Decimal system
That Decimal system or Zehnersystem (lat. decimus = tenth) is in Notational system to the representation of Numbers. It uses the basic figure (or basis) 10. The decimal system is today with that Binary system most spread the world-wide Number system, and comes originally out India.
The Persian mathematician Muhammad ibn Musa aluminium-Chwarizmi used it in 8. Century in its arithmetic book, by it in 10. Century to Europe arrived. With its penetration in 12. Century also the term came Arab numbers up. In Arab countries they become until today Indian numbers called.
Without zero however already with the decimal number idea (thus decimals, Hundred, Tausender etc.. one already counted in Old Egypt (see Hieroglyphics) and later with that Romans (see Roman numbers). Those Chinese numbers a mixing system from the numbers is one to nine (a zero were added later) and indications of the decimal steps.
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Definition and representation
Numbers
In the decimal system one uses 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.
The old Indian numbers also today still become in that Devan?gar?Writing uses.
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| ? | ? | ? | ? | ? | ? | ? | ? | ? | ? |
Definition
A decimal number becomes by the numbers zi represented. The numbers are written one behind the other without separators, their value corresponds power of ten to the fitting the place.
It becomes thus the most significant place with the value zm completely left and the niederwertigeren places with the values zm-1 to z0 in descending order right noted of it. To the representation of rational or real numbers follow then after a separating comma the places z -1 to z n, those the broken portion of the number represent. Numbers before the comma become also positive Exponent, after the comma with negative Exponent multiplied. In the English linguistic area instead of the comma one point is usually used.
- <math>z_m z_{m-1} \ldots z_0 \operatorname{,}z_{-1} z_{-2} \ldots z_{n}
\qquad \left(m,n \in \mathbb{N} \quad z_i \in \{0, \ldots, 9 \}\right) </math>
The value Z the decimal number arises as a result of addition and/or. Subtraction of these numbers, which before in each case with their value 10i are multiplied:
- <math>Z = \pm \sum_{i= n}^m z_i \cdot 10^i</math>.
One calls this representation also Decimal fraction development.
Example
- 723,48 = 7·102 + 2·101 + 3·100 + 4·10-1 + 8·10-2
Decimal fraction development
With the help of that Decimal fraction development one can do everyone real number by a sum of breaks as Powers of ten represent. This representation is a simple Series expansion. In particular with all irrational This row does not break numbers off; then a decimal fraction infinite decimal is present.
To the shaping periodically 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 becomes in each case into that Counter taken over. In Denominator stands as many Neunen, as the period has places. If necessary the developed break should be still shortened.
The calculation is somewhat more complicated, if the period on the comma does not follow directly:
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 the following formula can be set up:
- <math>p = \frac{x \cdot \left(10^n-1 \right) + y}{10^m \cdot \left(10^n - 1 \right)}</math>
Are p 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 of the period (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>
Period
In that Mathematics calls one Period one Decimal fraction one Number or number sequence, after the comma repeats itself again and again. Everything rational numbers have one periodic Decimal fraction development.
Examples:
Sofortperiodi: 1/3 = 0,33333... 1/7 = 0,142857142857... 1/9 = 0,11111... Nichtsofortperiodi: 2/55 = 0,036363636... 1/30 = 0,03333... 1/6 = 0,16666...
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 then, if itself that Denominator the which is the basis break not exclusive by those Prime factors 2 and 5 to produce leaves -- 2 and 5 is the prime factors of the number of 10, that Basis the decimal system. The denominator is a prime number (except 2 and 5), so the period has at the most one length, around one is lower than the value of the denominator (represented in the examples fat).
Periodic decimal fractions as limit values
That Limes- or limit value term that 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> is the limes of the following (infinite) geometrical row
- <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 with period line
For periodic decimal fraction developments a way of writing is usual, with the periodically repeating part of the right-of-comma positions by one re-painted over is marked. Examples are
- <math>1/6 = 0.1 \bar{6}</math>,
- <math>1/7 = 0, \overline{142857}</math>.
Not periodic Nachkommaziffern consequence
As in the article Notational system described, possess surds (also) in the decimal system an infinite nonperiodic Nachkommaziffern consequence. Surds cannot be represented thus by a finite number sequence. One can approximate with finite (or periodic) decimal fractions at will, 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, as for?2, Letter how ? or e, as well as mathematical expressions how infinite rows or Limit values
However is also like that not everyone real number representably, because it over countable many real numbers, but only countable many finite representations with a finite character set gives.
Special characteristic of the decimal fraction development
A special characteristic during the decimal fraction development is, that one rational number two different decimal fraction developments to possess knows. As described above, one knows the value of 0,999999... compute to 9/9. Thus one receives the statement surprising first
- <math>0{,}99999 \ldots = 0{,}\overline{9} = 1</math>.
This fact is descriptive with difficulty understandable, mathematically however correctly. It hangs closely with the definition of a decimal fraction as Limes (limit value) of one infinite row together. One can say, that the number with each further 9 moves close more near at 1. Since there is however infinitely many Neunen, come the partial sums arbitrarily close at 1 near; thus the limit value is 1!
Likewise 0.7999999 becomes... to 0.8 etc..
Conversion into other notational systems
Methods for the conversion from and into the decimal system become in the articles other notational systems and under Number basis change and Notational system described.
See also
- Number system, Roman numbers
- Binary system, Octal system, Duodezimalsystem, Hexadecimal number system, Vigesimalsystem, Sexagesimalsystem, Way of writing of numbers, Basis change
- Decimal currency, Metric system
- Floating-point number (floating POINT NUMBER)
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