Decimal place

The decimal places are the places behind the (to the right of) comma of a decimal number or, more generally, a non-whole number , which is represented as a decimal point using a place value system . In the first case, one also speaks of decimal places or decimals . Together they form the fractional part and are generally something different from the significant digits .

Examples

• For the number 223.5678, the decimal places are the four digits 5, 6, 7 and 8.
• The fraction has an infinite number of decimal places because its decimal representation never breaks off. Like all fractions of integers, it represents a periodic decimal number . ${\displaystyle {\tfrac {4}{7}}=0{,}{\overline {571428}}}$
• Odd powers of the golden section Φ²ⁿ⁺¹ and their reciprocal values each have the same decimal places , z. B. Φ³ = 4.2360679775…, 1 / Φ³ = 0.2360679775…

Fractional part

The fractional part (of English fractional part ) can be with the functions and determine ( round-off and Aufrundungsfunktionen ). ${\displaystyle \operatorname {frac} (x)}$ ${\displaystyle \lfloor x\rfloor }$ ${\displaystyle \lceil x\rceil }$

${\displaystyle \operatorname {frac} (x):={\begin{cases}x-\lfloor x\rfloor &x\geq 0\\x-\lceil x\rceil &x<0\end{cases}}}$

The notation is also used for this , but is usually avoided because it can be confused with the set from x. ${\displaystyle \{x\}}$

Examples:

• ${\displaystyle \operatorname {frac} (223{,}5678)=0{,}5678}$
• ${\displaystyle \operatorname {frac} (-223{,}5678)=-0{,}5678}$
• ${\displaystyle \operatorname {frac} (1{,}{\overline {571428}})=0{,}{\overline {571428}}}$

The equally common definition without distinction between cases

${\displaystyle \operatorname {frac} (x):=x-\lfloor x\rfloor }$

is not applicable to negative values. A fractional part determined in this way is then wrong, for example:

${\displaystyle \operatorname {frac} (-223{,}5678)=0{,}4322\neq -0{,}5678}$

Pronunciation and notation

According to the SI definition , numbers are summarized in groups of three and shown visually separated in the text by small spaces . However, this is not yet consistently correctly visualized by all display programs and is currently still a relatively complex form of representation, which is why the visual grouping is often dispensed with. Grouping points are no longer allowed.

The pronunciation of decimal places is the same as before the decimal separator , either by stringing the digits together, e.g. B. 123 000 , 123 000 “one-two-three-zero-zero-zero point one-two-three-zero-zero-zero”, or using the decimal fraction , usually in powers of three according to technical notation , but without naming the Separator: "One hundred and twenty-three thousand, one hundred and twenty-three thousandths, zero millionths" (the commas in the text are pure pauses and are not pronounced). The mixed form of grouping just before the comma is also common, but inconsistent and misleading.

To preserve the significant digits in the verbal transmission, it is also possible to split the decimal division into two: "One hundred and twenty-three thousand and one hundred and twenty-three thousand millionths".

Individual evidence

1. ^ JN Bronstein , KA Semendjajew : Taschenbuch der Mathematik . Ed .: Günter Grosche, Viktor Ziegler. 20th edition. Verlag Harri Deutsch, Thun / Frankfurt / Main 1981, ISBN 3-87144-492-8 , section 2.1.1., P. 150 (24th edition 1989: p. 98).
2. ↑ decimals . In: Meyer's large pocket dictionary in 24 volumes . tape 13 : Lat - Mand . Mannheim / Vienna / Zurich 1983, ISBN 3-411-02100-4 , pp. 218 (1992: vol. 5, p. 202).
3. ↑ Eric W. Weisstein : Fractional Part . In: MathWorld (English).