Decimal representation
A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum
where a_{0} is a nonnegative integer, and a_{1}, a_{2}, ... 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.^{[1]} 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
That is to say, a_{0} is the integer part of r, not necessarily between 0 and 9, and a_{1}, a_{2}, a_{3}, ... are the digits forming the fractional part of r.
Both notations above are, by definition, the following limit of a sequence:
- .
Contents
Finite decimal approximations
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 . (The fact that has a finite decimal representation is easily established.)
Non-uniqueness of decimal representation and notational conventions
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 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 , an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself if is an integer.
Certain procedures for constructing the decimal expansion of will avoid the problem of trailing 9's. For instance, the following algorithmic procedure will give the standard decimal representation: Given , we first define (the integer part of ) to be the largest integer such that (i.e., ). If the procedure terminates. Otherwise, for already found, we define inductively to be the largest integer such that
The procedure terminates whenever is found such that equality holds in ; otherwise, it continues indefinitely to give an infinite sequence of decimal digits. It can be shown that ^{[2]} (conventionally written as ), with and . This construction is extended to by applying the above procedure to and denoting the resultant decimal expansion by .
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 2^{n}5^{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 10^{n} = 2^{n}5^{n}.
Conversely, if the denominator of x is of the form 2^{n}5^{m}, for some p. While x is of the form , for some n. By , x will end in zeros.
Recurring decimal representations
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.
Conversion to fraction
Every decimal representation of a rational number can be converted to a fraction by summing up the integer, non-repeating and repeating parts as in the example below
where the exponents in the denominators are 3 (number of non-repeating digits after decimal point) and 4 (number of repeating digits). If there are no repeating digits assume there's a forever repeating 0 or 9, i.e. .
See also
References
- ^ Knuth, Donald Ervin (1973). The Art of Computer Programming. Volume 1: Fundamental Algorithms. Addison-Wesley. p. 21.
- ^ Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. p. 11. ISBN 0-07-054235-X.
Further reading
- Apostol, Tom (1974). Mathematical analysis (Second ed.). Addison-Wesley.
- Savard, John J. G. (2018) [2006]. "Decimal Representations". quadibloc. Archived from the original on 2018-07-16. Retrieved 2018-07-16.