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Concept
Decimal representation
Summary
A decimal representation of a non-negative real number r is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
Here is the decimal separator, k is a nonnegative integer, and are digits, which are symbols representing integers in the range 0, ..., 9.
Commonly, if The sequence of the —the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all 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:
Every nonnegative real number has at least one such representation; it has two such representations (with if ) 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.
The natural number , is called the integer part of r, and is denoted by a0 in the remainder of this article. The sequence of the represents the number
which belongs to the interval and is called the fractional part of r (except when all are ).
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.)
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 trailing 0's or 9's, respectively, are represented by "..."). Conventionally, the decimal representation without trailing 9's is preferred.
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Related concepts (19)
Rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction \tfrac p q of two integers, a numerator p and a non-zero denominator q. For example, \tfrac{-3}{7} is a rational number, as is every integer (e.g., 5 = 5/1). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface Q, or blackboard bold \Q. A rational number is a real number.
Real number
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.
Decimal representation
A decimal representation of a non-negative real number r is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: Here is the decimal separator, k is a nonnegative integer, and are digits, which are symbols representing integers in the range 0, ..., 9. Commonly, if The sequence of the —the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0.
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