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For complex numbers, the principal cube root is usually defined as the cube root that has the greatest real part, or, equivalently, the cube root whose argument has the least absolute value. It is related to the principal value of the natural logarithm by the formula If we write x as where r is a non-negative real number and θ lies in the range ,
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients a, b, c, and d of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions ).
In algebraic terms, doubling a unit cube requires the construction of a line segment of length x, where x3 = 2; in other words, x = , the cube root of two. This is because a cube of side length 1 has a volume of 13 = 1, and a cube of twice that volume (a volume of 2) has a side length of the cube root of 2.
The cube root law is an observation in political science that the number of members of a unicameral legislature, or the lower house of a bicameral legislature, is about the cube root of the population being represented.  The rule was devised by Rein Taagepera in his 1972 paper "The size of national assemblies". 
If f has three real roots, then K is called a totally real cubic field and it is an example of a totally real field. If, on the other hand, f has a non-real root, then K is called a complex cubic field . A cubic field K is called a cyclic cubic field if it contains all three roots of its generating polynomial f.
The cube function is the function x ↦ x3 (often denoted y = x3) that maps a number to its cube. It is an odd function, as (−n)3 = − (n3). 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.
Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots.