Decimal representation

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 in the form of a series, traditionally written as a sum

$r=\sum_{i=0}^\infty \frac{a_i}{10^i}$

where a₀ is a nonnegative integer, and a₁, a₂, ... are integers satisfying 0 ≤ a_i ≤ 9, called the digits of the decimal representation. The sequence of digits specified may be finite, in which case any further digits a_i are assumed to be 0. Some authors forbid decimal representations with a trailing infinite sequence of "9"s. This restriction still allows a decimal representation for each non-negative real number, but additionally makes such a representation unique. The number defined by a decimal representation is often written more briefly as

$r=a_0.a_1 a_2 a_3\dots.\,$

That is to say, a₀ is the integer part of r, not necessarily between 0 and 9, and a₁, a₂, a₃, ... are the digits forming the fractional part of r.

Both notations above are, by definition, the following limit of a sequence:

$r=\lim_{n\to \infty} \sum_{i=0}^n \frac{a_i}{10^i}$.

Finite decimal approximations

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

Assume $x\geq 0$. Then for every integer $n\geq 1$ there is a finite decimal $r_n=a_0.a_1a_2\cdots a_n$ such that

$r_n\leq x < r_n+\frac{1}{10^n}.\,$

Proof:

Let $r_n = \textstyle\frac{p}{10^n}$, where $p = \lfloor 10^nx\rfloor$. Then $p \leq 10^nx < p+1$, and the result follows from dividing all sides by $10^n$. (The fact that $r_n$ has a finite decimal representation is easily established.)

Non-uniqueness of decimal representation

Main article: 0.999...

Some real numbers have two infinite decimal representations. For example, the number 1 may be equally represented by 1.000... as by 0.999... (where the infinite sequences of digits 0 and 9, respectively, are represented 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.

Finite decimal representations

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 $x=\sum_{i=0}^n\frac{a_i}{10^i}=\sum_{i=0}^n10^{n-i}a_i/10^n$ 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, $x=\frac{p}{2^n5^m}=\frac{2^m5^np}{2^{n+m}5^{n+m}}= \frac{2^m5^np}{10^{n+m}}$ for some p. While x is of the form $\textstyle\frac{p}{10^k}$, $p=\sum_{i=0}^{n}10^ia_i$ for some n. By $x=\sum_{i=0}^n10^{n-i}a_i/10^n=\sum_{i=0}^n\frac{a_i}{10^i}$, x will end in zeros.

Recurring decimal representations

Main article: Repeating decimal

Some real numbers have decimal expansions that eventually get into loops, endlessly repeating a sequence of one or more digits:

¹/₃ = 0.33333...

¹/₇ = 0.142857142857...

¹³¹⁸/₁₈₅ = 7.1243243243...

Every time this happens the number is still a rational number (i.e. can alternatively be represented as a ratio of an integer and a positive integer). Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating.

See also

• Decimal
• Series (mathematics)
• IEEE 754
• Simon Stevin