Decimal

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The decimal (base ten or occasionally denary) numeral system has ten as its base. It is the most widely used numeral system, probably because humans normally have a total of ten fingers and thumbs on their hands. Template:Numeral systems

Contents

Decimal notation

Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) to represent numbers. These digits are frequently used with a decimal point 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.

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.

However, some cultures do or used to use other numeral systems, including the Tzotzil, who use a vigesimal system (using all twenty fingers and toes), some Nigerians who use several duodecimal systems, the Babylonians, who used sexagesimal, and the Yuki, who reportedly used octal.

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.

Computers commonly use a different system, binary, 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 by the computer to the equivalent decimal values for presentation to humans.

Nevertheless, sometimes computers do use internal representations which are equivalent to decimal for doing arithmetic. Frequently this arithmetic is done on data in the form of binary-coded decimal, but there are other decimal representations in use (see IEEE 754r). Decimal arithmetic is used in computers so that fractional results can be computed exactly, which is not possible using a binary fractional representation. This is often important for financial and other calculations [1].

Fractional numbers

Decimal fractions

A decimal fraction is a vulgar fraction where the denominator is a power of ten.

Decimal fractions can be expressed without a denominator, the decimal point being inserted into the numerator (with leading zeros 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 thus: 0.8, 8.33, 0.083, 0.0008 and 0.008.

Numbers which can be expressed in this way are called decimal numbers or regular numbers.

The integer and fractional parts of a decimal number are separated by a decimal point. In this article, as in most of the English speaking world, a dot (.) is used. It is usual for a decimal number which is less than one to have a leading zero. Trailing zeroes after the decimal point are not necessary, although in science, engineering and statistics they can be retained to show a level of confidence in the accuracy of the number: Whereas 0.080 and 0.08 are mathematically the same number, in engineering 0.080 suggests an error of up to 1 part in a thousand, while 0.08 suggests an error of up to 1 in a hundred (see Significant figures).

Other rational numbers

Any rational number which cannot be expressed as a decimal fraction has a unique infinite decimal expansion ending with recurring decimals.

Ten is the product of the first and third prime numbers, 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 sequences, see for instance 7, 13.

That a rational must produce a finite 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 remainders 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,

0.0123123123\cdots = \frac{123}{10000} \sum_{k=0}^\infty 0.001^k = \frac{123}{10000}\ \frac{1}{1-0.001} = \frac{123}{9990} = \frac{41}{3330}

The real numbers

Every real number has a (possibly infinite) decimal representation, i.e. it can be written as

x = \mathop{\rm sign}(x) \sum_{i\in\mathbb Z} a_i\,10^i

where

  • sign() is the sign function,
  • ai ∈ { 0,1,…,9 } for all i ∈ Z, are its decimal digits, equal to zero for all i greater than some number (the common logarithm of |x|).

Such a sum always makes sense (i.e. converges), even if there is an infinite number of ai (with negative indices), which is the case for all reals which are not decimal numbers, according to what precedes.

The representation is unique, if one excludes representations that end in a recurring 9.

Indeed, consider rational numbers which can be written as p/(2a5b) (i.e. the only prime factors in denominator are 2 and 5). In this case there is a terminating decimal representation. For instance 1/1=1, −1/2=−0.5, 3/5=0.6, 3/25=0.12 and 1306/1250=1.0448. Such numbers are the only real numbers which don't have a unique decimal representation, as they can also be written as a representation that has a recurring 9, for instance 1=0.99999…, −1/2=−0.499999…, etc.

Rational numbers p/q with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring decimal representation.

This leaves the irrational numbers. They also have unique infinite decimal representation, and can be characterised as the numbers whose decimal representations neither terminate nor recur.

Naturally, the same trichotomy holds for other base-n positional numeral systems:

  • Terminating representation: rational where the denominator divides some nk
  • Recurring representation: other rational
  • Non-terminating, non-recurring representation: irrational

and a version of this even holds for irrational-base numeration systems, such as golden mean base representation.

History

Decimal writers

See also

External links

de:Dezimalsystem es:Sistema decimal eo:Dekuma sistemo fr:Nombre décimal ko:십진법 it:Sistema numerico decimale he:השיטה העשרונית nl:Decimaal ja:十進記数法 no:Titallsystemet pl:Dziesiętny system liczbowy pt:Sistema decimal ru:Десятичная система счисления sl:Desetiški številski sistem fi:Kymmenjärjestelmä sv:Decimala talsystemet th:เลขฐานสิบ uk:Десяткова система числення zh:十进制

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