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The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution). Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25).
Rules for calculating the periods of repeating decimals from rational fractions were given by James Whitbread Lee Glaisher in 1878. [5] For a prime p, the period of its reciprocal divides p − 1. [6] The sequence of recurrence periods of the reciprocal primes (sequence A002371 in the OEIS) appears in the 1973 Handbook of Integer Sequences.
The sum of the reciprocals of the pentatope numbers is 4 / 3 . Sylvester's sequence is an integer sequence in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are 2, 3, 7, 43, 1807 . The sum of the reciprocals of the numbers in Sylvester's sequence is 1.
Then for , the expansion of () contains at least one term for each reciprocal of a positive integer with exactly prime factors (counting multiplicities) only from the set {+, +,}. It follows that the geometric series ∑ i = 0 ∞ ( x k ) i {\textstyle \sum _{i=0}^{\infty }(x_{k})^{i}} contains at least one term for each reciprocal of a ...
Reciprocal polynomial, a polynomial obtained from another polynomial by reversing its coefficients; Reciprocal rule, a technique in calculus for calculating derivatives of reciprocal functions; Reciprocal spiral, a plane curve; Reciprocal averaging, a statistical technique for aggregating categorical data
5.3 In finance. 5.4 In geometry. 5.5 In other sciences. ... [3] The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, ...
The reciprocal of a fraction is another fraction with the numerator and denominator exchanged. The reciprocal of 3 / 7 , for instance, is 7 / 3 . The product of a non-zero fraction and its reciprocal is 1, hence the reciprocal is the multiplicative inverse of a fraction. The reciprocal of a proper fraction is improper, and the ...
Noting [16] that it is easier to state for −3 and +5 than it is for +3 or −5, he states [17] the general theorem in the form: If p is a prime of the form 4 n + 1 then p , but if p is of the form 4 n + 3 then − p , is a quadratic residue (resp. nonresidue) of every prime, which, with a positive sign, is a residue (resp. nonresidue) of p .