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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.
Conversely, a decimal expansion that terminates or repeats must be a rational number. These are provable properties of rational numbers and positional number systems and are not used as definitions in mathematics. Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic), and in many ...
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".
An irrational number stays aperiodic (with an infinite number of non-repeating digits) in all integral bases. Thus, for example in base 2, π = 3.1415926... 10 can be written as the aperiodic 11.001001000011111... 2. Putting overscores, n, or dots, ṅ, above the common digits is a convention used to represent repeating rational expansions. Thus:
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.
Every rational number corresponds to a finite or a repeating decimal. [21] [c] Irrational numbers are sometimes required to describe magnitudes in geometry. For example, the length of the hypotenuse of a right triangle is irrational if its legs have a length of 1. Irrational numbers are numbers that cannot be expressed through the ratio of two ...
For example, the constant π may be defined as the ratio of the length of a circle's circumference to its diameter. The following list includes a decimal expansion and set containing each number, ordered by year of discovery. The column headings may be clicked to sort the table alphabetically, by decimal value, or by set.
For a repdigit to be prime, it must be a repunit (i.e. the repeating digit is 1) and have a prime number of digits in its base (except trivial single-digit numbers), since, for example, the repdigit 77777 is divisible by 7, in any base > 7.