’.  I’m not sure why.  One thousand thousands is called ‘one million’ after the Latin term ‘mil’ meaning ‘one thousand’.  ‘Billion’ is one thousand multiplied by one thousand to the second power (‘bi’ meaning ‘two’).  ‘Trillion’, in turn, is one thousand multiplied by one thousand to the third power (‘tri’ meaning ‘three’).   I went in a different direction with my number sets – more visual-based as well as having a rational mathematics base.  In our decimal notation, ‘one billion’, as I said above, is one thousand times one thousand to the second power.  While making sense in this context, a simplification of this process leaves you with ‘one thousand to the third power’ – this reduction strips away the rational basis for the name ‘billion’. What I have done is similar, but more rational.  We currently like to deal with very big numbers by demarcating them with three-digit sets.  Thus, one million, two hundred and four thousand, seven hundred and three looks like ‘1 204 703’.  Likewise, 1 billion is ‘1 000 000 000’.  So when I first started formulating my system, I wanted the name to relate to the number of full three-digit sets following the first one to three digits.  Thus, the number that looks like ‘1 000’ would be named for the one set of three following the first digit.  Likewise, the number that looks like ‘1 000 000’ would be named for the two sets of three following the first digit.  The number scheme I use was borrowed from several sources including Modern English, Old English, and Latin.  For the number ‘1 000’ (1 728 in decimal), I used the Old English word for ‘one’ – mon – to create the word ‘monan’.  For the second set (1 000 000), I use the term ‘bi’ meaning ‘two’ to create the word ‘bian’.  And so it goes: ‘thrian’ (1 000 000 000) is derived from the word for ‘three’, ‘fouran’ from ‘four’, ‘fifan’ from the Old English word for ‘five’, and ‘sixan’ from ‘six’.  ‘Septan’, ‘octan’, and ‘novan’ are from the Latin for ‘seven’, ‘eight’, and ‘nine’.  ‘Tennan’ is derived from ‘ten’, ‘eolfan’ is based on the ‘eleven’ that I use for my single-digit numbers, and ‘donan’ is loosely based on the dozen.

The following chart lists the names for the numbers in the dozenal system that I propose.

Dozenal Value

Name

Decimal Value

0

Neen (also: ‘Zero’)

0

1

One

1

2

Two

2

3

Three

3

4

Four

4

5

Five

5

6

Six

6

7

Seof

7

8

Eight

8

9

Nine

9

# X

Ten

10

E

Eolf

11

10

One Dozen (also ‘Twelve’)

12

11

One Dozen One (formal: ‘one dozen and one’)

13

12

One Dozen Two

14

13

One Dozen Three

15

14

One Dozen Four

16

15

One Dozen Five

17

16

One Dozen Six

18

17

One Dozen Seof

19

18

One Dozen Eight

20

19

One Dozen Nine

21

1X

One Dozen Ten

22

1E

One Dozen Eolf

23

20

Two Dozen

24

30

Three Dozen

36

40

Four Dozen

48

50

Five Dozen

60

60

Six Dozen

72

70

Seof Dozen

84

80

Eight Dozen

96

90

Nine Dozen

108

X0

Ten Dozen

120

E0

Eolf Dozen

132

100

One Grosan (also: ‘one gross’)

144

1 000

One Monan

1 728

10 000

One Dozen Monan

20 736

100 000

One Gross Monan

248 832

1 000 000

One Bian

2 985 984

10 000 000

One Dozen Bian

35 831 808

100 000 000

One Gross Bian

429 981 696

1 000 000 000

One Thrian

5 159 780 352

10 000 000 000

One Dozen Thrian

61 917 364 224

100 000 000 000

One Gross Thrian

743 008 370 688

1 000 000 000 000

One Fouran

8 916 100 448 256

10 000 000 000 000

One Dozen Fouran

106 993 205 379 072

100 000 000 000 000

One Gross Fouran

1 283 918 464 548 864

1 000 000 000 000 000

One Fifan

15 407 021 574 586 368

10 000 000 000 000 000

One Dozen Fifan

184 884 258 895 036 416

100 000 000 000 000 000

One Gross Fifan

2 218 611 106 740 436 992

1 000 000 000 000 000 000

One Sixan

26 623 333 280 885 243 904

10 000 000 000 000 000 000

One Dozen Sixan

319 479 999 370 622 926 848

100 000 000 000 000 000 000

One Gross Sixan

3 833 759 992 447 475 122 176

1 000 000 000 000 000 000 000

One Septan

46 005 119 909 369 701 466 112

1 000 000 000 000 000 000 000 000

One Octan

79 496 847 203 390 844 133 441 536

1 000 000 000 000 000 000 000 000 000

One Novan

137 370 551 967 459 378 662 586 974 208

1 000 000 000 000 000 000 000 000 000 000

One Tennan

237 376 313 799 769 806 328 950 291 431 424

1 000 000 000 000 000 000 000 000 000 000 000

One Eolfan

410 186 270 246 002 225 336 426 103 593 501 000

1 000 000 000 000 000 000 000 000 000 000 000 000

One Donan

7.088 018 749 850 918 453 813 443 070 095 7 e+38

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Okay, now we have numbers that we can use.  But what about all those nifty prefixes we have in our decimal system such as ‘kilo-’ for 1 000 and ‘milli-’ for 1/1 000?  Decimalists are always lauding their prefixes, talking about how convenient they are, etc.  Well, it just so happens that I have devised a list of such prefixes.  I follow the same pattern that we do in base ten – the only difference being that we are working with one dozen as the base and multiplying by positive or negative powers of twelve.

All of the prefixes are derived from the number names I assigned in the chart above.  The exception is ‘septan’ and ‘septanth’.  I assigned the prefixes ‘petta-’ and ‘pecco-’ to those because there is a ‘p’ in ‘sept’ and I wanted a rational explanation for the abbreviation I used.  (The same explanation applies for ‘zona-’ and ‘zoco-’ which are the abbreviations for ‘donan’ and ‘donanth’ respectively.)

The abbreviations for the prefixes are simple to explain.  Abbreviations are always one letter.  For powers of twelve, the abbreviation is always capitalized.  For negative powers of twelve, the abbreviation is lower case.  The rule is that the abbreviation shall be the first letter of the prefix.   The exceptions are when a first letter repeats.  Some, I had to alter the prefix so it did not closely resemble the word it was originally supposed to represent (see previous paragraph).  Others had a convenient Roman numeral (five and ten), so I just used that.

The following chart lists the names for the prefixes in the dozenal system that I propose.

Prefix

Symbol

Scientific Notation*

Ordinary Notation*

### Ordinary Name

zoco-

z

x10-30

0.000 000 000 000 000 000 000 000 000 000 000 001

### donanth

elco-

e

x10-29

0.000 000 000 000 000 000 000 000 000 000 001

### eolfanth

tecco-

x

x10-26

0.000 000 000 000 000 000 000 000 000 001

### tennanth

noco-

n

x10-23

0.000 000 000 000 000 000 000 000 001

### novanth

occo-

o

x10-20

0.000 000 000 000 000 000 000 001

octanth

pecco-

p

x10-19

0.000 000 000 000 000 000 001

septanth

sicco-

s

x10-16

0.000 000 000 000 000 001

sixanth

ficco-

v

x10-13

0.000 000 000 000 001

fifanth

furco-

f

x10-10

0.000 000 000 001

fouranth

tricco-

t

x10-9

0.000 000 001

thrianth

bicco-

b

x10-6

0.000 001

bianth

micco-

m

x10-3

0.001

monanth

groco-

g

x10-2

0.01

grosanth

doco-

d

x10-1

0.1

dozenth

x100

1

one

dola-

D

x101

10

twelve

grola-

G

x102

100

grosan

milla-

M

x103

1 000

monan

billa-

B

x106

1 000 000

bian

trilla-

T

x109

1 000 000 000

thrian

furla-

F

x1010

1 000 000 000 000

fouran

fitha-

V

x1013

1 000 000 000 000 000

fifan

sicca-

S

x1016

1 000 000 000 000 000 000

sixan

petta-

P

x1019

1 000 000 000 000 000 000 000

septan

octa-

O

x1020

1 000 000 000 000 000 000 000 000

octan

nova-

N

x1023

1 000 000 000 000 000 000 000 000 000

novan

tenna-

X

x1026

1 000 000 000 000 000 000 000 000 000 000

tennan

elfa-

E

x1029

1 000 000 000 000 000 000 000 000 000 000 000

eolfan

zona-

Z

x1030

1 000 000 000 000 000 000 000 000 000 000 000 000

donan

*Remember: ‘10’ is one dozen; also called ‘twelve’.

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