In mathematics, a cube root of a number is a number such that . All nonzero have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero have three distinct complex cube roots. For example, the real cube root of , denoted $\backslash sqrt38$, is , because , while the other cube roots of are $1+i\backslash sqrt\; 3$ and $1i\backslash sqrt\; 3$. The three cube roots of are:
In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign $\backslash sqrt3\{~^~\}.$ The cube root is the inverse function of the cube function if considering only real numbers, but not if considering also complex numbers: although one has always $\backslash left(\backslash sqrt3x\backslash right)^3\; =x,$ the cube of a nonzero number has more than one complex cube root and its principal cube root may not be the number that was cubed. For example, $(1+i\backslash sqrt\; 3)^3=8$, but $\backslash sqrt38=2.$
If x and y are allowed to be complex number, then there are three solutions (if x is nonzero) and so x has three cube roots. A real number has one real cube root and two further cube roots which form a complex conjugate pair. For instance, the cube roots of 1 are:
The last two of these roots lead to a relationship between all roots of any real or complex number. If a number is one cube root of a particular real or complex number, the other two cube roots can be found by multiplying that cube root by one or the other of the two complex cube roots of 1.
If we write x as
where r is a nonnegative real number and θ lies in the range
then the principal complex cube root is
This means that in polar coordinates, we are taking the cube root of the radius and dividing the polar angle by three in order to define a cube root. With this definition, the principal cube root of a negative number is a complex number, and for instance will not be −2, but rather .
This difficulty can also be solved by considering the cube root as a multivalued function: if we write the original complex number x in three equivalent forms, namely
The principal complex cube roots of these three forms are then respectively
Unless , these three complex numbers are distinct, even though the three representations of x were equivalent. For example, may then be calculated to be −2, , or .
This is related with the concept of monodromy: if one follows by continuity the function cube root along a closed path around zero, after a turn the value of the cube root is multiplied (or divided) by $e^\{2i\backslash pi/3\}.$
The method is simply averaging three factors chosen such that
Halley's method improves upon this with an algorithm that converges more quickly with each iteration, albeit with more work per iteration:
This converges cubically, so two iterations do as much work as three iterations of Newton's method. Each iteration of Newton's method costs two multiplications, one addition and one division, assuming that is precomputed, so three iterations plus the precomputation require seven multiplications, three additions, and three divisions.
Each iteration of Halley's method requires three multiplications, three additions, and one division, so two iterations cost six multiplications, six additions, and two divisions. Thus, Halley's method has the potential to be faster if one division is more expensive than three additions.
With either method a poor initial approximation of can give very poor algorithm performance, and coming up with a good initial approximation is somewhat of a black art. Some implementations manipulate the exponent bits of the floatingpoint number; i.e. they arrive at an initial approximation by dividing the exponent by 3.
Also useful is this generalized continued fraction, based on the nth root method:
If x is a good first approximation to the cube root of a and y = a − x^{3}, then:
The second equation combines each pair of fractions from the first into a single fraction, thus doubling the speed of convergence.
can also be solved in terms of cube roots and square roots.
A method for extracting cube roots appears in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the second century BCE and commented on by Liu Hui in the third century CE.
The Greek mathematician Hero of Alexandria devised a method for calculating cube roots in the first century CE. His formula is again mentioned by Eutokios in a commentary on Archimedes. In 499 CE Aryabhata, a mathematicianastronomer from the classical age of Indian mathematics and Indian astronomy, gave a method for finding the cube root of numbers having many digits in the Aryabhatiya (section 2.5). Aryabhatiya आर्यभटीय, Mohan Apte, Pune, India, Rajhans Publications, 2009, p. 62,

