Decimal

The decimal numeral system (also called base ten or occasionally denary) has ten as its base. It is the numerical base most widely used by modern civilizations.^[1]^[2]

Decimal notation often refers to a base-10 positional notation such as the Hindu-Arabic numeral system; however, it can also be used more generally to refer to non-positional systems such as Roman or Chinese numerals which are also based on powers of ten.

Decimals also refer to decimal fractions, either separately or in contrast to vulgar fractions. In this context, a decimal is a tenth part, and decimals become a series of nested tenths. There was a notation in use like 'tenth-metre', meaning the tenth decimal of the metre, currently an Angstrom. The contrast here is between decimals and vulgar fractions, and decimal divisions and other divisions of measures, like the inch. It is possible to follow a decimal expansion with a vulgar fraction; this is done with the recent divisions of the troy ounce, which has three places of decimals, followed by a trinary place.

Decimal notation

Decimal notation is the writing of numbers in a base-10 numeral system. Examples are Roman numerals, Brahmi numerals

, and Chinese numerals, as well as the Hindu-Arabic numerals used by speakers of many European languages. Roman numerals have symbols for the decimal powers (1, 10, 100, 1000) and secondary symbols for half these values (5, 50, 500). Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100 and another for 1000. Chinese numerals have symbols for 1–9, and fourteen additional symbols for higher powers of 10, which in modern usage reach 10⁴⁴.

However, when people who use Hindu-Arabic numerals speak of decimal notation, they often mean not just decimal numeration, as above, but also decimal fractions, all conveyed as part of a positional system. Positional decimal systems include a zero and use symbols (called digits) for the ten values (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to represent any number, no matter how large or how small. These digits are often used with a decimal separator which indicates the start of a fractional part, and with a symbol such as the plus sign + (for positive) or minus sign − (for negative) adjacent to the numeral to indicate whether it is greater or less than zero, respectively.

Positional notation uses positions for each power of ten: units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier (power of ten) multiplied with that digit—each position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization: the Chinese counting rod system and the Hindu-Arabic numeral system(the latter descended from Brahmi numerals).

Ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). The English word digit as well as its translation in many languages is also the anatomical term for 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.

Decimal fractions

A decimal fraction is a fraction whose denominator is a power of ten.

Decimal fractions are commonly expressed without a denominator, the decimal separator being inserted into the numerator (with leading zerosadded if needed) at the position from the right corresponding to the power of ten of the denominator; e.g., 8/10, 83/100, 83/1000, and 8/10000 are expressed as 0.8, 0.83, 0.083, and 0.0008. In English-speaking and many Asian countries, a period (.) or raised period (·) is used as the decimal separator; in many other countries, a comma is used.

The integer part or integral part of a decimal number is the part to the left of the decimal separator (see also truncation). The part from the decimal separator to the right is the fractional part; if considered as a separate number, a zero is often written in front. Especially for negative numbers, we have to distinguish between the fractional part of the notation and the fractional part of the number itself, because the latter gets its own decimal sign. It is usual for a decimal number whose absolute value is less than one to have a leading zero.

Trailing zeros after the decimal point are not necessary, although in science, engineering and statistics they can be retained to indicate a required precision or to show a level of confidence in the accuracy of the number: Although 0.080 and 0.08 are numerically equal, in engineering 0.080 suggests a measurement with an error of up to one part in two thousand (±0.0005), while 0.08 suggests a measurement with an error of up to one in two hundred (see significant figures).

[Other rational numbers

Any rational number with a denominator whose only prime factors are 2 and/or 5 may be precisely expressed as a decimal fraction and has a finite decimal expansion.^[3]

1/2 = 0.5

1/20 = 0.05

1/5 = 0.2

1/50 = 0.02

1/4 = 0.25

1/40 = 0.025

1/25 = 0.04

1/8 = 0.125

1/125= 0.008

1/10 = 0.1

If the rational number's denominator has any prime factors other than 2 or 5, it cannot be expressed as a finite decimal fraction,^[3] and has a unique infinite decimal expansion ending with recurring decimals.

1/3 = 0.333333… (with 3 repeating)

1/9 = 0.111111… (with 1 repeating)

100-1=99=9×11

1/11 = 0.090909… (with 09 repeating)

1000-1=9×111=27×37

1/27 = 0.037037037…

1/37 = 0.027027027…

1/111 = 0 .009009009…

also:

1/81= 0.012345679012… (with 012345679 repeating)

Other prime factors in the denominator will give longer recurring sequences; see for instance 1/7, and 1/13.

That a rational number must have 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. For instance, to find 3/7 by long division:

     0.4 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

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,

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

Real numbers

Further information: Decimal representation

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, and

· a_i ∈ { 0,1,…,9 } for all i ∈ Z are its decimal digits, equal to zero for all i greater than some number (that number being the common logarithm of |x|).

Such a sum converges as i increases, even if there are infinitely many non-zero a_i.

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

Non-uniqueness of decimal representation

Consider those rational numbers which have only the factors 2 and 5 in the denominator, i.e., which can be written as p/(2^a5^b). 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 do not 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. The number 0 = 0/1 is special in that it has no representation with recurring 9.

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

So in general the decimal representation is unique, if one excludes representations that end in a recurring 9.

The same trichotomy

holds for other base-n positional numeral systems:

· Terminating representation: rational where the denominator divides some n^k

· Recurring representation: other rational

· Non-terminating, non-recurring representation: irrational

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

Decimal computation

Decimal computation was carried out in ancient times in many ways, typically on sand tables or with a variety of abaci.

Modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such as theENIAC

or the IBM 650, used decimal representation internally).^[4] 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 or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.)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 some variant of binary-coded decimal,^[5] especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for Floating-Point Arithmetic).^[6]

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.^[7]