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Conversely the period of the repeating decimal of a fraction c / d will be (at most) the smallest number n such that 10 n − 1 is divisible by d. For example, the fraction 2 / 7 has d = 7, and the smallest k that makes 10 k − 1 divisible by 7 is k = 6, because 999999 = 7 × 142857. The period of the fraction 2 / 7 is ...
The extended Midy's theorem [2] states that if the repeating portion of the decimal expansion of a/p is divided into k-digit numbers, then their sum is a multiple of 10 k − 1. For example, 1 19 = 0. 052631578947368421 ¯ {\displaystyle {\frac {1}{19}}=0.{\overline {052631578947368421}}}
It was also used to mark Roman numerals whose values are multiplied by 1,000. [2] Today, however, the common usage of a vinculum to indicate the repetend of a repeating decimal [ 3 ] [ 4 ] is a significant exception and reflects the original usage.
The result that 0.999... = 1 generalizes readily in two ways. 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]
Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of 1/(L + 1). Conversely, if the digital period of 1/p (where p is prime) is p − 1, then the digits represent a cyclic number. For example: 1/7 = 0.142857 142857...
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.
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.
While an optimally packed fractal appears only for a defined value of r, i.e., r opt, it is possible to play the chaos game using other values as well.If r>1 (the point x k+1 jumps at a greater distance than the distance between the point x k and the vertex v), the generated figure extends outside the initial polygon. [5]