Decimal representation

Integer and fractional parts

The natural number \sum_{i=0}^k b_i 10^i, is called the integer part of r, and is denoted by a0 in the remainder of this article. The sequence of the a_i represents the number 0.a_1a_2\ldots = \sum_{i=1}^\infty \frac{a_i}{10^i}, which belongs to the interval 0,1), and is called the fractional part of r (except when all a_i are equal to ).

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

Proof: Let r_n = \textstyle\frac{p}{10^n}, where p = \lfloor 10^n x\rfloor. Then p \leq 10^nx , 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 and notational conventions

Main article: 0.999...

Some real numbers x 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 trailing 0's or 9's, respectively, are represented by "..."). Conventionally, the decimal representation without trailing 9's is preferred. Moreover, in the standard decimal representation of x, an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself if x is an integer.

Certain procedures for constructing the decimal expansion of x will avoid the problem of trailing 9's. For instance, the following algorithmic procedure will give the standard decimal representation: Given x\geq 0, we first define a_0 (the integer part of x) to be the largest integer such that a_0\leq x (i.e., a_0 = \lfloor x\rfloor). If x=a_0 the procedure terminates. Otherwise, for (a_i){i=0}^{k-1} already found, we define a_k inductively to be the largest integer such that: The procedure terminates whenever a_k is found such that equality holds in ; otherwise, it continues indefinitely to give an infinite sequence of decimal digits. It can be shown that x = \sup_k \left{\sum{i=0}^{k} \frac{a_i}{10^i}\right} (conventionally written as x=a_0.a_1a_2a_3\cdots), where a_1,a_2,a_3\ldots \in {0,1,2,\ldots, 9}, and the nonnegative integer a_0 is represented in decimal notation. This construction is extended to x by applying the above procedure to -x0 and denoting the resultant decimal expansion by -a_0.a_1a_2a_3\cdots.

Types

Finite

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 2n5m, where m and n are non-negative integers.

Proof:

If the decimal expansion of x will end in zeros, one has x=\sum_{i=0}^n\frac{a_i}{10^i} = \frac{\sum_{i=0}^n 10^{n-i}a_i}{10^n} for some n. So, one has a denominator equal to .

Conversely, if the denominator of x is of the form 2n5m, one has x = \frac{p}{2^n5^m}=\frac{2^m5^np}{2^{n+m}5^{n+m}} = \frac{2^m 5^np}{10^{n+m}} for some p. The decimal representation of the integer has the form 2^m 5^np = \sum_{i=0}^{k} 10^ia_i for some and some . So, the decimal expansion of is (up to the order of the terms) x = \sum_{i=0}^{k} 10^{i-m-n}a_i , which is finite.

Infinite

Repeating 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: : = 0.33333... : = 0.142857142857... : = 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.

Finite decimal representations can also be seen as a special case of infinite repeating decimal representations. For example, = 1.44 = 1.4400000...; the endlessly repeated sequence is the one-digit sequence "0".

Non-repeating decimal representations

Other real numbers have decimal expansions that never repeat. These are precisely the irrational numbers, numbers that cannot be represented as a ratio of integers. Some well-known examples are: : = 1.41421356237309504880... :  e = 2.71828182845904523536... :  π = 3.14159265358979323846...

Conversion to fraction

Every decimal representation of a rational number can be converted to a fraction by converting it into a sum of the integer, non-repeating, and repeating parts and then converting that sum to a single fraction with a common denominator.

For example, to convert \pm 8.123\overline{4567} to a fraction one notes the lemma: \begin{align} 0.000\overline{4567} & = 4567\times0.000\overline{0001} \ & = 4567\times0.\overline{0001}\times\frac{1}{10^3} \ & = 4567\times\frac{1}{9999}\times\frac{1}{10^3} \ & = \frac{4567}{9999}\times\frac{1}{10^3} \ & = \frac{4567}{(10^4 - 1)\times10^3}& \text{The exponents are the number of non-repeating digits after the decimal point (3) and the number of repeating digits (4).} \end{align}

Thus one converts as follows: \begin{align} \pm 8.123\overline{4567} & = \pm \left(8 + \frac{123}{10^3} + \frac{4567}{(10^4 - 1) \times 10^3}\right) & \text{from above} \ & = \pm \frac{8\times(10^4-1)\times10^3+123\times(10^4-1)+4567}{(10^4 - 1) \times 10^3} & \text{common denominator}\ & = \pm \frac{81226444}{9999000} & \text{multiplying, and summing the numerator}\ & = \pm \frac{20306611}{2499750} & \text{reducing}\ \end{align}

If there are no repeating digits one assumes that there is a forever repeating 0, e.g. 1.9 = 1.9\overline{0}, although since that makes the repeating term zero the sum simplifies to two terms and a simpler conversion.

For example: \begin{align} \pm 8.1234 & = \pm \left(8 + \frac{1234}{10^4}\right) & \ & = \pm \frac{8\times10^4+1234}{10^4} & \text{common denominator}\ & = \pm \frac{81234}{10000} & \text{multiplying, and summing the numerator}\ & = \pm \frac{40617}{5000} & \text{reducing}\ \end{align}

References

References

1. (1973). "The Art of Computer Programming". [[Addison-Wesley]].
2. (1976). "Principles of Mathematical Analysis". [[McGraw-Hill]].