Cube root

Last updated March 23, 2024

In mathematics, a cube root of a number $x$ is a number $y$ such that $y 3 = x$ . All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of $8$ , denoted ${\sqrt[{3}]{8}}$ , is $2$ , because $23 = 8$ , while the other cube roots of $8$ are $-1+i{\sqrt {3}}$ and $-1-i{\sqrt {3}}$ . The three cube roots of $-27 i$ 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 ${\sqrt[{3}]{~^{~}}}.$ 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 $\left({\sqrt[{3}]{x}}\right)^{3}=x,$ 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, $(-1+i{\sqrt {3}})^{3}=8$ , but ${\sqrt[{3}]{8}}=2.$

Formal 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 one real number y such that y³ = x. The cube function is increasing, so does not give the same result for two different inputs, and 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.

If x and y are allowed to be 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:

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

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

x^{\frac {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 θ 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 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 ³√−8 will not be −2, but rather 1 + i√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

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

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 of x were equivalent. For example, ³√−8 may then be calculated to be −2, 1 + i√3, or 1 − i√3.

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 $e^{2i\pi /3}.$

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 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:

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 method improves 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).

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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1/3a$ 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 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 − x³, 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 are polynomial equations 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 complex cube root of a complex number.

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

History

The calculation of cube roots can be traced back to Babylonian mathematicians from as early as 1800 BCE.^[2] In the fourth century BCE Plato posed the problem of doubling the cube, which required a compass-and-straightedge construction of the edge of a cube with twice the volume of a given cube; this required the construction, now known to be impossible, of the length ³√2.

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.^[3] 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.^[4] 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).^[5]

Related Research Articles

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In geometry and algebra, a real number $is constructible if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only integers and the operations for addition, subtraction, multiplication, division, and square roots.$

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<span class="mw-page-title-main">Exponential function</span> Mathematical function, denoted exp(x) or e^x

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The natural logarithm of a number is its logarithm to the base of the mathematical constant $e$ , which is an irrational and transcendental number approximately equal to $2.718 281828459$ . The natural logarithm of $x$ is generally written as $ln x$ , $log e x$ , or sometimes, if the base $e$ is implicit, simply $log x$ . Parentheses are sometimes added for clarity, giving $ln(x)$ , $log e (x)$ , or $log(x)$ . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

In algebra, a quadratic equation is any equation that can be rearranged in standard form as

In mathematics, a square root of a number $x$ is a number $y$ such that $; in other words, a number y whose square is x . For example, 4 and -4 are square roots of 16 because .$

In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

<span class="mw-page-title-main">Imaginary unit</span> Principal square root of −1

The imaginary unit or unit imaginary number is a solution to the quadratic equation $x 2 + 1 = 0.$ Although there is no real number with this property, $i$ can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of $i$ in a complex number is $2 + 3 i .$

<span class="mw-page-title-main">Exponentiation</span> Arithmetic operation

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<span class="mw-page-title-main">Cubic equation</span> Polynomial equation of degree 3

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In mathematics, taking the nth root is an operation involving two numbers, the radicand and the index or degree. Taking the nth root is written as $, where x is the radicand and n is the index. This is pronounced as "the nth root of x". The definition then of an n th root of a number x is a number r which, when raised to the power of the positive integer n, yields x :$

<span class="mw-page-title-main">Cubic function</span> Polynomial function of degree 3

In mathematics, a cubic function is a function of the form $that is, a polynomial function of degree three. In many texts, the coefficients a, b, c, and d are supposed to be real numbers, and the function is considered as a real function that maps real numbers to real numbers or as a complex function that maps complex numbers to complex numbers. In other cases, the coefficients may be complex numbers, and the function is a complex function that has the set of the complex numbers as its codomain, even when the domain is restricted to the real numbers.$

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

<span class="mw-page-title-main">Tetration</span> Arithmetic operation

In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though $and the left-exponent x b are common.$

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Methods of computing square roots are algorithms for approximating the non-negative square root $of a positive real number . Since all square roots of natural numbers, other than of perfect squares, are irrational, square roots can usually only be computed to some finite precision: these methods typically construct a series of increasingly accurate approximations.$

$<span class="mw-page-title-main">Newton fractal</span> Boundary set in the complex plane$

The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial $p (z) \in$ $[z] or transcendental function. It is the Julia set of the meromorphic function z ↦ z − p(z)/p′(z) which is given by Newton's method. When there are no attractive cycles (of order greater than 1), it divides the complex plane into regions Gk, each of which is associated with a root ζk of the polynomial, k = 1, …, deg(p). In this way the Newton fractal is similar to the Mandelbrot set, and like other fractals it exhibits an intricate appearance arising from a simple description. It is relevant to numerical analysis because it shows that (outside the region of quadratic convergence) the Newton method can be very sensitive to its choice of start point.$

In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form

In algebra, casus irreducibilis is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically, i.e., by obtaining roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of casus irreducibilis is in the case of cubic polynomials that have three real roots, which was proven by Pierre Wantzel in 1843. One can see whether a given cubic polynomial is in so-called casus irreducibilis by looking at the discriminant, via Cardano's formula.

References

1 2 "In Search of a Fast Cube Root". metamerist.com. 2008. Archived from the original on 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 Archived 15 August 2011 at archive.today Marathi : आर्यभटीय, Mohan Apte, Pune, India, Rajhans Publications, 2009, p. 62, ISBN 978-81-7434-480-9

External links

Cube root calculator reduces any number to simplest radical form
Computing the Cube Root, Ken Turkowski, Apple Technical Report #KT-32, 1998. Includes C source code.
Weisstein, Eric W. "Cube Root". MathWorld .

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[metamerist-1] 1 2 "In Search of a Fast Cube Root". metamerist.com. 2008. Archived from the original on 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 Archived 15 August 2011 at archive.today Marathi : आर्यभटीय, Mohan Apte, Pune, India, Rajhans Publications, 2009, p. 62, ISBN 978-81-7434-480-9

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