Information AboutBase 10 |
| CATEGORIES ABOUT DECIMAL | |
| elementary arithmetic | |
| fractions | |
| positional numeral systems | |
| 10 | |
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DECIMAL NOTATION Decimal notation is the writing of Number s in the base-ten Numeral System , which uses various symbols (called Digits ) for ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent numbers. These digits are often used with a Decimal Separator which indicates the start of a fractional part, and with one of the sign symbols + (plus) or − (minus) to indicate sign. The Decimal System is a Positional Numeral System ; it has positions for units, tens, hundreds, ''etc.'' The position of each digit conveys the multiplier (a power of ten) to be used with that digit—each position has a value ten times that of the position to its right. Ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). In many languages the word Digit or its translation is also the anatomical term referring to fingers and toes. In English, decimal (decimus < Lat. ) means ''tenth'', decimate means ''reduce by a tenth'', and denary (denarius < Lat.) means ''the Unit of ten''. The symbols for the digits in common use around the Globe today are called Arabic Numerals by Europeans and Indian Numerals by Arabs, the two groups' terms both referring to the culture from which they learned the system. However, the symbols used in different areas are not identical; for instance, Western Arabic numerals (from which the European numerals are derived) differ from the forms used by other Arab cultures. Alternative notations Some cultures do, or used to, use other numeral systems, including the Tzotzil , who use a Vigesimal system (using all twenty fingers and Toe s), some Nigeria ns who use several Duodecimal systems, the Babylonia ns, who used Sexagesimal , and the Yuki , who reportedly used Octal . Computer hardware and software systems commonly use a Binary Repesentation , 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 the equivalent decimal values for presentation to and manipulation by humans. 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 Binary-coded Decimal , but there are other decimal representations in use (see IEEE 754r ), especially in database implementations. Decimal arithmetic is used in computers so that decimal fractional results can be computed exactly, which is not possible using a binary fractional representation. This is often important for financial and other calculations {Link without Title} . Decimal fractions A decimal fraction is a Fraction where the Denominator is a Power of ten. Decimal fractions are commonly expressed without a denominator, the Decimal Separator being inserted into the numerator (with Leading Zero s added if needed), at the position from the right corresponding to the power of ten of the denominator. e.g., 8/10, 833/100, 83/1000, 8/10000 and 80/10000 are expressed as: 0.8, 8.33, 0.083, 0.0008 and 0.008. Numbers which can be expressed exactly in this way are called decimal numbers or '''regular numbers'''. The Integer and Fraction al parts of a decimal number are separated by a Decimal Separator . In English a dot (.) or period (.) is used as the separator. In most other languages a comma is used. It is usual for a decimal number which is less than one to have a leading zero. Trailing zeros after the decimal point are not necessary, although in science, engineering and ''). Grouping of digits Numbers with many digits before and/or after the decimal point may be divided into groups of three, starting from the decimal separator in both directions. This is done with a comma or a space. The latter is recommended in the SI/ISO 31-0 . A point may be used in notations with a decimal comma. For details, see Decimal Separator . Notations like "12,345", "12.345", "12,345.678", and "12.345,678" are ambiguous if the notational system is not known. Other rational numbers Any Rational Number which cannot be expressed as a decimal fraction has a unique infinite decimal expansion ending with Recurring Decimal s. Ten is the product of the first and third Prime Number s, is one greater than the square of the second prime number, and is one less than the fifth prime number. This leads to plenty of simple decimal fractions: :1/2 = 0.5 :1/3 = 0.333333… (with 3 recurring) :1/4 = 0.25 :1/5 = 0.2 :1/6 = 0.166666… (with 6 recurring) :1/8 = 0.125 :1/9 = 0.111111… (with 1 recurring) :1/10 = 0.1 :1/11 = 0.090909… (with 09 recurring) :1/12 = 0.083333… (with 3 recurring) :1/81 = 0.012345679012… (with 012345679 recurring) Other prime factors in the denominator will give longer recurring Sequence s, see for instance 7 , 13 . That a rational must produce a Finite Set or recurring decimal expansion can be seen to be a consequence of the Long Division Algorithm , in that there are only (q-1) possible nonzero Remainder s on division by q, so that the recurring pattern will have a period less than q-1. For instance to find 3/7 by long division: .4 2 8 5 7 1 4 ... 7 ) 3.0 0 0 0 0 0 0 0 2 8 30/7 = 4 r 2 2 0 1 4 20/7 = 2 r 6 6 0 5 6 60/7 = 8 r 4 4 0 3 5 40/7 = 5 r 5 5 0 4 9 50/7 = 7 r 1 1 0 7 10/7 = 1 r 3 3 0 2 8 30/7 = 4 r 2 (again) 2 0 etc The converse to this observation is that every Recurring Decimal represents a rational number ''p''/''q''. This is a consequence of the fact the recurring part of a decimal representation is, in fact, an infinite Geometric Series which will sum to a rational number. For instance, : Real numbers Every Real Number has a (possibly infinite) decimal representation, i.e., it can be written as : where
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