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Möbius μ function: Sum of the nth primitive roots of unity, it depends on the prime factorization of n. Prime omega functions; Chebyshev functions; Liouville function, λ(n) = (–1) Ω(n) Von Mangoldt function, Λ(n) = log p if n is a positive power of the prime p; Carmichael function
The principal cube root is the cube root with the largest real part. In the case of negative real numbers, the largest real part is shared by the two nonreal cube roots, and the principal cube root is the one with positive imaginary part. So, for negative real numbers, the real cube root is not the principal cube root. For positive real numbers ...
A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an n th root is a root extraction. For example, 3 is a square root of 9, since 3 2 = 9, and −3 is also a square root of 9, since (−3) 2 = 9.
The cube function is the function x ↦ x 3 (often denoted y = x 3) that maps a number to its cube. It is an odd function, as (−n) 3 = −(n 3). The volume of a geometric cube is the cube of its side length, giving rise to the name. The inverse operation that consists of finding a number whose cube is n is called extracting the cube root of n ...
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is . This formula for the roots is always correct except when p = q = 0 , with the proviso that if p = 0 , the square root is chosen so that C ≠ 0 .