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 |
* * *
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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