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A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.
First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense. [41] [9] Second, a comparable theorem applies in each radix or base.
A vinculum can indicate the repetend of a repeating decimal value: 1 ⁄ 7 = 0. 142857 = 0.1428571428571428571... A vinculum can indicate the complex conjugate of a complex number: + ¯ = Logarithm of a number less than 1 can conveniently be represented using vinculum:
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, 36 ⁄ 25 = 1.44 = 1.4400000...; the endlessly repeated sequence is the one-digit sequence "0".
A repeating decimal is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., 5.123144144144144... = 5.123 144). [4] An infinite decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.
The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum , as Leonhard Euler proved in 1737. Like rational numbers , the reciprocals of primes have repeating decimal representations.
The vinculum can indicate a line segment: [4] ¯ The vinculum can indicate a repeating decimal value: = ¯ = When it is not possible to format the number so that the overline is over the digit(s) that repeat, one overline character is placed to the left of the digit(s) that repeat: 3.
Representing other real numbers as decimals would require an infinite sequence of digits to the right of the decimal point. If this infinite sequence of digits follows a pattern, it can be written with an ellipsis or another notation that indicates the repeating pattern. Such a decimal is called a repeating decimal.