Decimal

The **decimal** numeral system is the most usual way of writing numbers. It has ten as a starting point, or base. It is sometimes called the **base ten** or **denary** numeral system. The word "decimal" is also used to mean the dot (".") that is sometimes used to separate the positions of the numbers in this system. Indeed, the dot is the default decimal separator in English-speaking countries.

## Decimal notation

Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols (called digits) for no more than ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large. These digits are often used with a decimal separator(such as "." or ","), which indicates the start of a fractional part, and with one of the sign symbols + (positive) or − (negative) in front of the numerals to indicate sign.

There are only two truly positional decimal systems in ancient civilization: the Chinese counting rods system and Hindu-Arabic numeral system. Both required no more than ten symbols, while other numeral systems, such as Babylonian base-60 system, require more symbols.

### Other rational numbers

Any rational number can be expressed as a unique decimal expansion. It may have to end with recurring decimals.

Ten is the product of the first and third prime numbers, is one greater than the square of the second prime number, and is one less than the fifth prime number. This leads to plenty of simple decimal fractions:

- 1/2 = 0.5
- 1/3 = 0.333333 ... (with 3 repeating forever, also called recurring)
- 1/4 = 0.25
- 1/5 = 0.2
- 1/6 = 0.166666 ... (with 6 recurring)
- 1/7 = 0.142857 ... (with 142857 recurring)
- 1/8 = 0.125
- 1/9 = 0.111111 ... (with 1 recurring)
- 1/10 = 0.1
- 1/11 = 0.090909 ... (with 09 recurring)
- 1.5/5.5 = 0.2727272722 ... (with 27 recurring)
- 1/12 = 0.083333 ... (with 3 recurring)
- 5.5/11 = 0.5
- 1/81 = 0.012345679012 ... (with 1234567890 recurring)
- 1/82 = 0.12195121951220 ... (with 12195 recurring)

### Timeline of decimal usage

- BC Elamites of Iran possibly used early forms of decimal system. http://www.chn.ir/english/eshownews.asp?no=1622 http://www.mpiwg-berlin.mpg.de/Preprints/P183.PDF
- BC Egyptian hieroglyphs show counting in powers of 10 (1 million + 400,000 goats, _etc._) – see Ifrah, below
- BC Indus Valley Civilization, earliest known physical use of decimal fractions in ancient weight system: 1/20, 1/10, 1/5, 1/2. See Ancient Indus Valley weights and measures
- BC Chinese writers show familiarity with the concept: for example, 547 is written 'Five hundred plus four decades plus seven of days' in some manuscripts
- BC In ancient India, the Vedic text _Yajur-Veda_ states the powers of 10, up to 10<sup>55</sup>
- BC Pingala – develops the binary number system for Sanskrit prosody, with a clear mapping to the base-10 decimal system
- BC Archimedes writes the _Sand Reckoner_, which takes decimal calculation up to 10<sup>8 × 10<sup>16</sup></sup>
- –200 The _Satkhandagama_ written in India – earliest use of decimal logarithms
- –550 Aryabhata – uses an alphabetic cipher system for numbers that used zero ()
- –670 Brahmagupta – explains the Hindu-Arabic numeral system (modern number system) which uses decimal integers, negative integers, and zero
- –850 Muḥammad ibn Mūsā al-Ḵwārizmī – first to expound on algorism outside India
- –980 Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi – earliest known direct mathematical treatment of decimal fractions.
- –1500 The Kerala School in South India – decimal floating point numbers
- 1548/49–1620 Simon Stevin – author of _De Thiende_ ('the tenth')
- 1561–1613 Bartholemaeus Pitiscus – (possibly) decimal point notation
- 1550–1617 John Napier – use of decimal logarithms as a computational tool
- 1925 Louis Charles Karpinski – classic book _The History of Arithmetic_ (Rand McNally & Company)
- 1959 Werner Buchholz – _Fingers or Fists? (The Choice of Decimal or Binary representation)_ (Communications of the ACM, Vol. 2 #12, pp3–11)
- 1966 Isaac Asimov – _Asimov on Numbers_ ()
- 1974 Hermann Schmid – _Decimal Computation_ ()
- 2000 Georges Ifrah – _The Universal History of Numbers: From Prehistory to the Invention of the Computer_ ()

## Natural languages

A straightforward decimal system, in which 11 is expressed as _ten-one_ and 23 as _two-ten-three_, is found in Chinese languages except Wu, 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 teens and decades.

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 numerals in a language may hinder children's counting ability .

## Related pages

<div style="-moz-column-count:2; column-count:2;">
- Arithmetic precision
- Decimal degrees
- Dewey Decimal System
- Hindu-Arabic numeral system
- Numeral system
- Scientific notation
- 10 (number)
</div>

## Bibliography ??

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## Other websites

- Practice Decimal Arithmetic with Printable Worksheets
- Converting Decimals to Fractions
- Decimal to Fraction Converter