Wikipedia / Decimal representation
Wikipedia / Decimal representation
A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits Here is the decimal separator, is a nonnegative integer are digits, which are symbols representing integers in the range 0, ..., 9.
—the digits after the dot—is generally infinite are , the separator is also omitted, resulting in a finite sequence of digits, which represents a natural number.
The decimal representation represents the infinite sum
) if and only if one has a trailing infinite sequence of , and the other has a trailing infinite sequence of . For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of are sometimes excluded.
Integer and fractional parts
, is called the integer part of , and is denoted by which belongs to the interval and is called the fractional part of 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.
such that:
Proof has a finite decimal representation is easily established.)
Non-uniqueness of decimal representation and notational conventions
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, an infinite sequence of trailing 0's appearing after the decimal point is an integer.
(the integer part inductively to be the largest integer such that: is found such that equality holds in is represented in decimal notation.
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 for some n. So, one has a denominator equal to .
Conversely, if the denominator of is of the form for some p. The decimal representation of the integer for some and some . So, the decimal expansion of is (up to which is finite.
Infinite
Repeating decimal representations
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:
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.
, although since that makes the repeating term zero the sum simplifies to two terms and a simpler conversion.
See also
References
Further reading