The
vigesimal or
base 20 numeral
system is based on twenty (in the same way in which the ordinary
decimal numeral system is based on
ten).
Places
In a vigesimal
place system, twenty individual
numerals (or digit symbols) are used, ten more than in the usual
decimal system. One modern method of finding the extra needed
symbols is to write
ten as the letter
A
_{20} (the
_{20} means
base- ), to write
nineteen as J
_{20}, and the numbers
between with the corresponding letters of the alphabet. This is
similar to the common
computer-science practice of writing
hexadecimal numerals over 9 with the
letters "A-F". Another method skips over the letter "I", in order
to avoid confusion between I
_{20} as
eighteen and 1 (
one),
so that the number eighteen is written as J
_{20}, and
nineteen is written as K
_{20}. The number twenty is written
as 10
_{20}.
According to this notation:
- 20_{20} means forty in
decimal {= (2 × 20^{1} + (0 × 20^{0})}
- DA_{20} means two hundred [and]
seventy in decimal {= (13 × 20^{1}) + (10 ×
20^{0}}
- 100_{20} means four hundred
in decimal {= (1 × 20^{2}) + (0 × 20^{1}) + (0 ×
20^{0})}.
In the rest of this article below, numbers are expressed in decimal
notation, unless specified otherwise. For example, 10 means
ten, 20 means
twenty.
Vigesimal fractions
As with decimal, any number with a
prime factor other than
2 or
5 will have a
repeating expansion in vigesimal. However, the forms of familiar
fractions are very different from those in decimal. The following
table gives a list of the vigesimal expansion for some small
reciprocals and for a few other denominators (listed as fractions
in their
decimal form) that yield very short vigesimal
periods.
Note that J
_{20} = 18
_{10} and K
_{20} =
19
_{10}.
- \frac{1}{3} = .6D6D6D6D6D6D6D6D6D6D6D6D6D6D6D6D6D6D
- \frac{1}{7} = .2H2H2H2H2H2H2H2H2H2H2H2H2H2H2H2H2H2H
- \frac{1}{11} = .1G7591G7591G7591G7591G7591G7591G759
- \frac{1}{13} = .1AF7DGH94C631AF7DGH94C631AF7DGH94C63
- \frac{1}{421} = .00K00K00K00K00K00K00K00K00K00K00K00K
- \frac{1}{401} = .00KK00KK00KK00KK00KK00KK00KK00KK00KK
- \frac{1}{127} = .032KGH032KGH032KGH032KGH032KGH032KGH
- \frac{1}{29} = .0DFH4GB0DFH4GB0DFH4GB0DFH4GB0DFH4GB
- \frac{1}{71} = .05CDA8905CDA8905CDA8905CDA8905CDA89
- \frac{1}{32719} = .0004HG10004HG10004HG10004HG10004HG1
- \frac{1}{160001} = .0000KKKK0000KKKK0000KKKK0000KKKK
The number 6D
_{20}, equivalent to
133 in decimal, is a
cyclic number analogous to
142857 in decimal:
- 2_{20} × 6D_{20} = D6_{20}
1AF7DGH94C63
_{20} is also a cyclic number. It is equivalent
to 315,076,919,876,923 in decimal.
160,001
_{10} is a vigesimal
generalized Fermat prime. In
vigesimal it is 10001
_{20} or, to describe its status as a
generalized Fermat number,
20
^{22} + 1.
Use
In many
languages, especially in
Europe,
20 is a base, at
least with respect to the linguistic structure of the names of
certain numbers (though a thoroughgoing consistent vigesimal
system, based on the powers 20, 400, 8000 etc., is not generally
used).
Asia and America
- In
Santali, a Munda language of India, "fifty" is
expressed by the phrase bār isī gäl, literally "two twenty
ten." Likewise, in Didei,
another Munda language spoken in India, complex numerals are
decimal to 19 and decimal-vigesimal to 399.
- In East Asia, the Ainu language also uses a counting system that
is based around the number 20. “ ” is 20, “ ” (ten more until two
twenties) is 30, “ ” (two twenties) is
40, “ ” (five twenties) is 100.
Subtraction is also heavily used, e.g. “ ” (one more until ten) is
9.
- Twenty was a base in the Maya
number systems. The Maya used the following names for the powers of
twenty: (20), (20^{2} = 400), (20^{3} = 8,000),
(20^{4} = 160,000), (20^{5} = 3,200,000) and
(20^{6} = 64,000,000). See also Maya numerals and Maya calendar, Mayan languages, Yucatec. The Aztec called them: (1 × 20), (1 × 400),
(1 × 8,000), (1 × 20 × 8,000 = 160,000), (1 × 400 × 8,000 =
3,200,000) and (1 × 20 × 400 × 8,000 = 64,000,000). Note that the
prefix at the beginning means "one" (as in "one hundred" and "one
thousand") and is replaced with the corresponding number to get the
names of other multiples of the power. For example, (2) × (20) =
(40), (2) × (400) = (800). Note also that the in (and ) and the in
are grammatical noun suffixes that are appended only at the end of
the word; thus , and compound together as (instead of * ). (See
also Nahuatl language.)
In Europe
According to German linguist
Theo
Vennemann, the vigesimal system in Europe is of
Basque (Vasconic) origin and spread from
Vasconic languages to other European tongues, such as many
Celtic languages, French and Danish.
According to Menninger, the vigesimal system originated with the
Normans and spread through them to Western Europe, the evidence
being that
Celtic languages often
use vigesimal counting systems. Others believe that this theory is
unlikely, however.
- Twenty
( ) is used as a base number in the French language names of numbers from 70 to
99, except in the French of Switzerland, Belgium, the
Democratic
Republic of the Congo, Rwanda, the
Aosta
Valley and the Channel
Islands. For example, , the French word for 80, literally means "four twenties",
soixante-dix, the word for 70,
is literally "sixty-ten", (75) is
literally "sixty-fifteen", quatre-vingt-sept (87) is literally "four-twenties-seven",
quatre-vingt-dix (90) is
literally "four-twenties-ten", and quatre-vingt-seize
(96) is literally
"four-twenties-sixteen". However, in the French of Belgium, the
Democratic Republic of the Congo, Rwanda, the Aosta Valley, the
Channel Islands and the Swiss cantons of Berne, Geneva, Jura, Vaud, and
Neuchâtel, the numbers 70 and 90 generally have the names
septante and nonante (but 80 is
quatre-vingts except in Switzerland where 80 is sometimes
huitante). So, the year 1996 is "mille neuf cent
quatre-vingt-seize" in Parisian French, but it is "mille neuf cent
nonante-six" in e.g. Belgian French.
- Twenty ( ) is used as a base number in the Danish language names of numbers from 50 to
99. For example, (short for ) means 3 times 20, i.e. 60. For details, see Danish numerals.
- Twenty ( ) is used as a base number in the Breton language names of numbers from 40 to
49 and from 60 to 99. For example, means 2 times 20, i.e. 40, and (literally "three-six and four-twenty")
means 3×6 + 4×20, i.e. 98. However, 30 is and not * ("ten
and twenty"), and 50 is ("half-hundred").
- Twenty ( ) is used as a base number in the Welsh language, although in the latter part
of the twentieth century a decimal counting system has come to be
preferred (particularly in the
South), with the vigesimal system becoming 'traditional' and
more popular in North Welsh.
means 2 times 20 i.e. 40, means 3 times 20 i.e. 60. Prior to the
currency decimalisation in 1971, (6 times 20 paper) was the
nickname for the 10 shilling (= 120
pence) note. A vigesimal system (Yan Tan
Tethera) for counting sheep has also been recorded in areas of
Britain that today are no longer Celtic-speaking.
- Twenty ( ) is used in an older counting system in Irish Gaelic, though most people nowadays use a
decimal system, and this is what is taught in schools. Thirty is
(originally fiche agus deich), literally twenty and
ten. Forty is , literally two twenties (retained in
the decimal system as daichead). is sixty (three
twenties) and is eighty (literally four twenties).
Similarly, Scottish Gaelic has
traditionally used a vigesimal system, with ( ) being the word for
twenty. A decimal system is now taught in schools.
- Twenty ( ) is used as a base number in the Albanian language. The word for 40 ( )
means two times 20.
- Twenty ( ) is used as a base number in the Georgian language. For example, 31 ( ) literally means,
twenty-and-eleven. 67 ( ) is
said as, “three-twenty-and-seven”.
- Twenty ( ) is used as a base number in the Basque language for numbers up to 100 ( ).
The words for 40 ( ), 60 ( ) and 80 ( ) mean "two-score",
"three-score" and "four-score", respectively. The number 75 is
called , lit. "three-score-and ten-five". The Basque nationalist
Sabino Arana proposed a vigesimal digit
system to match the spoken language, and, as an alternative, a
reform of the spoken language to make it decimal, but both are
mostly forgotten.
- Twenty
(dwisti) is used as a base number in the Resian dialect of the Slovenian language in Italy's Resia valley. 60 is expressed by trïkart
dwisti (3×20), 70 by trïkart dwisti nu dësat
(3x20 + 10), 80 by štirikrat dwisti (4×20) and
90 by štirikrat dwisti nu dësat
(4×20 + 10).
- In the
old British currency
system (pre-1971), there were 20 shillings
to the pound. This was still
the case under the decimal system introduced in 1971 for those
shilling coins still in circulation (no more were minted and the
shilling coin was demonetised in 1990), because the shilling –
which was valued at 12 pence in the old currency – was re-valued at
5 pence in the new system. Thus, the old shilling coins still
accumulate 20 to the pound, because 20 × 5 new pence =
100 new pence = 1 pound (whereas in the old system, 1 pound
equalled 240 pence instead of 100 pence).
- In the imperial weight system there are twenty hundredweight in a ton.
- In English, counting by the
score has been used historically, as in the famous opening of the
Gettysburg Address "Four
score and seven years ago…", meaning eighty-seven (87) years ago. This method has fallen into
disuse, however.
Related observations
- Among multiples of 10, 20 is described in a special way in some
languages. For example, the Spanish
words (30) and (40) consist of " (10
times)", " (10 times)", but the word (20) is not presently connected to any word
meaning "two" (although historically it isThe diachronic view is like this. , the IE etymology of which ( view) connects it to the roots meaning '2' and 10'. (The etymological databases of the Tower of
Babel project are referred here.)). Similarly, in Semitic languages such as Arabic and
Hebrew, the numbers 30, 40 ... 90 are expressed by morphologically
plural forms of the words for the numbers 3, 4 ... 9, but the
number 20 is expressed by a morphologically plural form of the word
for 10.
- In some languages (e.g. English, Slavic languages), the names of the
two-digit numbers from 11 to 19 consist of one word, but the names of the
two-digit numbers from 21 on consist of
two words. So for example, the English words eleven (11), twelve (12),
thirteen (13) etc., as opposed to
twenty-one (21),
twenty-two (22),
twenty-three (23), etc. In
French, this is true up to 16. In a number of other languages (such
as Hebrew), the names of the numbers
from 11-19 contain two words, but one of these words is a special
"teen" form which is different from the ordinary form of the word
for the number 10, and may in fact be only found in these names of
the numbers 11-19.
- The term vicesimal (from the Latin vicesimus)
is sometimes used
Further reading
- Karl Menninger:
Number words and number symbols: a cultural history of
numbers; translated by Paul Broneer from the revised German
edition. Cambridge, Mass.: M.I.T. Press, 1969 (also available in
paperback: New York: Dover, 1992 ISBN 0-486-27096-3)
- Levi Leonard Conant: The Number Concept: Its Origin and
Development; New York, New York: MacMillon & Co, 1931.
Project Gutenberg EBook
Notes
- Gvozdanović, Jadranka. Numeral Types and Changes
Worldwide (1999), p.223.
- Chatterjee, Suhas. 1963. On Didei nouns, pronouns, numerals,
and demonstratives. Chicago: mimeo., 1963. (cf. Munda Bibliography at the University of Hawaii
Department of Linguistics)
- Artículos publicados en la 1.ª época de "Euzkadi" : revista
de Ciencias, Bellas Artes y Letras de Bilbao por Arana-Goiri´taŕ
Sabin: 1901, Artículos publicados en la 1 época de
"Euskadi" : revista de Ciencias, Bellas Artes y Letras de Bilbao
por Arana-Goiri´ttarr Sabin : 1901, Sabino Arana, 1908, Bilbao, Eléxpuru
Hermanos. 102–112
- Artículos ..., Sabino Arana, 112–118
- Efemérides Vascas y Reforma d ela Numeración
Euzkérica, Sabino Arana, Biblioteca de la Gran
Enciclopedia Vasca, Bilbao, 1969. Extracted from the magazine
Euskal-Erria, 1880 and 1881.