Cube root
In mathematics, a cube root of a number x is a number y such that y3 = 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 , is 2, because 23 = 8, while the other cube roots of 8 are and . The three cube roots of −27i 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
Formal definition
The cube roots of a number x are the numbers y which satisfy the equation
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:
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
If we write x as
where r is a non-negative real number and θ lies in the range
- ,
then the principal complex cube root is
This means that in
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
The principal complex cube roots of these three forms are then respectively
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
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 (
Numerical methods
The method is simply averaging three factors chosen such that
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:
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 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 x0 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:
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
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
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]
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
- ^ a b "In Search of a Fast Cube Root". metamerist.com. 2008. Archived from the original on 27 December 2013.
- ISBN 978-0-300-05031-8.
- ISBN 978-0-19-853936-0.
- JSTOR 23037103.
- ISBN 978-81-7434-480-9