Decimal

Information about Decimal

Numeral systems by culture
Hindu-Arabic numerals
Western Arabic
Eastern Arabic
Khmer		Indian family
Brahmi
Thai
East Asian numerals
Chinese
Chinese counting rods		Korean
Japanese 
Alphabetic numerals
Abjad
Armenian
Cyrillic
Ge'ez		Hebrew
Ionian/Greek
Sanskrit
 
Other systems
Attic
Etruscan
Urnfield
Roman		Babylonian
Egyptian
Mayan
List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64
3, 9, 12, 24, 30, 36, 60,
 •  • [ e]
The decimal (base ten or occasionally denary) numeral system has ten as its base. It is the most widely used numeral system, perhaps because humans have four fingers and a thumb on each hand, giving a total of ten digits over both hands.

Decimal notation

Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols (called digits) for no more than ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large. These digits are often used with a decimal separator which indicates the start of a fractional part, and with one of the sign symbols + (positive) or − (negative) in front of the numerals to indicate sign. There are only two truly positional decimal systems in ancient civilization, the Chinese counting rods system and Hindu-Arabic numeric system, both required no more than ten symbols. Other numeric systems require more symbols.

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 pre-Columbian Mesoamerican cultures such as the Maya, who use a vigesimal system (using all twenty fingers and toes), some Nigerians who use several duodecimal (base 12) systems, the Babylonians, who used sexagesimal (base 60), and the Yuki, who reportedly used octal (base 8).

Computer hardware and software systems commonly use a binary representation, 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, and 8/10000 are expressed as: 0.8, 8.33, 0.083, and 0.0008. In English-speaking countries, a dot (·) or period (.) is used as the decimal separator; in most other languages a comma is used.

The integer part or integral part of a decimal number is the part to the left of the decimal separator (see also floor function). The part from the decimal separator to the right is the fractional part; if considered as a separate number, a zero is often written in front. Especially for negative numbers, we have to distinguish between the fractional part of the notation and the fractional part of the number itself, because the latter gets its own minus sign. 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 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 repeating)
1/4 = 0.25
1/5 = 0.2
1/6 = 0.166666… (with 6 repeating)
1/8 = 0.125
1/9 = 0.111111… (with 1 repeating)
1/10 = 0.1
1/11 = 0.090909… (with 09 repeating)
1/12 = 0.083333… (with 3 repeating)
1/81 = 0.012345679012… (with 012345679 repeating)


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. 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,

Real numbers

Further information: Decimal representation


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

where
  • sign() is the sign function,
  • ai ∈ { 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 ai.

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/(2a5b). 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 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

There follows a chronological list of recorded decimal writers.

Decimal writers

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.

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

See also

References

  • id="CITEREFAzar1999">Azar, Beth (1999), "English words may hinder math skills development", American Psychology Association Monitor 30 (4), <[2].

    External links

    numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the context that allows the numeral "11" to be interpreted as the binary numeral for three
    ..... Click the link for more information.
    Hindu-Arabic numeral system (also called Algorism) is a positional decimal numeral system documented from the 9th century.

    The symbols (glyphs) used to represent the system are in principle independent of the system itself.
    ..... Click the link for more information.
    Arabic numerals, known formally as Hindu-Arabic numerals, and also as Indian numerals, Hindu numerals, Western Arabic numerals, European numerals, or Western numerals, are the most common symbolic representation of numbers around the world.
    ..... Click the link for more information.
    The Eastern Arabic numerals (also called Arabic-Indic numerals, Arabic Eastern Numerals) are the symbols (glyphs) used to represent the Hindu-Arabic numeral system in conjunction with the Arabic alphabet in Egypt, Iran, Afghanistan, Pakistan and parts of India, and also in
    ..... Click the link for more information.
    Khmer numerals are the numerals used in the Khmer language of Cambodia. In informal spoken language one can ignore the last "sep" (30 to 90) and it is still understood.
    e.g.
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    symbols used in various modern Indian scripts for the numbers from zero to nine:

    Variant 0 1 2 3 4 5 6 7 8 9 Used in
    Eastern Nagari numerals ০ ১ ২ ৩ ৪ ৫ ৬ ৭ ৮ ? Bengali language
    Assamese language

    ..... Click the link for more information.
    Brahmi numerals are an indigenous Indian numeral system attested from the 3rd century BCE (somewhat later in the case of most of the tens). They are the direct graphic ancestors of the modern Indic and Hindu-Arabic numerals.
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    Thai numerals (ตัวเลขไทย) are traditionally used in Thailand, although the Arabic numerals (also known as Western numerals) are more common.
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    This page contains Chinese text.
    Without proper rendering support, you may see question marks, boxes, or other symbols instead of Chinese characters.

    Numeral systems by culture
    Hindu-Arabic numerals
    Western Arabic
    Eastern Arabic
    Khmer Indian family
    ..... Click the link for more information.
    Counting rods (Traditional Chinese: 籌; Simplified Chinese: 筹; Pinyin: chou2
    ..... Click the link for more information.
      sset
    • 여덟 권 yeodeolgwon (eight (books)) is pronounced like [여덜꿘] yeodeolkkwon
    Several numerals have long vowels, namely 둘 (two), 셋 (three) and 넷 (four), but these become short when
    ..... Click the link for more information.
    Japanese numerals is the system of number names used in the Japanese language. The Japanese numerals in writing are entirely based on the Chinese numerals and the grouping of large numbers follow the Chinese tradition of grouping by 10,000.
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    Abjad numerals are a decimal numeral system which was used in the Arabic-speaking world prior to the use of the Hindu-Arabic numerals from the 8th century, and in parallel with the latter until Modern times.
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    Armenian numerals is a historic numeral system created using the majuscules (uppercase letters) of the Armenian alphabet.

    There was no notation for zero in the old system, and the numeric values for individual letters were added together.
    ..... Click the link for more information.
    Cyrillic numerals was a numbering system derived from the Cyrillic alphabet, used by South and East Slavic peoples. The system was used in Russia as late as the 1700s when Peter the Great replaced it with the Hindu-Arabic numeral system.
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    Hebrew numerals is a quasi-decimal alphabetic numeral system using the letters of the Hebrew alphabet.

    In this system, there is no notation for zero, and the numeric values for individual letters are added together. Each unit (1, 2, ...
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    Greek numerals are a system of representing numbers using letters of the Greek alphabet. They are also known by the names Milesian numerals, Alexandrian numerals, or alphabetic numerals.
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    Attic numerals were used by ancient Greeks, possibly from the 7th century BC. They were also known as Herodianic numerals because they were first described in a 2nd century manuscript by Herodian.
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    Etruscan numerals were used by the ancient Etruscans. The system was adapted from the Greek Attic numerals and formed the inspiration for the later Roman numerals.

    Etruscan Decimal Symbol *
    θu 1 I
    ma? 5 ?
    śar 10 X
    muval? 50
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    /» and the fifths place with a stroke from the top-left to the bottom-right «\». The numbers from 1 = / to 29 = ////\\\\\ have been found.

    Interpretation

    These embossed marks, unique in objects from the Bronze Age, were introduced in cast-iron molds and were not
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    Roman numerals is a numeral system originating in ancient Rome, adapted from Etruscan numerals. The system used in classical antiquity was slightly modified in the Middle Ages to produce the system we use today. It is based on certain letters which are given values as numerals.
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    Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.
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    Egyptian numerals was a numeral system used in ancient Egypt. It was a decimal system, often rounded off to the higher power, written in hieroglyphs. The hieratic form of numerals stressed an exact finite series notation, being ciphered one:one onto the Egyptian alphabet.
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    Maya numerals is very simple. [1]
    Addition is performed by combining the numeric symbols at each level:

    If five or more dots result from the combination, five dots are removed and replaced by a bar.
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    This is a list of numeral system topics (and "numeric representations"), by Wikipedia page. It does not systematically list computer formats for storing numbers (computer numbering formats). See also number names.
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    A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system.
    ..... Click the link for more information.
    base or radix is usually the number of various unique digits, including zero, that a positional numeral system uses to represent numbers. For example, the decimal system, the most common system in use today, uses base ten, hence the maximum number a single digit will ever
    ..... Click the link for more information.
    binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2.
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    Quaternary is the base-4 numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number.

    It shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the
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    octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7.

    Octal numerals can be made from binary numerals by grouping consecutive digits into groups of three (starting from the right).
    ..... Click the link for more information.

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