Decimal system
the decimal system or Zehnersystem (lat. decimus = tenth) is a notational system for the representation of numbers. It uses the basic figure (or basis) 10. The decimal system is today the number system most spread world-wide, and originally originates from India.
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definition and representation
to numbers
in the decimal system uses one the 10 numbers
- 0 (zero), 1 (unity), 2 (two), 3 (three), 4 (four), 5 (five), 6 (six), 7 (filters), 8 (eight), 9 (nine).
These numbers are differently written however in different parts of the world. See in addition the articles Arab numbers and Indian numbers.
Indian number characters also today still become inthe different Indian writings (Devanagari, Bengali writing, Tamili writing etc.) uses. They differ strongly from each other.
definition
a decimal number becomes in the form
- < math> \ pm z_m z_ {m-1} \ ldots z_0 \ operator name {,} z_ {- 1} z_ {- 2} \ ldots z_ {- n}
\ qquad \ left (m, n \ in \ mathbb {N} \ quadz_i \ in \ {0, \ ldots, 9 \} \ right) </math>
noted. Each z i_{ is } one of the numbers specified above. The index i describes the value of the respective number, the priority of a number is the power of ten 10^{ i}. The numbers are written one behind the other without separators, howthe most significant place with the number z_{ m} completely left and the niederwertigeren places with the numbers z_{ m-1} to z_{ 0} in descending sequence right of it stand. For the representation of rational with nonperiodic development follow then, after a separatingComma, the numbers z_{ -1} to z_{ - n}.
Numbers before the comma are multiplied by the basis 10 and a positive exponent and after the comma by 10 and a negative exponent. In the English linguistic area becomes instead of the commausually one point uses. That is, the value Z of the decimal number arises as a result of addition and/or. Subtraction of these numbers, which by their value 10 i are multiplied^{ before} in each case:
- <math> Z = \ pm \ sum_ {i=-n} ^m z_i \ cdot 10^i< /math>.
One calls this representationalso decimal fraction development.
Example
- 723.48 = 7·10^{ 2} + 2·10^{ 1} + 3·10^{ 0} + 4·10^{ -1} + 8·10^{ -2}
decimal fraction development
with the help of the decimal fraction development one can assign a consequence of numbers to each real number. Each finitePart this consequence defines a decimal fraction, which is an approximation of the real number. One receives the real number, if one turns into from the finite sums of the parts to the infinite row over all numbers. This representation is an example of one Series expansion. One says that the decimal fraction development breaks off, if the number sequence consists starting from a place only of zeros, which are already represented real number thus a decimal fraction. In particular with all surds the number sequence does not break off; ita decimal fraction infinite decimal is present.
For the shaping of periodic decimal fraction developments (see further below) one uses the relations:
- <math> 0 {,} \ overline {1} = \ frac {1} {9}; \ quad
0 {,} \ overline {01} = \ frac {1} {99}; \ quad 0 {,} \ overline {001} = \ frac {1} {999}; \ quad \ ldots< /math>.
Examples:
- <math> 0 {,} 55555 \ ldots = 0 {,} \ overline {5} = \ frac {5} {9}< /math>
- <math> 0 {,} 33333 \ ldots = 0 {,} \ overline {3} = \ frac {3} {9}= \ frac {1} {3}< /math>
- <math> 0 {,} 424242 \ ldots = 0 {,} \ overline {42} = \ frac {42} {99} = \ frac {14} {33}< /math>
- <math> 0 {,} 081081081 \ ldots = 0 {,} \ overline {081} = \ frac {81} {999} = \ frac {3} {37}< /math>
The period is transferred in each case to the counter. In the denominator as many Neunen stands, as the period has places. If necessary the developed break should stillare shortened.
The calculation is somewhat more complicated, if the period does not follow directly after the comma:
Examples:
- <math> 0 {,} 83333 \ ldots = 0 {,} 8 \ overline {3} = 8 {,} \ overline {3}: 10 = 8 \ frac {3} {9}: 10 = 8 \ frac {1} {3}: 10 = \ frac {25} {3}: 10 = \ frac {25} {30} = \ frac {5} {6}< /math>
- <math> 0 {,} 48363636 \ ldots = 0 {,} 48 \ overline {36} = 48 {,} \ overline {36}: 100 = 48 \ frac {36} {99}: 100 = 48 \ frac {4} {11}: 100 = \ frac {532} {11}: 100 = \ frac {532} {1100} = \ frac {133} {275}< /math>
formula
for infinite decimals with a zero before the comma leaves itself the following formulaset up:
- <math> p = \ frac {x \ cdot \ left (10^n-1 \ right) + y} {10^m \ cdot \ left (10^n - 1 \ right)}</math>
P are the periodic number, x the number before beginning of the period (as integer), m the number of numbers before beginning of the period, y the number sequence thatPeriod (as integer) and n the length of the period.
The application of this formula is to be demonstrated on the basis the last example:
- <math> p = 0 {,} 48363636 \ ldots = 0 {,} 48 \ overline {36}< /math>
- <math> x = 48; \ quad m = 2; \ quad y = 36; \ quad n = 2< /math>
- <math> p = \ frac {48 \ cdot \ left (10^2-1 \ right) + 36} {10^2 \ cdot \ left (10^2 - 1 \ right)} = \ frac {48 \ cdot 99 + 36} {100 \ cdot 99} = \ frac {4788} {9900} = \ frac {133} {275}< /math>
one
calls period in mathematics period of a decimal fraction a number or a number sequence,after the comma repeats itself again and again. All rational numbers, and only these, have a periodic decimal fraction development.
Examples:
- Sofortperiodi:
- 1/3 = 0, 3 3333…
- 1/7 = 0, 142857 142857…
- 1/9 = 0, 1 1111…
- Nichtsofortperiodi:
- 2/55 = 0,036 363636…
- 1/30 = 0.0 3 333…
- 1/6 = 0.1 6 666…
Also finite decimal fractions like 0.12 can be understood as periodic decimal fractions: 0,12 = 0,12000…
Periods arise in the decimal system exactly if itself the denominator are the basisBreak not exclusively by the prime factors 2 and 5 to produce leaves -- 2 and 5 is the prime factors of the number of 10, the basis of the decimal system. If the denominator is a prime number (except 2 and 5), then the period has at the most oneLength, those around one is lower than the value of the denominator (represented in the examples fat).
periodic decimal fractions as limit values
the Limes - or limit value term of the analysis permits an accurate definition of periodic decimal fractions. Thus applies for example:
- <math> 0 {,} \ overline {7} = 0 {,} 7 + 0 {,} 07 + 0 {,} 007 + \ ldots< /math>
<math> 0 {,} \ overline {7}< /math> the Limes of the following (infinite) geometrical row is
- <math> 0 {,} 7 + 0 {,} 7 \ cdot \ frac {1} {10} + 0 {,} 7 \ cdot \ left (\ frac {1} {10} \ right) ^2
+ 0 {,} 7 \ cdot \ left (\ frac {1} {10} \ right) ^3 + \ ldots< /math>
notation
forperiodic decimal fraction developments is usual a way of writing, with which the periodically repeating part of the right-of-comma positions re-painted over by one is marked. Examples are
- <math> 1/6 = 0 {,} 1 \ bar {6}< /math>,
- <math> 1/7 = 0 {,} \ overline {142857}< /math>.
Due to technical restrictions also different conventions exist. Thus can re-painted over placed in front,a typographic emphasis (fat, italically, underlined) of the periodic part to be selected or for this in parentheses set:
- 1/6 = 0,1¯6 = 0.1 6 = 0.1 6 = 0.1 6 = 0.1 (6)
- 1/7 = 0,1¯142857 = 0.1 142857 = 0.1 142857 = 0.1 142857 = 0.1 (142857)
, surds
(also) in the decimal system an infinite nonperiodic Nachkommaziffern consequence possess not periodic Nachkommaziffern consequence as described in the article notational system. Surds cannot be represented thus by a finite number sequence. One can itselfwith finite (or periodic) decimal fractions at will approximate, however a finite representation is never accurate. It is thus only possible for assistance of additional symbols to indicate surds by finite representations.
Examples of such symbols are radical signs, like for √2, letter as π or e, as well as mathematical expressions like infinite rows or limit values
however also so not each real number is representable, because there are over countable many real numbers, but only countable many finite representations with a finite character set.
Special characteristic of the decimal fraction development
a special characteristic during the decimal fraction development is that a rational number can possess two different decimal fraction developments. As described above, one knows the value of 0,999999… compute to 9/9. Thus one receives the statement surprising first
- < math> to 0 {,} 99999 \ ldots= 0 {,} \ overline {9} = 1< /math>.
This fact is descriptive with difficulty understandably, mathematically however correct. It is connected closely with the definition of a decimal fraction as Limes (limit value) of an infinite row. One can say that the number with each further 9 more near on1 moves close. Since there is however infinitely many Neunen, the partial sums arbitrarily close approach at 1; thus the limit value is 1!
Likewise 0,7999999 becomes… to 0.8 etc.
conversion into other notational systems
methods for the conversion of andinto the decimal system in the articles to other notational systems and under number basis changes and notational system are described.
history
without zero, but with the decimal number idea (thus decimal, hundred, thousands of etc.) one already already counted in the old person Egypt (see hieroglyphics) and later with the Romans (see Roman numbers). The Chinese numbers are a mixing system from the numbers one to nine (a zero were added later) and indication of the decimal steps.
The decimal system is Indian origin, which by inscriptionsand mentions is provable. The Indian have the number of zero over approx. 600 n.Chr. invented and in further consequence the decimal system develops. From the mathematician and astronomer Brahmagupta is from the year 628 the earliest well-known text over the fundamental calculation specificationswritten in this number system, as at that time in India usual in verse form, the Brahmasphutasiddhanta, on German Sindhind. Its contemporary Bhaskara I. uses 629 the first nine Brahmi numbers, from which our numbers descend, and a small circle forthe zero.
With the revision of the Sindhind by the Persian mathematician Muhammad ibn Musa aluminium-Chwarizmi in 8. Century and with its arithmetic book became the Indian number system in the Arab world admits. Its text book “Hisab aluminium-jabr” became only in 12. Century in latin translates and arrived in such a way to Europe, where it was continued to spread by the mathematician Leonardo Fibonacci. On journeys he had become acquainted with Arab mathematics, which he in its computing book “dear one abaci “(book of the abacus)mediated. With its penetration in 12. Century arose also the term Arab numbers . In Arab countries they are called to today Indian numbers.
In Europe counting on the ten numbers spread very slowly and sat down onlywith the invention of the printing through.
see also
- number system, Roman numbers
- binary system, octal system, Duodezimalsystem, hexadecimal number system, Vigesimalsystem, Sexagesimalsystem, way of writing of numbers, of number basis change
- decimal currency, metric system
- floating-point number (floating POINT NUMBER)
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