Cube root

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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 x1/3, 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{cases} \ \ \ \ \ \ \ \ \ \ \ 3i \\ \ \ \frac{3\sqrt{3}}{2}-\frac{3}{2}i \\ -\frac{3\sqrt{3}}{2}-\frac{3}{2}i. \end{cases}

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:

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

but

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

Formal definition[edit]

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

y^3 = x.\

Real numbers[edit]

For any real number y, there is one real number x such that x3 = y. The 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.

The three cube roots of 1

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{cases} \ \ 1 \\ -\frac{1}{2}+\frac{\sqrt{3}}{2}i \\ -\frac{1}{2}-\frac{\sqrt{3}}{2}i. \end{cases}

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[edit]

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 ( \tfrac13 \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 ( \tfrac13 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{cases} r \exp \bigl(i (\theta) \bigr), \\ r \exp \bigl(i (\theta + 2\pi) \bigr),  \\ r \exp \bigl( i (\theta - 2\pi) \bigr). \end{cases}

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

\sqrt[3]{x} = \begin{cases} \sqrt[3]{r}\exp \bigl( i ( \tfrac13 \theta) \bigr), \\ \sqrt[3]{r}\exp \bigl( i ( \tfrac13 \theta + \tfrac23 \pi ) \bigr), \\ \sqrt[3]{r}\exp \bigl( i ( \tfrac13 \theta - \tfrac23 \pi ) \bigr). \end{cases}

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[edit]

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} \left(\frac{a}{x_i^2} + 2x_i\right).

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 = zx3, then:

\sqrt[3]{z} = \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(2z-y)-y-\cfrac{2\cdot 4y^2} {9(2z-y)-\cfrac{5\cdot 7y^2} {15(2z-y)-\cfrac{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[edit]

The calculation of cube roots can be to traced back to Babylonian mathematicians from as early as 1800 BCE.[1] 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.[2] 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.[3] 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).[4]

See also[edit]

References[edit]

  1. ^ Saggs, H. W. F. (1989). Civilization Before Greece and Rome. Yale University Press. p. 227. ISBN 978-0-300-05031-8. 
  2. ^ 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. 
  3. ^ Smyly, J. Gilbart (1920). "Heron's Formula for Cube Root". Hermathena (Trinity College Dublin) 19 (42): 64–67. 
  4. ^ Aryabhatiya Marathi: आर्यभटीय, Mohan Apte, Pune, India, Rajhans Publications, 2009, p.62, ISBN 978-81-7434-480-9

External links[edit]