Decimal

A decimal system (from medieval Latin decimalis to Latin decem ' ten' ), also known as the decimal system or decadic system is a number system that the number used as the base 10.

In general, it is understood specifically with the decimal place value system, which developed in the Indian number font, was passed through Arab mediation to European countries and is now established worldwide as an international standard.

As Dezimalsysteme but is also referred to number systems on the base 10 without place value system, as they are based partly in conjunction with the five-component, vigesimalen or otherwise based number systems, the number of words of many natural languages ​​and the number older writings.

The emergence of Dezimalsystemen well as those of Quinärsystemen is anthropologically associated with the number of fingers as counting and computing assistance, in conjunction, a statement which is also supported by the fact that in some languages ​​than numerals for 5 and 10 expressions meaning " hand " (5) or " two hands " (10 ) are needed. [s 1]

With decimal place value system

Digits

In the decimal system using the ten digits

Referred to as decimal digits.

These digits are written differently in different parts of the world. See the article Arabic and Indian numerals.

Indian numeral characters are still used in various Indian scripts ( Devanagari, Bengali Font, Tamil font, etc.). They differ greatly from each other.

Definition

A decimal number is in German speaking countries mostly in the form

Written; exist alongside depending on the purpose and place additional spellings. In this case, each zi is one of the above paragraphs. The index i describes the significance of each digit, the value of a digit is the power of ten 10i. The digits are consecutively written without delimiters, with the most significant digit are with the number zm far left and the lower-order places with the number zm- 1 to z0 in descending order to the right. For the presentation of rational numbers with non- periodic development follow, according to a separating comma, the number z -1 to z -n. In the English -speaking world, a point is used instead of the comma usually.

Digits before the comma with a power of ten with a positive exponent multiplied (except for the first digit to the left of the decimal point belongs to the exponent zero), the digits after the decimal point, however, with a power of ten with a negative exponent. The value Z of the decimal so obtained by summation of these numbers, which are pre- multiplied by their importance 10i; addition (at least for negative numbers ) preceded the sign:

This representation is also called decimal - development.

Example

Solving the potencies in the formula is clear:

Decimal expansion (periodic decimals to fractions reshape )

Using the decimal expansion can be assigned to each of the real number, a sequence of digits. Every finite part of this sequence defines a decimal fraction, which is an approximation of the real number. The real number is obtained even if one goes by the finite sums of the parts to the infinite number of all digits. This diagram is an example of a series expansion.

Formally, therefore the value of the series is denoted by.

It is said that the decimal expansion terminates if the number sequence from a point n only consists of zeros, which represented real number that is itself already is a decimal fraction. In particular, for all irrational numbers breaks the digit sequence not starting; there exists an infinite decimal expansion.

For reshaping periodic Dezimalbruchentwicklungen (see below ), use the relations:

These identities follow from the calculation rules for geometric series, which is valid for. The first example is chosen and begins the summation until the first follower.

Examples:

The period is chosen in each case in the counter. In the denominator are so many nines, as the period has locations. If necessary, the resulting fraction should be reduced.

Somewhat more complicated is the invoice, if the period does not immediately follow the comma:

Examples:

Ambiguity of the representation

A special feature in the decimal expansion is that many rational numbers have two different Dezimalbruchentwicklungen. As described above, can be transformed and of the message

Access, see the article 0.999 ...

The representation either just as a finite decimal fraction with period 0, or infinite with period 9 To clear: From this identity one can further conclude that many rational numbers (namely all finite decimal expansion with the exception of 0) can be represented in two different ways to make, you can see the period 9 (or more rarely the period 0 ), however, prohibit simple and limited to finite decimal fractions.

Formula

For periodic decimal fractions with a zero before the decimal point, the following formula can be set up:

The number p is the number x before the start of the period ( as an integer ), m is the number of digits before the start of the period, y is the sequence of digits of the period ( as an integer ) and n is the length of the period.

The application of this formula is to be demonstrated on the basis of the last example:

Period

In mathematics is called a period of a decimal fraction digit or sequence of digits that repeats itself over and over again after the decimal point. All rational numbers, and only these, have a periodic decimal expansion.

Examples:

Also finite decimal fractions among the periodic decimal fractions; after insertion of infinitely many zeros, for example 0.12 = 0.12000 ...

True periods (ie no finite decimal fractions ) occur in the decimal system on iff can be generated exclusively through the prime factors 2 and 5, the denominator of the underlying fracture. 2 and 5 are the prime factors of the number of 10, based on the decimal system. The denominator is a prime number (other than 2 and 5), the period has a maximum length that is lower by one than the value of the denominator ( in the examples shown in bold).

The exact length of the period from (if the prime is neither 2 nor 5) corresponds to the smallest natural number at which occurs in the prime factorization of.

Example of period length 6: ( 106-1 ) = 999.999:

999,999 = 3 · 3 · 3 · 7 · 11 · 13 · 37, 1/ 7 = 0.142857142857 ... or 1/ 13 = 0.076923076923 ...

Both 1/7 and 1/13 have a period length of 6, 7 and 13 because the first time in its prime factorization of Rn = 106 - 1 appear. However, 1/37 has a period length of only 3, because already ( 103-1 ) = 999 = 3 · 3 · 3 · 37

If the denominator is not a prime number, then the period length is determined according as the number at which the denominator is the first time a divisor of; the prime factors 2 and 5 of the denominator are not considered.

Examples: 1/185 = 1 / (5.37) have the same period length, such as 1/37, namely, 3

1/143 = 1 / (11.13) has the period length of 6 because 999,999 = 3 · 3 · 3 · 7 · 37 · 143 (see above)

1/260 = 1 / (2.2.5.13) have the same period length, such as 1/13, that is 6

To determine the period length efficient the determination of the prime factorizations of the rapidly growing number sequence 9, 99, 999, 9999, etc. can be avoided by the equivalent relationship will be used, so repeatedly multiplying ( starting from 1) to 10 modulo the given denominator to this back 1 results. For example, for:

So has 1/13 the period length of 6

Notation

For periodic Dezimalbruchentwicklungen a notation is common in the periodically repeating part of the decimal point is marked by an overline. Examples are

• ,
• .

Due to technical limitations and other conventions exist. So the crossbeam can precede, elected a typographic highlighting (bold, italic, underline) of the periodic part or all of these are enclosed in parentheses:

Non- periodic sequence Nachkommaziffern

As explained in the article place value system, have irrational numbers (also) in the decimal an infinite, non-periodic Nachkommaziffern sequence. Irrational numbers can therefore not be represented by a finite and not by a periodic sequence of digits. Well, you can with finite (or periodic ) decimal fractions approximate arbitrary, but such a finite representation is never exact. So it is only by using additional symbols possible to specify irrational numbers by finite representations.

Examples of such symbols are root sign, as for, letters like π or e, as well as mathematical expressions as infinite series or limits.

Conversion to other value systems

Methods for converting to and from the decimal system are described in the references to different value systems and Number System Transform and place value system.

History

Decimal number systems without place value system and without representation of zero were in ancient times including the number of writings of the Egyptians, Greeks and Romans basis. These were to additive number fonts, numbers, though written with those in computing as a memory aid, but arithmetic operations essentially could not in writing be made: these were rather with mental arithmetic or with other tools such as the computing stones (Greek psephoi, Latin calculi, in the late Middle ages Rechenpfennige or French jetons called ) on the abacus and may be liable to the finger numbers.

The spread in Roman and medieval times, in slightly different form in the Arab world used finger numbers was a decimal system for the representation of numbers between 1 and 9999 based with no sign for zero and with a positional system of its own kind These were precisely defined finger positions on the left hand with a small, ring and middle fingers the one 1-9 and with the forefinger and thumb the ten 10 to 90 shown, while on the right hand the hundred with your thumb and forefinger a mirror image of the tens and thousands with the three other fingers were shown a mirror image of the units have been [m 1] [ 2 s ] [u 1] This finger numbers are used not only for counting and memorizing numbers, but also for computation. ; However, the contemporary written sources are limited to the description of the finger positions and give no precise information about the resulting feasible computational operations.

On the computational boards of the Greco-Roman antiquity and the Christian Middle Ages stood in contrast to the representation of integers a full decimal place value system, by reducing the number of its units, tens, hundreds, etc. was represented by counting stones in corresponding vertical decimal columns for a given number. On the ancient abacus this was done by dropping or pushing a corresponding number of calculi in the respective decimal column, with an additional five bundling was practiced by five units were represented by a single calculus in a side or top special area of ​​the decimal column. [M 2] [ u 2 ] [p 1] the monastery Abacus the early Middle Ages, which is now mostly associated with the name of Gerbert and from the 10th to the 12th century was in use, instead, the number of units in each decimal column was only by a single stone shown, which was quantified with a number from 1 to 9, [m 3 ] [u 3] [s 3] during the later middle Ages and the early modern period came back again for use unquantified counting stones and columns and now horizontal solid lines for either decimal arithmetic with integers to the base number 10 ( with five bundling), [s 4] [ m 4 ] [p 2] or for the financial Computing at the from the Carolingian coinage (1 libra = 20 solidi = 240 denarii ) not inherited monetary base units imal lined up. [s 5 ] [m 5] [ p 3 ] [u 4] on the ancient and the medieval versions of this tool was the representation of the value zero in each case by leaving the relevant decimal column or line, even at the monastery Abacus that said, although a calculation stone with a derived from the Arabic digit ( cifra ) for zero available, but was used for other purposes at the abazistischen arithmetic operations. Using the ancient and medieval Abacuses could be considerably simplified by addition and subtraction, while they were not very suitable for multiplication and division or relatively complicated operations required, which have been described particularly for the Abacus monastery in medieval treatises and were notorious in their difficulty.

A number font with full priority system in which the position of the numeral determines its value, first developed by the Babylonians on the base 60 and supplemented it probably already before the 4th century BC also to its own sign of zero. [U 5] a number font with place value system on the base 10, but still no sign of the zero originated in China probably already some centuries before the Christian era ( in detail attested since the 2nd century BC), probably using computational chopsticks on a chessboard-like banded Chinese variant of the abacus, and was supplemented only under Indian influence since the 8th century also a sign of zero. [u 6]

In India, even the beginnings of the positional decimal system with signs for the zero are not known with certainty. The older Brahmi number font, which was in use from the 3rd to the 8th century, used a decimal system with approaches to positional sensitive, but still no sign of zero. [U 7]. The oldest Indian form of today's Indo- Arabic numerals, with to be derived from the Brahmi character for number 1 to 9 and a point or small circle for zero, by datable epigraphic testimony first outside India since the 7th century in Southeast Asia as Indian export and prove itself in India since the 9th century [u 8]; However, it is believed that the use of this numeric system in India began in the 5th century. [u 9] The same positional decimal with sign for zero was also the Indian in roughly contemporary scholarly numeral system astronomers based in the periphrastic expressions such as " the beginning " (1 ) " eyes " ( 2) ," the three time stages "(3) for the numbers 1 to 9 and " sky "," emptiness "," point " or other words for zero according to their decimal value as a linguistic description multi-digit numbers were ranked. [u 10] As early example of such a positional setting of in this case largely non-metaphorical linguistic terms is already the number 458 written in Prakrit Lokavibhaga [u 11], which is only obtained in a later Sanskrit translation. Fully trained to find the circumscribing numeral system then in Bhaskara I ( 7th century).

Of the Arabs and Arabized peoples of them was initially the decimal additive system of alphabetic Greek number font, initially mediated by Hebrew and Syrian model, adopted and transmitted to the 28 letters of the Arabic alphabet [u 12 ] for the case of numbers. However, the Indian numerals and based thereon computational methods have been known first in the Arab Middle East and in the course of the 9th century then also in North Africa and Al -Andalus at least since the 8th century. The earliest mention is found in the 7th century by the Syrian bishop Severus Sebokht which explicitly praises the Indian system. An important role in the spread in the Arab world and the West played Muhammad ibn Musa al - Khwarizmi, who used the new digits not only in his mathematical works, but 825 also preserved only in Latin transmission introduction Kitaab al - Djam ʿ wa - l - tafriq bi- Hisab al - Hind ( "On the calculation with Hindu numerals " ) wrote with a quality suitable for the beginner description of digital systems and the acts building upon written basic arithmetic.

After the 10th/11th. Century already western Arabic in the Latin West or derived digits ( apices called ) had appeared on the computing stones of the monastery Abacus, but have not even used in addition as a number font or even for written arithmetic, and had got together with the monastery Abacus into oblivion, it was the introduction Al- Chwarizmis that ATTACHING in Latin edits and it since the 12th century vernacular treatises the Indian numerals Convert to break through and by its opening words " Dixit Algorismi " also meant that " Algorismus ", the Latin rendition of his name, is widely regarded as Name this new arithmetic established. [u 13] Especially in Italy, where Leonardo da Pisa made ​​it in his Liber abbaci also from our own, acquired in North Africa knowledge known, could the Indian numerals arithmetic since the 13th century the Abacus ( with an unknown amount of computing stones) almost completely displace finance and commercial area and even its name ( abbaco ) take while though was in other countries, the subject of scientific and commercial classes until the Early Modern Period but on the abacus had an overpowering competitors in line arithmetic. Also called simple number font for the practical purposes of writing down numbers and Nummerierens for which the value system does not need the Indo- Arabic numerals were gradually prevail against the Roman numbers only since the early modern period.