Decimal
Ten fingers on two hands likely sparked the first human counting systems. Ancient Egyptian hieroglyphs from around 3000 BCE used a purely decimal system to record quantities. The Indus Valley Civilisation standardized weights based on ratios like one-tenth, one-fifth, and ten. A ruler found in Mohenjo-daro was divided into exactly ten equal parts for trade measurements. The Únětice culture in central Europe between 2300 BC and 1600 BC employed standardised weights alongside a decimal system for commerce. Classical Greece utilized powers of ten including an intermediate base of five. Roman numerals also relied on powers of ten despite their non-positional nature. Archimedes invented a decimal positional system in his Sand Reckoner around 287 BCE to 212 BCE. This early work was based on 10 raised to the power of eight.
The world's earliest positional decimal system emerged as Chinese rod calculus during the Warring States period. Bamboo slips dating from 305 BCE display the oldest known multiplication table using this method. By the third century CE, Chinese metrological units expressed decimal fractions of lengths without positionality. The Sunzi Suanjing text from the third to fifth centuries described calculations with decimal fractions using counting rods. Mathematician Zu Chongzhi calculated a seven-digit approximation of pi by the fifth century CE. Qin Jiushao published Mathematical Treatise in Nine Sections in 1247 explicitly writing a decimal fraction representing a number rather than a measurement. Al-Khwarizmi introduced fractions to Islamic countries in the early ninth century CE written with numerator above denominator. Abu'l-Hasan al-Uqlidisi wrote about positional decimal fractions for the first time in the tenth century. Simon Stevin published De Thiende in Dutch in 1585 introducing a forerunner of modern European decimal notation. John Napier used the period to separate integer and fractional parts in his logarithm tables published posthumously in 1620.
Decimal numerals do not allow an exact representation for all real numbers yet they approximate every real number with desired accuracy. A mass given as 1.32 milligrams implies true value lies between 1.315 milligrams and 1.325 milligrams. If that same mass is recorded as 1.320 milligrams, confidence increases that it falls between 1.3195 and 1.3205 milligrams. The difference between 4.69 and 4.690 remains zero despite the extra digit adding meaning to precision. Any real number that is not a decimal fraction possesses a unique infinite decimal expansion. Every decimal fraction has exactly two infinite expansions: one containing only zeros after some place and another containing only nines. Long division allows computing the infinite decimal expansion of any rational number. If the rational number is not a decimal fraction, the division continues indefinitely but repeats the same sequence of digits. The group 012345679 repeats indefinitely when dividing one by eighty-one.
Most modern computer hardware and software systems use binary representation internally though early computers like ENIAC used decimal representation. Many early computers such as IBM 650 also utilized decimal representation for internal operations. Computer programs express literals in decimal by default even if languages cannot encode those numbers precisely. Decimal arithmetic ensures fractional results of fixed length always compute to that same length of precision. This approach is especially important for financial calculations requiring integer multiples of the smallest currency unit. Binary negative powers have no finite binary fractional representation making exact decimal arithmetic impossible within standard binary formats. Database implementations often store values using variants of binary-coded decimal or decimal floating point standards. Newer revisions of the IEEE 754 Standard for Floating-Point Arithmetic include decimal floating point capabilities. Arbitrary-precision arithmetic enables exact calculations where standard binary methods fail to preserve decimal integrity.
A method expressing every possible natural number using ten symbols emerged in India. Dravidian languages express numbers between ten and twenty through regular addition patterns to ten. Hungarian forms all numbers between ten and twenty regularly with eleven expressed as tizenegy meaning one on ten. Chinese uses a straightforward decimal rank system where eighty-nine thousand three hundred forty-five breaks into eight ten-thousands, nine thousands, three hundreds, four tens, and five. Japanese, Korean, and Thai imported the Chinese decimal system but retain some irregularities. English names like eleven do not follow the pattern of ten-one or one-teen. Incan languages such as Quechua and Aymara maintain an almost straightforward decimal system where eleven means ten with one. Some psychologists suggest irregularities in English numeral names may hinder children's counting ability. Vietnamese shares the Chinese decimal structure though it contains a few irregularities compared to pure base-ten logic.
Pre-Columbian Mesoamerican cultures such as the Maya used a base-twenty system possibly based on using all twenty fingers and toes. The Yuki language in California counts using spaces between fingers resulting in octal base-eight systems. Germanic languages historically utilized long hundreds equaling 120 and long thousands equaling 1200 before Christian influence introduced small hundreds of 100. Chumashan languages originally employed base-four counting structured according to multiples of four and sixteen. Gumatj is the only true five-to-twenty-five language known where 25 represents the higher group of five. Nigerians use duodecimal systems while small communities in India and Nepal also adopted similar approaches. Huli language of Papua New Guinea reports base-fifteen numbers where ngui equals fifteen and ngui ki equals thirty. Umbu-Ungu known as Kakoli utilizes base-twenty-four numbers where tokapu talu equals forty-eight. Ngiti maintains a base-thirty-two number system with base-four cycles. Ndom language of Papua New Guinea uses base-six numerals where mer means six and nif thef equals seventy-two.
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Common questions
When did ancient Egyptian hieroglyphs use a purely decimal system to record quantities?
Ancient Egyptian hieroglyphs used a purely decimal system to record quantities around 3000 BCE. This early system relied on powers of ten without positional notation.
Who invented the first decimal positional system in his Sand Reckoner work?
Archimedes invented a decimal positional system in his Sand Reckoner between 287 BCE and 212 BCE. His early work was based on 10 raised to the power of eight.
What year did Simon Stevin publish De Thiende introducing modern European decimal notation?
Simon Stevin published De Thiende in Dutch in 1585 introducing a forerunner of modern European decimal notation. This publication established key principles for decimal fractions in Europe.
How does decimal representation handle real numbers that are not decimal fractions?
Any real number that is not a decimal fraction possesses a unique infinite decimal expansion. Long division allows computing this infinite expansion which repeats the same sequence of digits indefinitely if the number is rational.
Which computer systems historically utilized decimal representation for internal operations before binary became standard?
Early computers such as ENIAC and IBM 650 utilized decimal representation for internal operations. Modern database implementations often store values using variants of binary-coded decimal or decimal floating point standards instead.