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
selfexplanatory: 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 onesyllable 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 onesyllable 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 visualbased
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 threedigit 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 threedigit 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 singledigit 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 


x10^{0} 
1 
one 
dola 
D 
x10^{1} 
10 
twelve 
grola 
G 
x10^{2} 
100 
grosan 
milla 
M 
x10^{3} 
1 000 
monan 
billa 
B 
x10^{6} 
1 000 000 
bian 
trilla 
T 
x10^{9} 
1 000 000 000 
thrian 
furla 
F 
x10^{10} 
1 000 000 000
000 
fouran 
fitha 
V 
x10^{13} 
1 000 000 000
000 000 
fifan 
sicca 
S 
x10^{16} 
1 000 000 000
000 000 000 
sixan 
petta 
P 
x10^{19} 
1 000 000 000
000 000 000 000 
septan 
octa 
O 
x10^{20} 
1 000 000 000
000 000 000 000 000 
octan 
nova 
N 
x10^{23} 
1 000 000 000
000 000 000 000 000 000 
novan 
tenna 
X 
x10^{26} 
1 000 000 000
000 000 000 000 000 000 000 
tennan 
elfa 
E 
x10^{29} 
1 000 000 000
000 000 000 000 000 000 000 000 
eolfan 
zona 
Z 
x10^{30} 
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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