In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
cube root of a number, denoted
or x
1/3, is a number
a such that
a3 =
x. All
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s (except zero) have exactly one real cube root and a pair of
complex conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8 is 2, because 2
3 = 8. All the cube roots of −27
i are
The cube root operation is not
associativeIn mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
or distributive with
additionAddition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
or
subtractionIn arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
.
The cube root operation is associative with
exponentiationExponentiation is a mathematical operation, written as an, involving two numbers, the base a and the exponent n...
and
distributiveIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
with
multiplicationMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
and
divisionright|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
if considering only real numbers, but not always if considering complex numbers, for example:
but
Formal definition
The cube roots of a number
x are the numbers
y which satisfy the equation
Real numbers
If
x and
y are
realIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
, 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
complexA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
, 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 conjugateIn mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
pair. This can lead to some interesting results.
For instance, the cube roots of the number one 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 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
For complex numbers, the principal cube root is usually defined by
where ln(
x) is the principal branch of the
natural logarithmThe natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
. 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 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
will not be
, but rather
This limitation can easily be avoided 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
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.
Cube root on standard calculator
From the identity
there is a simple method to compute cube roots using a non-scientific calculator, using only the multiplication and square root buttons, after the number is on the display. No memory is required.
- Press the square root button once. (Note that the last step will take care of this strange start.)
- Press the multiplication button.
- Press the square root button twice.
- Press the multiplication button.
- Press the square root button four times.
- Press the multiplication button.
- Press the square root button eight times.
- Press the multiplication button...
This process continues until the number does not change after pressing the multiplication button because the repeated square root gives 1 (this means that the solution has been figured to as many significant digits as the calculator can handle). Then:
- Press the square root button one last time.
At this point an approximation of the cube root of the original number will be shown in the display.
If the first multiplication is replaced by division, instead of the cube root, the fifth root will be shown on the display.
Why this method works
After raising
x to the power in both sides of the above identity, one obtains:
(*)
The left hand side is the cube root of
x.
The steps shown in the method give:
After 2nd step:
After 4th step:
After 6th step:
After 8th step:
etc.
After computing the necessary terms according to the calculator precision, the
last square root finds the right hand of (*).
Alternative method
The above method requires the calculator to have a square root button. Having a simple method of calculating the square root the following function converges fast to the result:
Where with each iteration the result comes closer to the cube root of
a.
The method requires less iterations than Halley's method but needs more calculations, hidden in determining the square roots. Because of the fast converging an initial approximation of 1 suffices.
Numerical methods
Newton's methodIn numerical analysis, Newton's method , named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots of a real-valued function. The algorithm is first in the class of Householder's methods, succeeded by Halley's method...
is an
Iterative methodIn computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the 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
:
The method is simply averaging three factors chosen such that
at each iteration.
Halley's methodIn numerical analysis, Halley’s method is a root-finding algorithm used for functions of one real variable with a continuous second derivative, i.e., C2 functions. It is named after its inventor Edmond Halley, who also discovered Halley's Comet....
improves upon this with an algorithm that converges more
quickly with each step, albeit consuming more multiplication operations:
With either method a poor initial approximation of
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 fractionIn complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary real or complex values....
, based on the
nth rootIn mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
method:
If
x is a good first approximation to the cube root of
z and
y =
z −
x3, then:
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
In 499 CE
AryabhataAryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
, a great
mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
-
astronomerAn astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
from the classical age of
Indian mathematicsIndian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
and Indian astronomy, gave a method for finding the cube root of numbers having many digits in the
AryabhatiyaĀryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Āryabhaṭa.- Structure and style:...
(section 2.5).
See also
- Methods of computing square roots
There are several methods for calculating the principal square root of a nonnegative real number. For the square roots of a negative or complex number, see below.- Rough estimation :...
- List of polynomial topics
- Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...
- Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...
- Nested radical
- Root of unity
In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
- Shifting nth-root algorithm
The shifting nth root algorithm is an algorithm for extracting the nth root of a positive real number which proceeds iteratively by shifting in n digits of the radicand, starting with the most significant, and produces one digit of the root on each iteration, in a manner similar to long...
External links