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, probably because humans commonly have a total of ten digits on their hands.

Numeral systems List of numeral system topics
Hindu-Arabic system History Symbol sets: Western Arabic Eastern Arabic Indian family Thai Abjad Armenian Babylonian Brahmi Chinese Cyrillic Egyptian Etruscan Ge'ez Greek Attic Ionian Hebrew Japanese Khmer Korean Mayan Roman D'ni (fictitious)
Positional systems with various bases:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 20, 24, 26, 27, 30, 32, 36, 60, 64 Non-standard systems: 1, -2, -3, Balanced ternary, mixed, Factoradic, Fibonacci coding, bijective, 2i, φ
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Contents 1 Decimal notation 1.1 Alternative notations 1.2 Decimal fractions 1.3 Other rational numbers 1.4 Real numbers 2 History 2.1 Decimal writers 3 See also 4 External links

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 [1].

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.

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

The integer and 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 = 0.333333… (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,

<math>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}</math>

Real numbers

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

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

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 (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=0.99999…, −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.

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. [2] [3]
• 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

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: Decimal Place Value Sums Fractions
• Practice Decimal Arithmetic with Printable Worksheets
• Converting Decimals to Fractionsbe:Дзесятковая сыстэма зьлічэньня

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