Cube root

Inmathematics,acube rootof a number $x$ is a number $y$ that has the given number as itsthird power;that is $y^{3}=x.$ The number of cube roots of a number depends on thenumber systemthat is considered.

Every nonzeroreal number $x$ has exactly one real cube root that is denoted ${\textstyle {\sqrt[{3}]{x}}}$ and called thereal cube rootof $x$ or simplythe cube rootof $x$ in contexts wherecomplex numbersare not considered. For example, the real cube roots of $8$ and $-8$ are respectively $2$ and $-2$ .The real cube root of anintegeror of arational numberis generally not a rational number, neither aconstructible number.

Every nonzero real orcomplex numberhas exactly three cube roots that are complex numbers. If the number is real, one of the cube roots is real and the two other are nonrealcomplex conjugatenumbers. Otherwise, the three cube roots are all nonreal. For example, the real cube root of $8$ is $2$ and the other cube roots of $8$ are $-1+i{\sqrt {3}}$ and $-1-i{\sqrt {3}}$ .The three cube roots of $-27 i$ are $3i,{\tfrac {3{\sqrt {3}}}{2}}-{\tfrac {3}{2}}i,$ and $-{\tfrac {3{\sqrt {3}}}{2}}-{\tfrac {3}{2}}i.$ The number zero has a unique cube root, which is zero itself.

The cube root is amultivalued function.Theprincipal cube rootis itsprincipal value,that is a unique cube root that has been chosen once for all. The principal cube root is the cube root with the largestreal 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, the principal cube root is the real cube root.

If $y$ is any cube root of the complex number $x$ ,the other cube roots are $y\,{\tfrac {-1+i{\sqrt {3}}}{2}}$ and $y\,{\tfrac {-1-i{\sqrt {3}}}{2}}.$

In analgebraically closed fieldofcharacteristicdifferent from three, every nonzero element has exactly three cube roots, which can be obtained from any of them by multiplying it by eitherrootof the polynomial $x^{2}+x+1.$ In an algebraically closed field of characteristic three, every element has exactly one cube root.

In other number systems or otheralgebraic structures,a number or element may have more than three cube roots. For example, in thequaternions,a real number has infinitely many cube roots.

Definition

The cube roots of a number $x$ are the numbers $y$ which satisfy the equation $y^{3}=x.\$

Properties

Real numbers

For any real number $x$ ,there is exactly one real number $y$ such that $y^{3}=x$ .Indeed, thecube functionis increasing, so it does not give the same result for two different inputs, and covers all real numbers. In other words, it is abijectionor one-to-one correspondence.

That is, one can definethecube root of a real number as its unique cube root that is also real. With this definition, the cube root of a negative number is a negative number.

However this definition may be confusing when real numbers are considered as specific complex numbers, since, in this casethecube root is generally defined as the principal cube root, and the principal cube root of a negative real number is not real.

The three cube roots of 1

If $x$ and $y$ are allowed to becomplex,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 acomplex conjugatepair. For instance, the cube roots of1are:

1,\quad -{\frac {1}{2}}+{\frac {\sqrt {3}}{2}}i,\quad -{\frac {1}{2}}-{\frac {\sqrt {3}}{2}}i.

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

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.

Riemann surfaceof 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 greatestreal part,or, equivalently, the cube root whoseargumenthas the leastabsolute value.It is related to the principal value of thenatural logarithmby the formula

x^{1/3}=\exp \left({\frac {1}{3}}\ln {x}\right).

If we write $x$ as

x=r\exp(i\theta )\,

where $r$ is a non-negative real number and $\theta$ lies in the range

-\pi <\theta \leq \pi

,

then the principal complex cube root is

{\sqrt[{3}]{x}}={\sqrt[{3}]{r}}\exp \left({\frac {i\theta }{3}}\right).

This means that inpolar 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 ${\sqrt[{3}]{-8}}$ will not be −2, but rather $1+i{\sqrt {3}}$

This difficulty can also be solved by considering the cube root as amultivalued function:if we write the original complex number $x$ in three equivalent forms, namely

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

Geometric representation of the 2nd to 6th roots of a complex number

z

,in polar form

re iφ

where

r = | z |

and

φ = arg z

.If

z

is real,

φ = 0

or

π

.Principal roots are shown in black.

The principal complex cube roots of these three forms are then respectively

{\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}}

Unless $x = 0$ ,these three complex numbers are distinct, even though the three representations ofxwere equivalent. For example, ${\sqrt[{3}]{-8}}$ may then be calculated to be −2, $1+i{\sqrt {3}}$ ,or $1-i{\sqrt {3}}$ .

This is related with the concept ofmonodromy:if one follows bycontinuitythe functioncube rootalong a closed path around zero, after a turn the value of the cube root is multiplied (or divided) by $e^{2i\pi /3}.$

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 1837Pierre Wantzelproved that neither of these can be done with acompass-and-straightedge construction.

Numerical methods

Newton's methodis aniterative methodthat can be used to calculate the cube root. For realfloating-pointnumbers this method reduces to the following iterative algorithm to produce successively better approximations of the cube root of $a$ :

x_{n+1}={\frac {1}{3}}\left({\frac {a}{x_{n}^{2}}}+2x_{n}\right).

The method is simply averaging three factors chosen such that

x_{n}\times x_{n}\times {\frac {a}{x_{n}^{2}}}=a

at each iteration.

Halley's methodimproves upon this with an algorithm that converges more quickly with each iteration, albeit with more work per iteration:

x_{n+1}=x_{n}\left({\frac {x_{n}^{3}+2a}{2x_{n}^{3}+a}}\right).

Thisconverges 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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/3⁠a$ 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,^[1]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 $x 0$ 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.^[1]

Also useful is thisgeneralized continued fraction,based on thenth rootmethod:

If $x$ is a good first approximation to the cube root of $a$ and $y=a-x^{3}$ ,then:

{\sqrt[{3}]{a}}={\sqrt[{3}]{x^{3}+y}}=x+{\cfrac {y}{3x^{2}+{\cfrac {2y}{2x+{\cfrac {4y}{9x^{2}+{\cfrac {5y}{2x+{\cfrac {7y}{15x^{2}+{\cfrac {8y}{2x+\ddots }}}}}}}}}}}}

=x+{\cfrac {2x\cdot y}{3(2x^{3}+y)-y-{\cfrac {2\cdot 4y^{2}}{9(2x^{3}+y)-{\cfrac {5\cdot 7y^{2}}{15(2x^{3}+y)-{\cfrac {8\cdot 10y^{2}}{21(2x^{3}+y)-\ddots }}}}}}}}.

The second equation combines each pair of fractions from the first into a single fraction, thus doubling the speed of convergence.

Appearance in solutions of third and fourth degree equations

Cubic equations,which arepolynomial equationsof 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 arational 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 thecomplex cube root of a complex number.

Quartic equationscan also be solved in terms of cube roots and square roots.

History

The calculation of cube roots can be traced back toBabylonian mathematiciansfrom as early as 1800 BCE.^[2]In the fourth century BCEPlatoposed the problem ofdoubling the cube,which required acompass-and-straightedge constructionof the edge of acubewith twice the volume of a given cube; this required the construction, now known to be impossible, of the length ${\sqrt[{3}]{2}}$ .

A method for extracting cube roots appears inThe Nine Chapters on the Mathematical Art,aChinese mathematicaltext compiled around the second century BCE and commented on byLiu Huiin the third century CE.^[3]TheGreek mathematician Hero of Alexandriadevised a method for calculating cube roots in the first century CE. His formula is again mentioned by Eutokios in a commentary onArchimedes.^[4]In 499 CEAryabhata,amathematician-astronomerfrom the classical age ofIndian mathematicsandIndian astronomy,gave a method for finding the cube root of numbers having many digits in theAryabhatiya(section 2.5).^[5]

References

^^a ^b"In Search of a Fast Cube Root".metamerist.2008. Archived fromthe originalon 27 December 2013.
^Saggs, H. W. F. (1989).Civilization Before Greece and Rome.Yale University Press. p.227.ISBN 978-0-300-05031-8.
^Crossley, John; W.-C. Lun, Anthony (1999).The Nine Chapters on the Mathematical Art: Companion and Commentary.Oxford University Press. p. 213.ISBN 978-0-19-853936-0.
^Smyly, J. Gilbart (1920). "Heron's Formula for Cube Root".Hermathena.19(42). Trinity College Dublin:64–67.JSTOR 23037103.
^Aryabhatiya Archived15 August 2011 atarchive.today Marathi:आर्यभटीय,Mohan Apte, Pune, India, Rajhans Publications, 2009, p. 62,ISBN 978-81-7434-480-9

External links

[metamerist-1] "In Search of a Fast Cube Root".metamerist.2008. Archived fromthe originalon 27 December 2013.

[cbgr-2] Saggs, H. W. F. (1989).Civilization Before Greece and Rome.Yale University Press. p.227.ISBN 978-0-300-05031-8.

[oxf-3] Crossley, John; W.-C. Lun, Anthony (1999).The Nine Chapters on the Mathematical Art: Companion and Commentary.Oxford University Press. p. 213.ISBN 978-0-19-853936-0.

[4] Smyly, J. Gilbart (1920). "Heron's Formula for Cube Root".Hermathena.19(42). Trinity College Dublin:64–67.JSTOR 23037103.

[5] Aryabhatiya Archived15 August 2011 atarchive.today Marathi:आर्यभटीय,Mohan Apte, Pune, India, Rajhans Publications, 2009, p. 62,ISBN 978-81-7434-480-9

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