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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...
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 ...
Encarta Calculator: Lite version of Microsoft Math 3.0; part of Microsoft Student 2008 Microsoft Mathematics 4.0 (removed): The first freeware version, released in 32-bit and 64-bit editions in January 2011; [ 8 ] features a ribbon GUI
The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if a term is even, the next term is one half of it. If a term is odd, the next term is 3 times the previous term plus 1.
For set representing all notes of Western major scale: [1 9 ⁄ 8 5 ⁄ 4 4 ⁄ 3 3 ⁄ 2 5 ⁄ 3 15 ⁄ 8] the LCD is 24 therefore T = 24 ⁄ f. For set representing all notes of a major triad: [1 5 ⁄ 4 3 ⁄ 2] the LCD is 4 therefore T = 4 ⁄ f. For set representing all notes of a minor triad: [1 6 ⁄ 5 3 ⁄ 2] the LCD is 10 therefore T ...
A continued fraction is an expression of the form = + + + + + where the a n (n > 0) are the partial numerators, the b n are the partial denominators, and the leading term b 0 is called the integer part of the continued fraction.
A vinculum can indicate a line segment where A and B are the endpoints: ¯. A vinculum can indicate the repetend of a repeating decimal value: . 1 ⁄ 7 = 0. 142857 = 0.1428571428571428571...
For example, the repeating continued fraction [1;1,1,1,...] is the golden ratio, and the repeating continued fraction [1;2,2,2,...] is the square root of 2. In contrast, the decimal representations of quadratic irrationals are apparently random. The square roots of all (positive) integers that are not perfect squares are quadratic irrationals ...
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