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# Decimal

Decimal, or denary, notation is the most common way of writing 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) together with the decimal point and the sign symbols + (plus) and − (minus) to represent numbers.

Decimal is the principal numeral system used by humans. This is almost certainly because humans have ten fingers; digit is also the anatomical term referring to fingers and toes. (However, some cultures do or did use other number systems, including the Maya, the Babylonians, and the Yuki Indians of California, who used base 8 because they counted the spaces between their fingers rather than the fingers themselves.) The set of symbols for the digits is called Arabic numerals by Europeans and Indian numerals by Arabs, each term referring to the people that the users of the term got the symbols from.

Computers internally almost always use another numeral system, binary, because it is hugely more efficient to implement this system electronically. For external use, the 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.

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## Fractions

### Decimal fractions

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

Decimal fractions are usually expressed without a denominator, the decimal point being inserted into the numerator at a position 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 & 0.008.

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)

Other prime factors in the denominator will give longer recurring sequences, see for instance 7, 13.

That a rational must producing 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,

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

### Decimal representation of the real numbers

Every real number has a representation as a decimal fraction.

The representation is unique, except for rational numbers which can be written as p/(2a5b) (i.e. the only prime factors in denominator are 2 and 5). In all such cases 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.

### Decimal writers

• c. 2900 BC Egyptian hieroglyphs show counting in powers of 10 (1 million + 400,000 goats, etc.).
• 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 – first direct treatment of decimal fractions
• 1548–1620 Simon Stevin – author of De Thiende ('the tenth')
• 1550–1617 John Napier– decimal logarithms