Decimal

Zero-digits after a decimal separator serve the purpose of signifying the precision of a value.

These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers.

In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation.

An example of a fraction that cannot be represented by a decimal expression (with a finite number of digits) is ⁠1/3⁠, 3 not being a power of 10.

In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the error bounds.

In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see also significant figures).

or which is called an infinite decimal expansion of x. Conversely, for any integer [x]0 and any sequence of digits

Long division allows computing the infinite decimal expansion of a rational number.

[9] For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems.

This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes.

have no finite binary fractional representation; and is generally impossible for multiplication (or division).

The Únětice culture in central Europe (2300-1600 BC) used standardised weights and a decimal system in trade.

[22] The number system of classical Greece also used powers of ten, including an intermediate base of 5, as did Roman numerals.

[23] Notably, the polymath Archimedes (c. 287–212 BCE) invented a decimal positional system in his Sand Reckoner which was based on 108.

[26] The world's earliest positional decimal system was the Chinese rod calculus.

[27] Starting from the 2nd century BCE, some Chinese units for length were based on divisions into ten; by the 3rd century CE these metrological units were used to express decimal fractions of lengths, non-positionally.

[28] 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 π. 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.

[29] 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.

[27] 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.

[27][30] 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.

[31] The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350 but did not develop any notation to represent them.

[32] The Persian mathematician Jamshid al-Kashi used, and claimed to have discovered, decimal fractions in the 15th century.

[31] 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.

8, archive p. 32) A method of expressing every possible natural number using a set of ten symbols emerged in India.

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 十, 100 百, 1000 千, 10,000 万), and in which 11 is expressed as ten-one and 23 as two-ten-three, and 89,345 is expressed as 8 (ten thousands) 万 9 (thousand) 千 3 (hundred) 百 4 (tens) 十 5 is found in Chinese, and in Vietnamese with a few irregularities.

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.