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Decimal

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The '''decimal''' ('''base ten''' or occasionally '''denary''') numeral system has 10 (number) ten as its Base (mathematics) base. It is the most widely used numeral system, probably because humans commonly have a total of ten digits on their hands. {{numeral_systems}}

Decimal notation
Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols (called numerical digit digits) for ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent numbers. These digits are often used with a decimal separator which indicates the start of a fractional part, and with one of the sign symbols + (plus) or − (minus) to indicate sign. The algorism decimal system is a positional notation positional numeral system; it has positions for units, tens, hundreds, ''etc.'' The position of each digit conveys the multiplier (a power of ten) to be used with that digit—each position has a value ten times that of the position to its right. 10 (number) Ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). In many languages the word digit or its translation is also the anatomical term referring to fingers and toes. In English, decimal (decimus < Latin Lat.) means ''tenth'', decimate means ''reduce by a tenth'', and denary (denarius < Lat.) means ''the unit of ten''. The symbols for the digits in common use around the globe today are called Hindu-Arabic numerals Arabic numerals by Europeans and Indian numerals by Arabs, the two groups' terms both referring to the culture from which they learned the system. However, the symbols used in different areas are not identical; for instance, Western Arabic numerals (from which the European numerals are derived) differ from the forms used by other Arab cultures.

Alternative notations
Some cultures do, or used to, use other numeral systems, including the Tzotzil, who use a vigesimal system (using all twenty fingers and toes), some Nigerians who use several duodecimal systems, the Babylonians, who used sexagesimal, and the Yuki tribe Yuki, who reportedly used octal. Computer hardware and software systems commonly use a binary system binary repesentation, internally. For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems. For most purposes, however, binary values are converted to the equivalent decimal values for presentation to and manipulation by humans. Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using binary-coded decimal, but there are other decimal representations in use (see IEEE 754r), especially in database implementations. Decimal arithmetic is used in computers so that decimal fractional results can be computed exactly, which is not possible using a binary fractional representation. This is often important for financial and other calculations [http://www2.hursley.ibm.com/decimal/decifaq.html].

Decimal fractions
A '''decimal fraction''' is a vulgar fraction fraction where the denominator is a exponentiation power of ten. Decimal fractions are commonly expressed without a denominator, the decimal separator being inserted into the numerator (with leading zeros added if needed), at the position from the right corresponding to the power of ten of the denominator. e.g., 8/10, 833/100, 83/1000, 8/10000 and 80/10000 are expressed as: 0.8, 8.33, 0.083, 0.0008 and 0.008. Numbers which can be expressed exactly in this way are called '''decimal numbers''' or '''regular numbers'''. The integer and fraction (mathematics) fractional parts of a decimal number are separated by a decimal separator. In this article, as in most of the English speaking world, a dot (^'''.''') or period ('''.''') is used as the separator. It is usual for a decimal number which is less than one to have a leading zero. Trailing zeros after the decimal point are not necessary, although in science, engineering and statistics they can be retained to indicate a required precision or to show a level of confidence in the accuracy of the number: Whereas 0.080 and 0.08 are numerically equal, in engineering 0.080 suggests a measurement with an error of up to 1 part in two thousand (±0.0005), while 0.08 suggests a measurement with an error of up to 1 in a two hundred (see ''Significant figures'').

Other rational numbers
Any rational number which cannot be expressed as a decimal fraction has a unique infinite decimal expansion ending 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 = 0.333333â€¦ (with 3 recurring) :1/4 = 0.25 :1/5 = 0.2 :1/6 = 0.166666â€¦ (with 6 recurring) :1/8 = 0.125 :1/9 = 0.111111â€¦ (with 1 recurring) :1/10 = 0.1 :1/11 = 0.090909â€¦ (with 09 recurring) :1/12 = 0.083333â€¦ (with 3 recurring) :1/81 = 0.012345679012â€¦ (with 012345679 recurring) Other prime factors in the denominator will give longer recurring sequences, see for instance 7 (number) 7, 13 (number) 13. That a rational must produce a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only (q-1) possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q-1. For instance to find 3/7 by long division: .4 2 8 5 7 1 4 ... 7 ) 3.0 0 0 0 0 0 0 0 2 8 30/7 = 4 r 2 2 0 1 4 20/7 = 2 r 6 6 0 5 6 60/7 = 8 r 4 4 0 3 5 40/7 = 5 r 5 5 0 4 9 50/7 = 7 r 1 1 0 7 10/7 = 1 r 3 3 0 2 8 30/7 = 4 r 2 (again) 2 0 etc The converse to this observation is that every recurring decimal represents a rational number ''p''/''q''. This is a consequence of the fact the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number. For instance, :

0.0123123123\cdots = \frac{123}{10000} \sum_{k=0}^\infty 0.001^k = \frac{123}{10000}\ \frac{1}{1-0.001} = \frac{123}{9990} = \frac{41}{3330}

Real numbers
Every real number has a (possibly infinite) decimal representation, i.e., it can be written as :

x = \mathop{\rm sign}(x) \sum_{i\in\mathbb Z} a_i\,10^i

where * sign() is the sign function, * ''a_i'' ∈ { 0,1,â€¦,9 } for all ''i'' ∈ '''Z''', are its '''decimal digits''', equal to zero for all ''i'' greater than some number (the common logarithm of |x|). Such a sum converges, even if there is an infinite number of ''a_i'' (with negative indices), which is the case for all reals which are not decimal numbers, according to what precedes. Indeed, consider those rational numbers that can be written as p/(2^a5^b) (i.e. the only prime factors in denominator are 2 and 5). In this case there is a terminating decimal representation. For instance 1/1=1, −1/2=−0.5, 3/5=0.6, 3/25=0.12 and 1306/1250=1.0448. Such numbers are the only real numbers which don't have a unique decimal representation, as they can also be written as a representation that has a recurring 9, for instance 1=0.99999â€¦, −1/2=−0.499999â€¦, etc. Rational numbers p/q with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring decimal representation. This leaves the irrational numbers. They also have unique infinite decimal representation, and can be characterised as the numbers whose decimal representations neither terminate nor recur. So in general the decimal representation is unique, if one excludes representations that end in a recurring 9. Naturally, the same trichotomy holds for other base-n Positional notation positional numeral systems: * Terminating representation: rational where the denominator divides some n^k * Recurring representation: other rational * Non-terminating, non-recurring representation: irrational and a version of this even holds for irrational-base numeration systems, such as golden mean base representation.

History

Decimal writers
* ''c.'' 3500 - 2500 BC Elamite Empire Elamites of Iran possibly use early forms of decimal system. [http://www.chn.ir/english/eshownews.asp?no=1622] [http://www.mpiwg-berlin.mpg.de/Preprints/P183.PDF] * ''c.'' 2900 BC Egyptian hieroglyphs show counting in powers of 10 (1 million + 400,000 goats, ''etc.''). * ''c.'' 2600 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. * ''c.'' 1400 BC History of China Chinese writers show familiarity with the concept: for example, 547 is written 'Five hundred plus four decades plus seven of days' in some manuscripts. * ''c.'' 1200 BC In ancient India, the Vedic text ''Yajur-Veda'' states the Exponentiation powers of 10, upto 10⁵⁵. * ''c.'' 450 BC PÄ?á¹‡ini Panini – uses the null operator in his grammar of Sanskrit. * ''c.'' 400 BC Pingala – develops the binary number system for Sanskrit prosody, with a clear mapping to the base-10 decimal system. * ''c.'' 100–200 The ''Indian mathematics#Satkhandagama (100-200 CE) Satkhandagama'' written in India – earliest use of decimal logarithms. * ''c.'' 476–550 Aryabhata – uses an alphabetic cipher system for numbers that used zero. * ''c.'' 598–670 Brahmagupta – explains the Hindu-Arabic numerals (modern number system) which uses decimal integers, negative integers, and 0 (number) zero. * ''c.'' 780–850 Muá¸¥ammad ibn MÅ«sÄ? al-á¸´wÄ?rizmÄ« – first to expound on algorism outside India. * ''c.'' 920–980 Al-Uqlidisi Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi – earliest known direct mathematical treatment of decimal fractions. * ''c.'' 1300–1500 The Kerala School in South India – decimal floating point numbers. * 1548/1549 49–1620 Simon_Stevin#Decimal_fractions 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

See also
* Algorism * Decimal point * Decimal representation * Decimal sequences for cryptography * Numeral system * Binary-coded decimal * Dewey Decimal System * 10 (number)

External links

- Decimal arithmetic FAQ * Tests: [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1352 Decimal Place Value] [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1353&CurriculumID=5 Sums] [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=739&CurriculumID=5 Fractions]
- Practice Decimal Arithmetic with Printable Worksheets
- Converting Decimals to Fractions Category:Elementary arithmetic Category:Fractions Category:Positional numeral systems 10 be:Ð”Ð·ÐµÑ?Ñ?Ñ‚ÐºÐ¾Ð²Ð°Ñ? Ñ?Ñ‹Ñ?Ñ‚Ñ?Ð¼Ð° Ð·ÑŒÐ»Ñ–Ñ‡Ñ?Ð½ÑŒÐ½Ñ? da:Decimal de:Dezimalsystem es:Sistema decimal eo:Dekuma sistemo fr:Nombre dÃ©cimal ko:ì‹ì§„ë²• it:Sistema numerico decimale he:×”×©×™×˜×” ×”×¢×©×¨×•× ×™×ª nl:Decimaal ja:å??é€²è¨˜æ•°æ³• no:Titallsystemet pl:DziesiÄ™tny system liczbowy pt:Sistema decimal ru:Ð”ÐµÑ?Ñ?Ñ‚Ð¸Ñ‡Ð½Ð°Ñ? Ñ?Ð¸Ñ?Ñ‚ÐµÐ¼Ð° Ñ?Ñ‡Ð¸Ñ?Ð»ÐµÐ½Ð¸Ñ? sl:DesetiÅ¡ki Å¡tevilski sistem fi:KymmenjÃ¤rjestelmÃ¤ sv:Decimala talsystemet th:à¹€à¸¥à¸‚à¸?à¸²à¸™à¸ªà¸´à¸š uk:Ð”ÐµÑ?Ñ?Ñ‚ÐºÐ¾Ð²Ð° Ñ?Ð¸Ñ?Ñ‚ÐµÐ¼Ð° Ñ‡Ð¸Ñ?Ð»ÐµÐ½Ð½Ñ? zh:å??è¿›åˆ¶

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