# Decimal

Decimal, or less commonly, denary, usually refers to the base 10 numeral system. Template:Table Numeral Systems

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## Decimal notation

Decimal notation is the writing of numbers in the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) to represent numbers. These digits are frequently used with the decimal point and the sign symbols + (plus) and − (minus), which indicate fractional parts and sign.

Decimal is the most common numeral system used around the world. This is because humans have ten fingers; for example, in English, digit is also the anatomical term referring to fingers and toes. However, some cultures do or did historically use other number systems, including the Tzotzil, who use a base 20 system (using all 20 fingers and toes), some Nigerians who use several base 12 systems, the Babylonians, who used base 60, and the Yuki, who reportedly used base 8.

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" numbers differ from the forms used by Arab cultures.

Computers internally commonly use the binary system, because it is slightly easier to implement than base-10, and in the 1940s any simplification of circuitry implied a significant improvement in reliability. This advantage is now outweighed by the human factors issues, and so computing is moving towards decimal data types (see IEEE 754r).

For external use, this binary representation is sometimes presented in the related octal or hexadecimal systems. Decimal numerals can be encoded for computers using binary-coded decimal or more efficient schemes.

## Decimal representation of real numbers

### Fractions

#### Decimal fractions

A decimal fraction is a vulgar fraction where the denominator is a power of ten.

Decimal fractions can be expressed without a denominator, the decimal point 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 thus: 0.8, 8.33, 0.083, 0.0008 and 0.008.

Numbers which can be expressed in this way are called decimal numbers.

The integer and fractional parts of a decimal number are separated by a decimal point. In this article, as in most of the English speaking world, a dot (.) is used. It is usual for a decimal number which is less than one to have a leading zero. Trailing zeroes after the decimal point are not necessary, although in science, engineering and statistics they can be retained to show a level of confidence in the accuracy of the number: Whereas 0.080 and 0.08 are mathematically the same number, in engineering 0.080 suggests an error of up to 1 part in a thousand, while 0.08 suggests an error of up to 1 in a hundred.

#### Decimal representation of 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 number, 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,

[itex]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}[itex]

### Decimal representation of the real numbers

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

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

where

• sign() is the sign function,
• ai ∈ { 0,1,...,9 } for all iZ, are its decimal digits, equal to zero for all i greater than some number (the common logarithm of |x|).

Such a sum always makes sense (i.e. converges), even if there is an infinite number of ai (with negative indices), which is the case for all reals which are not decimal numbers, according to what precedes.

The representation is unique, if one excludes representations that end in a recurring 9.

Indeed, consider rational numbers which can be written as p/(2a5b) (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=0.99999..., −1/2=−0.499999..., 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.

Naturally, the same trichotomy holds for other base-n numeral systems:

• Terminating representation: rational where the denominator divides some nk
• 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.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: 0.05, 0.1, 0.2, 0.5. See Ancient 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. 598–670 Brahmagupta – decimal integers, negative integers, and zero
• c. 790–840 Abu Abdullah Muhammad bin Musa al-Khwarizmi – 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
• 1548/491620 Simon Stevin – author of De Thiende ('the tenth')
• 15501617 John Napier– decimal logarithms

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