Decimal

Decimal, or less commonly, denary, usually refers to the base 10 numeral system.

Decimal representation of fractional numbers

Decimal fractions

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

Related Topics:
Vulgar fraction - Denominator - Power

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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.

Related Topics:
Decimal point - Leading zero

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

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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).

Related Topics:
Integer - Fraction - Decimal point - Statistics - Significant figures

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Decimal representation of other rational numbers

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

Related Topics:
Rational number - Recurring decimal

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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:

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:1/2 = 0.5

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:1/3 = 0.333333? (with 3 recurring)

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:1/4 = 0.25

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:1/5 = 0.2

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:1/6 = 0.166666? (with 6 recurring)

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:1/8 = 0.125

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:1/9 = 0.111111? (with 1 recurring)

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:1/10 = 0.1

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:1/11 = 0.090909? (with 09 recurring)

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:1/12 = 0.083333? (with 3 recurring)

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:1/81 = 0.012345679012? (with 012345679 recurring)

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

Related Topics:
Sequence - 7 - 13

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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:

Related Topics:
Finite - Long division - Algorithm - Remainder

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.4 2 8 5 7 1 4 ...

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7 ) 3.0 0 0 0 0 0 0 0

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2 8 30/7 = 4 r 2

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2 0

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1 4 20/7 = 2 r 6

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6 0

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5 6 60/7 = 8 r 4

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4 0

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3 5 40/7 = 5 r 5

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5 0

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4 9 50/7 = 7 r 1

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1 0

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7 10/7 = 1 r 3

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3 0

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2 8 30/7 = 4 r 2 (again)

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2 0

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etc

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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,

Related Topics:
Recurring decimal - Geometric series

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

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Decimal representation of the real numbers

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

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: x = mathop{ m sign}(x) sum_{iinmathbb Z} a_i,10^i

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where

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• sign() is the sign function,
• a_i ∈ { 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.

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

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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.

Related Topics:
Rational number - Prime factor

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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.

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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.

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

Related Topics:
Trichotomy - Positional numeral system

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

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