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This article gives a mathematical definition. For a more accessible article see Decimal.

A decimal representation of a non-negative real number r is an expression of the form

where a₀ is a nonnegative integer, and are integers satisfying ; this is usually written more briefly as

That is to say, a₀ is the integer part of r, not necessarily between 0 and 9, and are the digits forming the fractional part of r.

Contents 1 Finite decimal approximations 2 Multiple decimal representations 3 Finite decimal representations 4 Recurring decimal representations 5 See also 6 External links

Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations.

Assume . Then for every integer there is a finite decimal such that

Proof:

Let , where . Then , and the result follows from dividing all sides by 10ⁿ. (The fact that r_n has a finite decimal representation is easily established.)

Main article: Proof that 0.999... equals 1

Some real numbers have two infinite decimal representations. For example, the number 1 may be equally represented by 1.00000... as by 0.99999... (where for the sake of brevity the infinite sequences of digits 0 and 9, respectively, have been replaced by "..."). Conventionally, the version with zero digits is preferred; by omitting the infinite sequence of zero digits, removing any final zero digits and a possible final decimal point, a normalized finite decimal representation is obtained.

The decimal expansion of non-negative real number x will end in zeros (or in nines) if, and only if, x is a rational number whose denominator is of the form 2ⁿ5^m, where m and n are non-negative integers.

Proof:

If the decimal expansion of x will end in zeros, or for some n, then the denominator of x is of the form 10ⁿ = 2ⁿ5ⁿ.

Conversely, if the denominator of x is of the form 2ⁿ5^m, for some p. While x is of the form , for some n. By , x will end in zeros.

Main article: Recurring decimal

Some real numbers have a decimal expansion that eventually gets into a loop, endlessly repeating a sequence of one or more digits:

¹/₃ = 0.33333...

¹/₇ = 0.142857142857...

¹³¹⁸/₁₈₅ = 7.1243243243...

This happens precisely when the number is a rational number. A special case of this phenomenon is where the expansion ends in all zeros (or nines).

• Decimal
• Series (mathematics)

• Plouffe's inverter describes a number given its decimal representation. For instance, it will describe 3.14159265 as π.