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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.
In this context, the usual decimals, with a finite number of non-zero digits after the decimal separator, are sometimes called terminating decimals. 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 ]
For example, in base 2 (the binary numeral system) 0.111... equals 1, and in base 3 (the ternary numeral system) 0.222... equals 1. In general, any terminating base expression has a counterpart with repeated trailing digits equal to − 1. Textbooks of real analysis are likely to skip the example of 0.999... and present one or both of these ...
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".
For example, the decimal number 123456789 cannot be exactly represented if only eight decimal digits of precision are available (it would be rounded to one of the two straddling representable values, 12345678 × 10 1 or 12345679 × 10 1), the same applies to non-terminating digits (. 5 to be rounded to either .55555555 or .55555556).
In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence. For example, the decimal representation of π starts with 3.14159, but no finite number of digits can represent π exactly, nor does it repeat. Conversely, a decimal expansion that terminates or repeats must be a rational number.
Conversely any recurring (or terminating) base-φ expansion is a non-negative element of Q[√ 5]. For recurring decimals, the recurring part has been overlined: 1 / 2 = 0. 010 φ 1 / 3 = 0. 00101000 φ 1 / 4 = 0. 001000 φ 1 / 5 = 0. 001001010100100100 φ 1 / 10 = 0 ...
The value of n is then the period of the decimal expansion of 1/p. [10] At present, more than fifty decimal unique primes or probable primes are known. However, there are only twenty-three unique primes below 10 100. The decimal unique primes are 3, 11, 37, 101, 9091, 9901, 333667, 909091, ... (sequence A040017 in the OEIS).
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