period in China. Many ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers. Standardized weights used in the
Indus Valley Civilisation () were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler – the
Mohenjo-daro ruler – was divided into ten equal parts.
Egyptian hieroglyphs, in evidence since around 3000 BCE, used a purely decimal system, as did the
Linear A script () of the
Minoans and the
Linear B script (c. 1400–1200 BCE) of the
Mycenaeans. The
Únětice culture in central Europe (2300-1600 BC) used standardised weights and a decimal system in trade. The number system of
classical Greece also used powers of ten, including an intermediate base of 5, as did
Roman numerals. Notably, the polymath
Archimedes (c. 287–212 BCE) invented a decimal positional system in his
Sand Reckoner which was based on 108.
Hittite hieroglyphs (since 15th century BCE) were also strictly decimal. The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1,000, 2,000, 3,000, 4,000, to 10,000. The world's earliest positional decimal system was the Chinese
rod calculus. Calculations with decimal fractions of lengths were
performed using positional counting rods, as described in the 3rd–5th century CE
Sunzi Suanjing. The 5th century CE mathematician
Zu Chongzhi calculated a 7-digit
approximation of.
Qin Jiushao's book
Mathematical Treatise in Nine Sections (1247) explicitly writes a decimal fraction representing a number rather than a measurement, using counting rods. The number 0.96644 is denoted : : . Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East.
Al-Khwarizmi introduced fractions to Islamic countries in the early 9th century CE, written with a numerator above and denominator below, without a horizontal bar. This form of fraction remained in use for centuries. Positional decimal fractions appear for the first time in a book by the Arab mathematician
Abu'l-Hasan al-Uqlidisi written in the 10th century. The Jewish mathematician
Immanuel Bonfils used decimal fractions around 1350 but did not develop any notation to represent them. The Persian mathematician
Jamshid al-Kashi significantly advanced the theory in the 15th century. In his work,
The Key to Arithmetic (Miftah al-Hisab), al-Kashi provided the first systematic and comprehensive treatment of decimal fractions as a complete system, predating similar European developments by nearly 175 years. A forerunner of modern European decimal notation was introduced by
Simon Stevin in the 16th century. Stevin's influential booklet
De Thiende ("the art of tenths") was first published in Dutch in 1585 and translated into French as
La Disme.
John Napier introduced using the period (.) to separate the integer part of a decimal number from the fractional part in his book on constructing tables of logarithms, published posthumously in 1620.
Natural languages A method of expressing every possible
natural number using a set of ten symbols emerged in India. Several Indian languages show a straightforward decimal system.
Dravidian languages have numbers between 10 and 20 expressed in a regular pattern of addition to 10. The
Hungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty"). A straightforward decimal rank system with a word for each order (10 , 100 , 1000 , 10,000 ), and in which 11 is expressed as
ten-one and 23 as
two-ten-three, and 89,345 is expressed as 8 (ten thousands) 9 (thousand) 3 (hundred) 4 (tens) 5 is found in
Chinese, and in
Vietnamese with a few irregularities.
Japanese,
Korean, and
Thai have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades. For example, in English 11 is "eleven" not "ten-one" or "one-teen". Incan languages such as
Quechua and
Aymara have an almost straightforward decimal system, in which 11 is expressed as
ten with one and 23 as
two-ten with three. Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.
Other bases Some cultures do, or did, use other bases of numbers. •
Pre-Columbian Mesoamerican cultures such as the
Maya used a
base-20 system (perhaps based on using all twenty fingers and
toes). • The
Yuki language in
California and the
Oto-Pamean languages in
Mexico have
octal (
base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves. • The existence of a non-decimal base in the earliest traces of the Germanic languages is attested by the presence of words and glosses meaning that the count is in decimal (cognates to "ten-count" or "tenty-wise"); such would be expected if normal counting is not decimal, and unusual if it were. Where this counting system is known, it is based on the "
long hundred" = 120, and a "long thousand" of 1200. The descriptions like "long" only appear after the "small hundred" of 100 appeared with the Christians.
E. V. Gordon's
Introduction to Old Norse gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240. Goodare details the use of the long hundred in Scotland in the Middle Ages, giving examples such as calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'. • Many or all of the
Chumashan languages originally used a
base-4 counting system, in which the names for numbers were structured according to multiples of 4 and
16. • Many languages use
quinary (base-5) number systems, including
Gumatj,
Nunggubuyu,
Kuurn Kopan Noot and
Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5. • Some
Nigerians use
duodecimal systems. So did some small communities in India and Nepal, as indicated by their languages. • The
Huli language of
Papua New Guinea is reported to have
base-15 numbers.
Ngui means 15,
ngui ki means 15 × 2 = 30, and
ngui ngui means 15 × 15 = 225. •
Umbu-Ungu, also known as Kakoli, is reported to have
base-24 numbers.
Tokapu means 24,
tokapu talu means 24 × 2 = 48, and
tokapu tokapu means 24 × 24 = 576. •
Ngiti is reported to have a
base-32 number system with base-4 cycles.
Mer means 6,
mer an thef means 6 × 2 = 12,
nif means 36, and
nif thef means 36×2 = 72. == See also ==