Decimal

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The decimal (base ten or occasionally denary) numeral system has ten as its base. It is the most widely used numeral system, perhaps because a human usually has four fingers and a thumb on each hand, giving a total of ten digits on both hands.

Decimal notation

Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols (called 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 decimal system is a 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.

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 < 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 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, who reportedly used octal.

Computer hardware and software systems commonly use a 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 ^*.

Decimal fractions

A decimal fraction is a fraction where the denominator is a 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.

The integer and fractional parts of a decimal number are separated by a decimal separator. In English a dot (^.) or period (.) is used as the separator. In most other languages a comma is used. 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).

Grouping of digits

Numbers with many digits before and/or after the decimal point may be divided into groups of three, starting from the decimal separator in both directions. This is done with a comma or a space. The latter is recommended in the SI/ISO 31-0. A point may be used in notations with a decimal comma. For details, see decimal separator. Notations like "12,345", "12.345", "12,345.678", and "12.345,678" are ambiguous if the notational system is not known.

Making groups of three digits also emphasizes that there is a base 1000 of the numeral system that is being used. See Decimal superbase.

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 = 0.333333… (with 3 recurring)

1/4 = 0.25

1/5 = 0.2

1/6 = 0.166666… (with 6 recurring)

1/8 = 0.125

1/9 = 0.111111… (with 1 recurring)

1/10 = 0.1

1/11 = 0.090909… (with 09 recurring)

1/12 = 0.083333… (with 3 recurring)

1/81 = 0.012345679012… (with 012345679 recurring)

Other prime factors in the denominator will give longer recurring sequences, see for instance 7, 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,…,9 } for all i ∈ Z, are its decimal digits, equal to zero for all i greater than some number (that number being the common logarithm of |x|).

Such a sum converges as i decreases, even if there are infinitely many nonzero a_i.

Rational numbers (e.g. p/q) with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring decimal representation.

Consider those rational numbers which have only the factors 2 and 5 in the denominator, i.e. which can be written as p/(2^a5^b). 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=0.99999…, 1/2=0.499999…, etc.

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 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 Elamites of Iran possibly use early forms of decimal system. [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 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 powers of 10, upto 10⁵⁵.
c. 450 BC 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 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 zero.
c. 780–850 Muḥammad ibn Mūsā al-Ḵwārizmī – first to expound on algorism outside India.
c. 920–980 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.
1548/49–1620 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

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.

Many psychologists suggest irregularities of numerals in a language may hinder children's counting ability.

External links

Elementary arithmetic | Fractions | Positional numeral systems

This article is licensed under the GNU Free Documentation License. It uses material from the "Decimal".

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