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Reciprocal polynomials have several connections with their original polynomials, including: is not 0. p(x) = xnp∗(x−1). [2] α is a root of a polynomial p if and only if α−1 is a root of p∗. [4] If p(x) ≠ x then p is irreducible if and only if p∗ is irreducible. [5] p is primitive if and only if p∗ is primitive.
An important application is Newton–Raphson division, which can be used to quickly find the reciprocal of a number a, using only multiplication and subtraction, that is to say the number x such that 1 / x = a. We can rephrase that as finding the zero of f(x) = 1 / x − a. We have f ′ (x) = − 1 / x 2 . Newton's ...
The reciprocals of prime numbers have been of interest to mathematicians for various reasons. They do not have a finite sum, as Leonhard Euler proved in 1737. Like rational numbers, the reciprocals of primes have repeating decimal representations. In his later years, George Salmon (1819–1904) concerned himself with the repeating periods of ...
Radius of curvature. Radius of the circle which best approximates a curve at a given point. Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point.
Reciprocal space (also called k-space) provides a way to visualize the results of the Fourier transform of a spatial function. It is similar in role to the frequency domain arising from the Fourier transform of a time dependent function; reciprocal space is a space over which the Fourier transform of a spatial function is represented at spatial frequencies or wavevectors of plane waves of the ...
The condition number is a property of the problem. Paired with the problem are any number of algorithms that can be used to solve the problem, that is, to calculate the solution. Some algorithms have a property called backward stability; in general, a backward stable algorithm can be expected to accurately solve well-conditioned problems.
In the physical sciences, the wavenumber (or wave number), also known as repetency, [1] is the spatial frequency of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). [2][3][4] It is analogous to temporal frequency, which is defined as the number of wave cycles per unit time ...
The quadratic reciprocity law can be formulated in terms of the Hilbert symbol where a and b are any two nonzero rational numbers and v runs over all the non-trivial absolute values of the rationals (the Archimedean one and the p -adic absolute values for primes p). The Hilbert symbol is 1 or −1.
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