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Cube root
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In mathematics, a cube root of a number, denoted or x1/3, is a number a such that a3 = x. All real numbers 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
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 consider only real numbers, but not always if considering complex numbers, for example:
but
i>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.
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In mathematics, a cube root of a number, denoted or x1/3, is a number a such that a3 = x. All real numbers 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
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 consider 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 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. The principal cube root of x is also represented by
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:
These two roots lead to a relationship between all roots. 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 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 logarithm. If we write x as
where r is a non-negative real number and ? lies in the range
,
then the 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 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
, or .
The three complex cube roots are then
, or .
In general, these three complex numbers are distinct, even though the three representations of x were the same. For example, may then be calculated to be , or .
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. (In Mathematica, this can be achieved by executing the following line <.)
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.
- 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 (*).
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 :
,
Halley's method 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.
The following optimized C programming language implementation
uses Halley's method. It demonstrates successively raising an initial
approximation by powers of 2, until it has a third as many binary digits
as the input. It works for non-negative integer inputs:
double cube_root(unsigned long a_)
See also
External links
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