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{{Use dmy dates|date=July 2013}}
[[File:Cube-root function.svg|thumb|288x288px|Plot of ''y'' = {{radic|''x''|3}}. The plot is symmetric with respect to origin, as it is an [[odd function]]. At ''x'' = 0 this graph has a [[vertical tangent]].]]
[[File:Cube and doubled cube.svg|thumb|A unit cube (side = 1) and a cube with twice the volume (side = {{radic|2|3}} = 1.2599... {{OEIS2C|A002580}}).]]

In [[mathematics]], a '''cube root''' of a number ''x'' is a number ''y'' such that ''y''<sup>3</sup>&nbsp;=&nbsp;''x''.  All [[real number]]s, except zero, have exactly one real cube root and a pair of [[complex conjugate]] cube roots, and all nonzero [[complex number]]s have three distinct complex cube roots.  For example, the real cube root of 8, denoted {{radic|8|3}}, is 2, because 2<sup>3</sup>&nbsp;=&nbsp;8, while the other cube roots of 8 are −1&nbsp;+&nbsp;{{sqrt|3}}''i'' and −1&nbsp;−&nbsp;{{sqrt|3}}''i''. The three cube roots of −27''i'' are  
:<math>3i, \quad \frac{3\sqrt{3}}{2}-\frac{3}{2}i, \quad  \text{and} \quad -\frac{3\sqrt{3}}{2}-\frac{3}{2}i.  </math>

The cube root operation is not [[distributivity|distributive]] with [[addition]] or [[subtraction]].

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 {{radic||3}}. The cube root operation is associative with [[exponentiation]] and distributive with [[multiplication]] and [[division (mathematics)|division]] if considering only real numbers, but not always if considering complex numbers: for example, the cube of any cube root of 8 is 8, but the three cube roots of 8{{sup|3}} are 8, −4&nbsp;+&nbsp;4''i''{{sqrt|3}}, and −4&nbsp;−&nbsp;4''i''{{sqrt|3}}.

==Formal definition==
The cube roots of a number ''x'' are the numbers ''y'' which satisfy the equation

:<math>y^3 = x.\ </math>

==Properties==

===Real numbers===
For any real number ''x'', there is ''one'' real number ''y'' such that ''y''<sup>3</sup>&nbsp;=&nbsp;''x''. The [[cube (algebra)|cube function]] is increasing, so does not give the same result for two different inputs, plus it covers all real numbers. In other words, it is a bijection, or one-to-one. Then we can define an inverse function that is also one-to-one. For real numbers, we can define a unique cube root of all real numbers. If this definition is used, the cube root of a negative number is a negative number.

[[Image:3rd roots of unity.svg|thumb|right|The three cube roots of 1]]
If ''x'' and ''y'' are allowed to be [[complex number|complex]], then there are three solutions (if ''x'' is non-zero) 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:

:<math> 1, \quad -\frac{1}{2}+\frac{\sqrt{3}}{2}i, \quad -\frac{1}{2}-\frac{\sqrt{3}}{2}i. </math>

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.

===Complex numbers===
[[Image:Complex cube root.jpg|right|thumb|350px|Plot of the complex cube root together with its two additional leaves. The first image shows the main branch, which is described in the text.]]
[[Image:Riemann surface cube root.svg|right|thumb|200px|[[Riemann surface]] of the cube root. One can see how all three leaves fit together.]]

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 (complex analysis)|argument]] has the least [[absolute value]]. It is related to the principal value of the [[natural logarithm]] by the formula

:<math>x^{\frac13} = \exp \left( \frac13 \ln{x} \right).</math>

If we write ''x'' as

:<math>x = r \exp(i \theta)\,</math>

where ''r'' is a non-negative real number and ''θ'' lies in the range

:<math>-\pi < \theta \le \pi</math>,

then the principal complex cube root is

:<math>\sqrt[3]{x} = \sqrt[3]{r}\exp \left(\frac {i\theta}{3} \right).</math>

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 {{radic|−8|3}} will not be −2, but rather {{nowrap|1 + ''i''{{sqrt|3}}}}.

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

:<math>x = \begin{cases} r \exp (i \theta ), \\[3px] r \exp (i \theta + 2i\pi ),  \\[3px] r \exp ( i \theta - 2i\pi ). \end{cases} </math>

{{visualisation_complex_number_roots.svg}}
The principal complex cube roots of these three forms are then respectively

:<math>\sqrt[3]{x} = \begin{cases} 
\sqrt[3]{r}\exp \left( \frac{i\theta}{3}\right), 
\\ \sqrt[3]{r}\exp \left(\frac{i\theta}{3} + \frac{2i \pi}{3}  \right), 
\\ \sqrt[3]{r}\exp \left(\frac{i\theta}{3} - \frac{2i \pi}{3} \right). \end{cases} </math>

Unless {{nowrap|1=''x'' = 0}}, these three complex numbers are distinct, even though the three representations of ''x'' were equivalent. For example, {{radic|−8|3}} may then be calculated to be −2, {{nowrap|1 + ''i''{{sqrt|3}}}}, or {{nowrap|1 − ''i''{{sqrt|3}}}}.

This is related with the concept of [[monodromy]]: if one follows by [[continuous function|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 <math>e^{2i\pi/3}.</math>

==Impossibility of compass-and-straightedge construction==

Cube roots arise in the problem of finding an angle whose measure is one third that of a given angle ([[angle trisection]]) and in the problem of finding the edge of a cube whose volume is twice that of a cube with a given edge ([[doubling the cube]]). In 1837 [[Pierre Wantzel]] proved that neither of these can be done with a [[compass-and-straightedge construction]].

==Numerical methods==
[[Newton's method]] is an [[iterative method]] that can be used to calculate the cube root.  For real [[floating-point]] numbers this method reduces to the following iterative algorithm to produce successively better approximations of the cube root of ''a'':

:<math>x_{n+1} = \frac{1}{3} \left(\frac{a}{x_n^2} + 2x_n\right).</math>

The method is simply averaging three factors chosen such that
:<math> x_n \times x_n \times \frac{a}{x_n^2}=a </math>
at each iteration.

[[Halley's method]] improves upon this with an algorithm that converges more quickly with each step, albeit consuming more multiplication operations:

:<math>x_{n+1} = x_n \left(\frac{x_n^3 + 2a}{2x_n^3 + a}\right).</math>

With either method a poor initial approximation of ''x''{{sub|0}} 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 floating-point number; i.e. they arrive at an
initial approximation by dividing the exponent by 3.

==Appearance in solutions of third and fourth degree equations==

[[Cubic equation]]s, which are [[polynomial equation]]s of the third degree (meaning the highest power of the unknown is 3) can always be solved for their three solutions in terms of cube roots and square roots (although simpler expressions only in terms of square roots exist for all three solutions, if at least one of them is a [[rational number]]). If two of the solutions are complex numbers, then all three solution expressions involve the real cube root of a real number, while if all three solutions are real numbers then they may be expressed in terms of the [[Casus irreducibilis|complex cube root of a complex number]].

[[Quartic equation]]s can also be solved in terms of cube roots and square roots.

==History==

The calculation of cube roots can be traced back to [[Babylonian mathematics|Babylonian mathematicians]] from as early as 1800 BCE.<ref name="cbgr">{{cite book|last=Saggs|first=H. W. F.|title=Civilization Before Greece and Rome|url=https://books.google.com/books?id=R28oab-7jLcC&pg=PA227|year=1989|publisher=Yale University Press|isbn=978-0-300-05031-8|page=227}}</ref> In the fourth century BCE [[Plato]] posed the problem of [[doubling the cube#History|doubling the cube]], which required a [[compass-and-straightedge construction]] of the edge of a [[cube (geometry)|cube]] with twice the volume of a given cube; this required the construction, now known to be impossible, of  the length {{radic|2|3}}.

A method for extracting cube roots appears in ''[[The Nine Chapters on the Mathematical Art]]'', a [[Chinese mathematics|Chinese mathematical]] text compiled around the 2nd century BCE and commented on by [[Liu Hui]] in the 3rd century CE.<ref name="oxf">{{cite book|last=Crossley|first=John|last2=W.-C. Lun|first2=Anthony|title=The Nine Chapters on the Mathematical Art: Companion and Commentary|url=https://books.google.com/books?id=eiTJHRGTG6YC&pg=PA213|year=1999|publisher=Oxford University Press|isbn=978-0-19-853936-0|page=213}}</ref> The [[Greek mathematics|Greek mathematician]] [[Hero of Alexandria]] devised a method for calculating cube roots in the 1st century CE. His formula is again mentioned by Eutokios in a commentary on [[Archimedes]].<ref>{{cite journal|last=Smyly|first=J. Gilbart|title=Heron's Formula for Cube Root|journal=Hermathena|year=1920|volume=19|issue=42|pages=64–67|publisher=Trinity College Dublin|jstor=23037103}}</ref> In 499 CE [[Aryabhata]], a [[mathematician]]-[[astronomer]] 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).<ref>''[http://www.flipkart.com/aryabhatiya-mohan-apte-book-8174344802 Aryabhatiya] {{Webarchive|url=https://archive.is/20110815204559/http://www.flipkart.com/aryabhatiya-mohan-apte-book-8174344802 |date=15 August 2011 }} {{lang-mr|आर्यभटीय}}'', Mohan Apte, Pune, India, Rajhans Publications, 2009, p.62, {{ISBN|978-81-7434-480-9}}</ref>

==See also==
* [[Methods of computing square roots]]
* [[List of polynomial topics]]
* [[Nth root]]
* [[Square root]]
* [[Nested radical]]
* [[Root of unity]]
* [[Shifting nth-root algorithm]]

==References==
{{Reflist}}

==External links==
*[http://www.mathwarehouse.com/arithmetic/cube-root-calculator.php  Cube root calculator reduces any number to simplest radical form]
*[http://people.freebsd.org/~lstewart/references/apple_tr_kt32_cuberoot.pdf Computing the Cube Root, Ken Turkowski, Apple Technical Report #KT-32, 1998]. Includes C source code.
*{{planetmath reference|id=748|title=Cube root}}
*{{mathworld|urlname=CubeRoot|title=Cube Root}}

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[[Category:Elementary special functions]]
[[Category:Elementary algebra]]