Numbers' & Numeral
systems' history and curiosities
of the Numerals
1 / Page
2 / More...
Nombres – Numeri – Números – Zahlen – Liczba
– 数 – Число́ – Αριθμός
numbers, also called Hindu-Arabic numbers, are
a combination of just 10 symbols or digits: 1, 2, 3, 4, 5, 6, 7, 8, 9,
and 0. These digits were introduced
in Europe within the XII century by Leonardo
Pisano (aka Fibonacci),
an Italian mathematician. L. Pisano was educated in North
Africa, where he learned and later carried to Italy the
now popular Hindu-Arabic numerals.
numeral system is a pure place-value
system, that is why you need a zero.
Only the Hindus, within the context of Indo-European
civilisations, have consistently used a zero. The Arabs,
however, played an essential part in the dissemination
of this numeral system.
Earliest Evidence of The Use of Numbers
numeration systems started as simple tally marks, using single
strokes to represent each additional unit...
stick (recto/verso). Origin: Central Africa. Length: 10
prehistoric ancestors would have had a general sensibility
about amounts and quantities, and would have instinctively
known the difference between one, two or many antelopes.
But the intellectual leap from the concrete idea of two things
to the invention of a symbol for the abstract idea of "two" took
many ages to come about.
along with a formal system of numbers initially developed
when civilizations settled and developed agriculture - for
the measurement of plots of land, the taxation of individuals,
and so on - and this first occurred in the Sumerian and Babylonian
civilizations of Mesopotamia (roughly, modern Iraq) and in
10 cm long bone stick shown in the image above is 35'000-20'000
years old and is probably the first pocket calculator ever
known. The scores on the stick represent a sequence of prime
numbers and some duplication series (3-6, 4-8, 5-10, etc.).
It seems that the twelve-system of numeration was used to
make these operation tables, instead of our decimal one.
The prehistoric owner of this stick used it as a primitive
a Time Travel from India to Europe
discovery of zero and the place-value system were inventions
unique to the Indian civilization. As the Brahmi notation of
the first 9 whole numbers...
numeral set used in the Middle East today is a cousin of
the modern numeral set, with a common ancestor in the ancient
Hindu numerals. In fact, the Eastern Arabic numerals (also
called Arabic–Indic numerals) are
specific numerals currently used to represent the Hindu–Arabic
numeral system in conjunction with the Arabic alphabet
in the countries of the Arab east, and its variant (Persian
numerals) in other countries.
the first Western use of the digits, without
the zero, was reported in the Vth century by Beothius,
a Roman writer. Beothius explains, in one of his geometry
books, how to operate the abacus using marked small cones
instead of pebbles. Those cones, upon each of which was
drawn the symbol of one of the nine Hindu-Arabic digits,
were called apices. Thus, the early
representations of digits in Europe were called “apices”.
Each apex received also an individual name: Igin for
1, Andras for 2, Ormis for
3, Arbas for 4, Quimas (or
Quisnas) for 5 , Caltis (or Calctis)
for 6, Zenis (or Tenis) for 7, Temenisa for
8, and Celentis (or Scelentis) for 9.
The etymology of these names remains unclear, though
some of them were clearly Arab numbers. The Hindu-Arabic-like
figures reported by Beothius were reproduced almost everywhere
with the greatest fantasy! (see below)
adopting the Hindu-Arabic numeral system, people used
the Roman figures instead, which actually are
a legacy of the Etruscan period. The Roman numeration
is based on a biquinary (5) system.
write numbers the Romans used an additive system: V + I + I = VII (7)
or C + X + X + I (121), and
also a substractive system: IX (I before X =
9), XCIV (X before C = 90 and I before V =
4, 90 + 4 = 94). Latin numerals were used for reckoning
until late XVI century!
graphical origin of the Roman numbers
Sarcone & Waeber
systems of numeration
original systems of numeration were being used in the past.
earlier 13th century, the Archdeacon John of Basingstoke
introduced a notation for numbers between 1 and 99 based
on a vertical stroke provided with an appendage to the
left (representing units) and another to the right (tens).
Divers variants of the system turn up in various Cistercian
manuscripts, and were used for a variety of purposes, along
with Roman and Hindu-Arabic numerals.
In 1533, Agrippa
von Nettesheym included a description of a “vertical” variant
of the ciphers in his Occult Philosophy.
From then on and until the 19th century, the ciphers were
remembered as “Chaldaean”. In early 20th-century
Germany they turned Runic and Aryan. This original numeral
system later fell out of use and was forgotten.
Agrippa’s number-notation system called “Notae
Elegantissimae” allows to write numbers from 1 up
to 9999 and was primarily employed for indexing
purposes, where its compactness was a great advantage. But
it is also useful as a mnemonic aid, e.g. the symbol K in
the example further below may mean 1414 (the first 4 figures
of the square root of 2).
The Ba-Gua trigrams
(pron. pah-kwah, 八卦) and the Genji-Kô patterns
(源氏香), antique Chinese and
Japanese symbols, are strangely enough related to mathematics
and electronics. If all the entire lines of the trigrams
(___) are replaced with the digit 1 and the
broken lines (_ _) with the digit 0, each Ba-Gua
trigram will then represent a binary number from 0 to 7.
You can also notice that each
number is laid in front of its complementary: 0<>7,
1<>6, 2<>5, etc.
The "Genji-Kô" (源氏香)
symbols used for the chapters of the Tale of
Genji (early Japanese novel) indicate the
possible groupings and subgroupings of 5 elements. For
instance, if you write down "a", "b", "c", "d" and "e" beneath
the five small red sticks of each Genji-Kô pattern,
you will obtain 52 distinct ways to connect 5 variables
in Boolean algebra. The linked sticks form a "conjunction" (AND, ∨),
and the isolated sticks or groups of sticks form a "disjunction" (OR, ∧).
The pattern at the top left represents:
[("a" and "d") or ("b" and "e")
Sarcone&Waeber, all rights reserved
are encouraged to expand and/or improve this article. Send
your comments, feedback or suggestions to Gianni
A. Sarcone. Thanks!
welcome the re-use and distribution of the content published
on this Web page on the condition that you credit us by including
the following information: "Copyright © 1992-2015,
Gianni A. Sarcone, Archimedes-Lab.org. Used with the permission".
You may not use this editorial content for commercial purposes!
to next page