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Plot of y = $\sqrt[3]{x}$ for $x\ge 0$. Complete plot is symmetric with respect to origin, as it is an odd function. At x = 0 this graph has a vertical tangent.

In mathematics, a cube root of a number, denoted $\sqrt[3]{x}$ or x^1/3, is a number a such that a³ = x. All real numbers (except zero) have exactly one real cube root and a pair of complex conjugate roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8 is 2, because 2³ = 8. All the cube roots of −27i are

$\sqrt[3]{-27i}=\{\begin{array}{c}3i\\ \frac{3\sqrt{3}}{2}-\frac{3}{2}i\\ -\frac{3\sqrt{3}}{2}-\frac{3}{2}i.\end{array}$

The cube root operation is not associative or distributive with addition or subtraction.

The cube root operation is associative with exponentiation and distributive with multiplication and division if considering only real numbers, but not always if considering complex numbers, for example:

$(\sqrt[3]{8}{)}^3=8$

but

$\sqrt[3]{8^3}=\{\begin{array}{c}8\\ -4+4i\sqrt{3}\\ -4-4i\sqrt{3}.\end{array}$

Formal definition

The cube roots of a number x are the numbers y which satisfy the equation

$y^3=x.$

Real numbers

The three cube roots of 1

If x and y are real, then there is a unique solution and so the cube root of a real number is sometimes defined by this equation. 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. This can lead to some interesting results.

For instance, the cube roots of the number one are:

$\sqrt[3]{1}=\{\begin{array}{c}1\\ -\frac{1}{2}+\frac{\sqrt{3}}{2}i\\ -\frac{1}{2}-\frac{\sqrt{3}}{2}i.\end{array}$

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 any real or complex number, the other two cube roots can be found by multiplying that number by one or the other of the two complex cube roots of one.

Complex numbers

Plot of the complex cube root together with its two additional leaves. The first picture shows the main branch which is described in the text

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 by

$x^{1/3}=\exp (\frac{1}{3}\ln x)$

where ln(x) is the principal branch of the natural logarithm. If we write x as

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

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

$-\pi <\theta \le \pi$,

then the principal complex cube root is

$\sqrt[3]{x}=\sqrt[3]{r}\exp (\frac{1}{3}i\theta ).$

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 $\sqrt[3]{-8}$ will not be $-2$, but rather $1+i\sqrt{3}.$

This limitation can easily be avoided if we write the original complex number x in three equivalent forms, namely

$x=\{\begin{array}{c}r\exp (i(\theta )),\\ r\exp (i(\theta +2\pi )),\\ r\exp (i(\theta -2\pi )).\end{array}$

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

$\sqrt[3]{x}=\{\begin{array}{c}\sqrt[3]{r}\exp (i(\frac{1}{3}\theta )),\\ \sqrt[3]{r}\exp (i(\frac{1}{3}\theta +\frac{2}{3}\pi )),\\ \sqrt[3]{r}\exp (i(\frac{1}{3}\theta -\frac{2}{3}\pi )).\end{array}$

In general, these three complex numbers are distinct, even though the three representations of x were the same. For example, ∛-8 may then be calculated to be −2, 1 + i√3, or 1 − i√3.

In programs that are aware of the imaginary plane, the graph of the cube root of x on the real plane will not display any output for negative values of x. To also include negative roots, these programs must be explicitly instructed to only use real numbers.

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 $a$:

$x_{i+1}=\frac{1}{3}(\frac{a}{x_i^2}+2x_i).$

The method is simply averaging three factors chosen such that $x_i\times x_i\times \frac{a}{x_i^2}=a$ at each iteration.

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

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

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. This has the disadvantage of requiring knowledge of the internal representation of the floating point number, and therefore a single implementation is not guaranteed to work across all computing platforms.

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 z and y = z − x³, then:

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

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

The second equation combines each pair of fractions from the first into a single fraction, thus doubling the speed of convergence. The advantage is that x and y are only computed once.

History

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. The calculation of cube roots can be to traced back to Babylonian mathematicians from as early as 1800 BCE. A method for extracting cube roots appears in The Nine Chapters on the Mathematical Art, a Chinese mathematical text compiled around the 2nd century BCE and commented on by Liu Hui in the 3rd century CE. The 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. 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).

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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

• Cube root calculator reduces any number to simplest radical form
• Computing the Cube Root, K. Turkowski, Apple Technical Report #KT-32, 1998. Includes C source code.
• Template:Planetmath reference
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