Английская Википедия:Decimal

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Файл:Decimal digit.png

Place value of number in decimal system

The decimal numeral system (also called the base-ten positional numeral system and denary Шаблон:IPAc-en or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (decimal fractions) of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.

A decimal numeral (also often just decimal or, less correctly, decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in Шаблон:Math or Шаблон:Math). Decimal may also refer specifically to the digits after the decimal separator, such as in "Шаблон:Math is the approximation of Шаблон:Pi to two decimals". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value.

The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form Шаблон:Math, where Шаблон:Math is an integer, and Шаблон:Math is a non-negative integer. Decimal fractions also result from the addition of an integer and a fractional part; the resulting sum sometimes is called a fractional number.

Decimals are commonly used to approximate real numbers. By increasing the number of digits after the decimal separator, one can make the approximation errors as small as one wants, when one has a method for computing the new digits.

Шаблон:Anchor Originally and in most uses, a decimal has only a finite number of digits after the decimal seperator. However, the decimal system has been extended to infinite decimals for representing any real number, by using an infinite sequence of digits after the decimal separator (see decimal representation). In this context, the usual decimals, with a finite number of non-zero digits after the decimal separator, are sometimes called terminating decimals. A repeating decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., Шаблон:Math). An infinite decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.

Origin

Файл:Two hand, ten fingers.jpg

Ten digits on two hands, the possible origin of decimal counting

Many numeral systems of ancient civilizations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly the Egyptian numerals, then the Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers, for forming the decimal numeral system.

Decimal notation

For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign "−". The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; the decimal separator is the dot "Шаблон:Math" in many countries (mostly English-speaking), and a comma "Шаблон:Math" in other countries.

For representing a non-negative number, a decimal numeral consists of

• either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer:

  <math>a_ma_{m-1}\ldots a_0</math>

• or a decimal mark separating two sequences of digits (such as "20.70828")

<math>a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n</math>.

If Шаблон:Math, that is, if the first sequence contains at least two digits, it is generally assumed that the first digit Шаблон:Math is not zero. In some circumstances it may be useful to have one or more 0's on the left; this does not change the value represented by the decimal: for example, Шаблон:Math. Similarly, if the final digit on the right of the decimal mark is zero—that is, if Шаблон:Math—it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; Шаблон:NoteTag for example, Шаблон:Math and Шаблон:Math.

For representing a negative number, a minus sign is placed before Шаблон:Math.

The numeral <math>a_ma_{m-1}\ldots a_0.b_1b_2\ldots b_n</math> represents the number

<math>a_m10^m+a_{m-1}10^{m-1}+\cdots+a_{0}10^0+\frac{b_1}{10^1}+\frac{b_2}{10^2}+\cdots+\frac{b_n}{10^n}</math>.

The integer part or integral part of a decimal numeral is the integer written to the left of the decimal separator (see also truncation). For a non-negative decimal numeral, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the fractional part, which equals the difference between the numeral and its integer part.

When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example, Шаблон:Math, instead of Шаблон:Math). In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation.

In brief, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal system is a positional numeral system.

Decimal fractions

Шаблон:Table Numeral Systems Decimal fractions (sometimes called decimal numbers, especially in contexts involving explicit fractions) are the rational numbers that may be expressed as a fraction whose denominator is a power of ten. For example, the decimal expressions <math>0.8, 14.89, 0.00079, 1.618, 3.14159</math> represent the fractions Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math and Шаблон:Math, and therefore denote decimal fractions. An example of a fraction that cannot be represented by a decimal expression (with a finite number of digits) is Шаблон:Math, 3 not being a power of 10.

More generally, a decimal with Шаблон:Math digits after the separator (a point or comma) represents the fraction with denominator Шаблон:Math, whose numerator is the integer obtained by removing the separator.

It follows that a number is a decimal fraction if and only if it has a finite decimal representation.

Expressed as fully reduced fractions, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are

<math>1=2^0\cdot 5^0, 2=2^1\cdot 5^0, 4=2^2\cdot 5^0, 5=2^0\cdot 5^1, 8=2^3\cdot 5^0, 10=2^1\cdot 5^1, 16=2^4\cdot 5^0, 20=2^2\cdot5^1, 25=2^0\cdot 5^2, \ldots</math>

Approximation using decimal numbers

Decimal numerals do not allow an exact representation for all real numbers. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates Шаблон:Pi, being less than 10⁻⁵ off; so decimals are widely used in science, engineering and everyday life.

More precisely, for every real number Шаблон:Mvar and every positive integer Шаблон:Mvar, there are two decimals Шаблон:Mvar and Шаблон:Mvar with at most Шаблон:Mvar digits after the decimal mark such that Шаблон:Math and Шаблон:Math.

Numbers are very often obtained as the result of measurement. As measurements are subject to measurement uncertainty with a known upper bound, the result of a measurement is well-represented by a decimal with Шаблон:Math digits after the decimal mark, as soon as the absolute measurement error is bounded from above by Шаблон:Math. In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the error bounds. For example, although 0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see also significant figures).

Infinite decimal expansion

For a real number Шаблон:Mvar and an integer Шаблон:Math, let Шаблон:Math denote the (finite) decimal expansion of the greatest number that is not greater than Шаблон:Mvar that has exactly Шаблон:Mvar digits after the decimal mark. Let Шаблон:Math denote the last digit of Шаблон:Math. It is straightforward to see that Шаблон:Math may be obtained by appending Шаблон:Math to the right of Шаблон:Math. This way one has

Шаблон:Math,

and the difference of Шаблон:Math and Шаблон:Math amounts to

<math>\left\vert \left [ x \right ]_n-\left [ x \right ]_{n-1} \right\vert=d_n\cdot10^{-n}<10^{-n+1}</math>,

which is either 0, if Шаблон:Math, or gets arbitrarily small as Шаблон:Mvar tends to infinity. According to the definition of a limit, Шаблон:Mvar is the limit of Шаблон:Math when Шаблон:Mvar tends to infinity. This is written as<math display="inline">\; x = \lim_{n\rightarrow\infty} [x]_n \;</math>or

Шаблон:Math,

which is called an infinite decimal expansion of Шаблон:Mvar.

Conversely, for any integer Шаблон:Math and any sequence of digits<math display="inline">\;(d_n)_{n=1}^{\infty}</math> the (infinite) expression Шаблон:Math is an infinite decimal expansion of a real number Шаблон:Mvar. This expansion is unique if neither all Шаблон:Math are equal to 9 nor all Шаблон:Math are equal to 0 for Шаблон:Mvar large enough (for all Шаблон:Mvar greater than some natural number Шаблон:Mvar).

If all Шаблон:Math for Шаблон:Math equal to 9 and Шаблон:Math, the limit of the sequence<math display="inline">\;([x]_n)_{n=1}^{\infty}</math> is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: Шаблон:Math, by Шаблон:Math, and replacing all subsequent 9s by 0s (see 0.999...).

Any such decimal fraction, i.e.: Шаблон:Math for Шаблон:Math, may be converted to its equivalent infinite decimal expansion by replacing Шаблон:Math by Шаблон:Math and replacing all subsequent 0s by 9s (see 0.999...).

In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of Шаблон:Math, and the other containing only 9s after some place, which is obtained by defining Шаблон:Math as the greatest number that is less than Шаблон:Mvar, having exactly Шаблон:Mvar digits after the decimal mark.

Rational numbers

Long division allows computing the infinite decimal expansion of a rational number. If the rational number is a decimal fraction, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a repeating decimal. For example,

Шаблон:Sfrac = 0.Шаблон:Thin space012345679Шаблон:Thin space012... (with the group 012345679 indefinitely repeating).

The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational.

For example, if x is	Шаблон:Figure space0.4156156156...
then 10,000x is	Шаблон:Figure space4156.156156156...
and 10x is	Шаблон:Figure space4.156156156...
so 10,000x − 10x, i.e. 9,990x, is	Шаблон:Figure space4152.000000000...
and x is	Шаблон:Figure spaceШаблон:Sfrac

or, dividing both numerator and denominator by 6, Шаблон:Sfrac.

Decimal computation

Файл:Decimal multiplication table.JPG

Diagram of the world's earliest known multiplica­tion table (Шаблон:Circa) from the Warring States period

Most modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such as the ENIAC or the IBM 650, used decimal 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 or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.)

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 some variant of binary-coded decimal, especially in database implementations, but there are other decimal representations in use (including decimal floating point such as in newer revisions of the IEEE 754 Standard for Floating-Point Arithmetic).

Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This is not possible in binary, because the negative powers of <math>10</math> have no finite binary fractional representation; and is generally impossible for multiplication (or division). See Arbitrary-precision arithmetic for exact calculations.

History

Файл:Qinghuajian, Suan Biao.jpg

The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BCE, during the Warring States period in China.

Many ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers. Standardized weights used in the Indus Valley civilization (Шаблон:Circa) were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler – the Mohenjo-daro ruler – was divided into ten equal parts. Egyptian hieroglyphs, in evidence since around 3000 BCE, used a purely decimal system, as did the Linear A script (Шаблон:Circa) of the Minoans and the Linear B script (c. 1400–1200 BCE) of the Mycenaeans. The number system of classical Greece also used powers of ten, including an intermediate base of 5, as did Roman numerals. Notably, the polymath Archimedes (c. 287–212 BCE) invented a decimal positional system in his Sand Reckoner which was based on 10⁸. Hittite hieroglyphs (since 15th century BCE) were also strictly decimal.

Some non-mathematical ancient texts such as the Vedas, dating back to 1700–900 BCE make use of decimals and mathematical decimal fractions.

The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1000, 2000, 3000, 4000, to 10,000. The world's earliest positional decimal system was the Chinese rod calculus.

Файл:Chounumerals.svg

The world's earliest positional decimal system
Upper row vertical form
Lower row horizontal form

History of decimal fractions

Файл:Rod fraction.jpg

counting rod decimal fraction 1/7

Starting from the 2nd century BC, some Chinese units for length were based on divisions into ten; by the 3rd century AD, these metrological units were used to express decimal fractions of lengths, non-positionally. Calculations with decimal fractions of lengths were performed using positional counting rods, as described in the 3rd–5th century Sunzi Suanjing. The 5th century mathematician Zu Chongzhi calculated a 7-digit [[approximations of π|approximation of Шаблон:Mvar]]. Qin Jiushao's book Mathematical Treatise in Nine Sections (1247) explicitly writes a decimal fraction representing a number rather than a measurement, using counting rods. The number 0.96644 is denoted

Шаблон:Lang

Файл:Counting rod 0.png Файл:Counting rod h9 num.png Файл:Counting rod v6.png Файл:Counting rod h6.png Файл:Counting rod v4.png Файл:Counting rod h4.png.

Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East.

Al Khwarizmi introduced fractions to Islamic countries in the early 9th century, written with a numerator above and denominator below, without a horizontal bar. This form of fraction remained in use for centuries.

Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century. The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350 but did not develop any notation to represent them. The Persian mathematician Jamshīd al-Kāshī used, and claimed to have discovered, decimal fractions in the 15th century.

Файл:Stevin-decimal notation.svg

A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. Stevin's influential booklet De Thiende ("the art of tenths") was first published in Dutch in 1585 and translated into French as La Disme.

John Napier introduced using the period (.) to separate the integer part of a decimal number from the fractional part in his book on constructing tables of logarithms, published posthumously in 1620.Шаблон:Rp

Natural languages

A method of expressing every possible natural number using a set of ten symbols emerged in India. Several Indian languages show a straightforward decimal system. Many Indo-Aryan and Dravidian languages have numbers between 10 and 20 expressed in a regular pattern of addition to 10.

The Hungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty").

A straightforward decimal rank system with a word for each order (10 Шаблон:Lang, 100 Шаблон:Lang, 1000 Шаблон:Lang, 10,000 Шаблон:Lang), and in which 11 is expressed as ten-one and 23 as two-ten-three, and 89,345 is expressed as 8 (ten thousands) Шаблон:Lang 9 (thousand) Шаблон:Lang 3 (hundred) Шаблон:Lang 4 (tens) Шаблон:Lang 5 is found in Chinese, 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 the numbers between 10 and 20, and decades. For example, in English 11 is "eleven" not "ten-one" or "one-teen".

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 the English names of numerals may hinder children's counting ability.

Other bases

Шаблон:Main Шаблон:Fundamental info units Some cultures do, or did, use other bases of numbers.

• Pre-Columbian Mesoamerican cultures such as the Maya used a base-20 system (perhaps based on using all twenty fingers and toes).
• The Yuki language in California and the Pamean languages in Mexico have octal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.
• The existence of a non-decimal base in the earliest traces of the Germanic languages is attested by the presence of words and glosses meaning that the count is in decimal (cognates to "ten-count" or "tenty-wise"); such would be expected if normal counting is not decimal, and unusual if it were. Where this counting system is known, it is based on the "long hundred" = 120, and a "long thousand" of 1200. The descriptions like "long" only appear after the "small hundred" of 100 appeared with the Christians. Gordon's Introduction to Old Norse Шаблон:Webarchive p. 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240. GoodareШаблон:Dead link details the use of the long hundred in Scotland in the Middle Ages, giving examples such as calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.
• Many or all of the Chumashan languages originally used a base-4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.
• Many languages use quinary (base-5) number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot and Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
• Some Nigerians use duodecimal systems. So did some small communities in India and Nepal, as indicated by their languages.
• The Huli language of Papua New Guinea is reported to have base-15 numbers. Ngui means 15, ngui ki means 15 × 2 = 30, and ngui ngui means 15 × 15 = 225.
• Umbu-Ungu, also known as Kakoli, is reported to have base-24 numbers. Tokapu means 24, tokapu talu means 24 × 2 = 48, and tokapu tokapu means 24 × 24 = 576.
• Ngiti is reported to have a base-32 number system with base-4 cycles.
• The Ndom language of Papua New Guinea is reported to have base-6 numerals. Mer means 6, mer an thef means 6 × 2 = 12, nif means 36, and nif thef means 36×2 = 72.