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Encyclopedia > Cube root
Plot of y =
Plot of y = sqrt[3]{x}

In mathematics, the cube root of a number, denoted sqrt[3]{x} or x1/3, is the number a such that a3 = x. All real numbers have exactly one real cube root and 2 complex roots, and all nonzero complex numbers have 3 distinct complex cube roots. For example, the real cube root of 8 is 2, because 23 = 8. All the cube roots of −27i are Image File history File links Cube_root. ... Image File history File links Cube_root. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

sqrt[3]{-27i} = begin{cases} 3i frac{3sqrt3}{2}-frac{3}{2}i -frac{3sqrt3}{2}-frac{3}{2}i end{cases}

The cube root operation is associative with exponentiation and distributive with multiplication and division, but is not associative or distributive with addition or subtraction. In mathematics, associativity is a property that a binary operation can have. ... In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... 5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ...

Contents

Formal definition

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

y^3 = x,

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 principle cube root of x is also represented by In mathematics, the real numbers may be described informally in several different ways. ...

sqrt[3]{x} = x^{1over3}

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. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...


For instance, the cube roots of the number one are: One redirects here. ...

sqrt[3]{1} = begin{cases} 1 -frac{1}{2}+frac{sqrt{3}}{2}i -frac{1}{2}-frac{sqrt{3}}{2}i end{cases}

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 principle cube root is usually defined by

x^{1over3} = exp left( {ln{x}over3} right)

where ln(x) is the principal branch of the natural logarithm. If we write x as The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e, where e is equal to 2. ...

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 complex cube root is

sqrt[3]{x} = sqrt[3]{r}exp left( {itheta over 3} right).

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 sqrt[3]{-8} will not be − 2, but rather 1 + isqrt{3}. This article describes some of the common coordinate systems that appear in elementary mathematics. ...


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 >>Miscellaneous`RealOnly`.) This article is about computer software. ...


Cube root on a standard calculator

From the identity:

frac{1}{3} = frac{1}{2^2} left(1 + frac{1}{2^2}right) left(1 + frac{1}{2^4}right) left(1 + frac{1}{2^8}right) left(1 + frac{1}{2^{16}}right) dots,

there is a simple method to compute the cube roots using a non-scientific calculator, which requires only the multiplication and square root buttons. No memory is required. The following method is used:

  • Press the square root button twice.
  • 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 is continued until the number does not change when the multiplication button is pressed, since the repeated square root gives 1 (this means that the solution has been determined 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 on both sides of the above identity:

x^{frac{1}{3}} = x^{frac{1}{2^2} left(1 + frac{1}{2^2}right) left(1 + frac{1}{2^4}right) left(1 + frac{1}{2^8}right) left(1 + frac{1}{2^{16}}right) ...} (*)

The left hand side is the cube root of x.


The steps shown in the method give:


After the second step:

x^{frac{1}{2^2}}

After the fourth step:

x^{frac{1}{2^2} (1 + frac{1}{2^2})}

After the sixth step:

x^{frac{1}{2^2} (1 + frac{1}{2^2}) (1 + frac{1}{2^4})}

After the eighth step:

x^{frac{1}{2^2} (1 + frac{1}{2^2}) (1 + frac{1}{2^4}) (1 + frac{1}{2^8})}

etc.


Once the value of the expression is equal to 1 to the accuracy of the calculator, the final square root will return the right hand of (*).


See also

In mathematics, an nth root of a number a is a number b, such that bn=a. ... In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ... In algebra, nested radicals are radical expressions that have another radical expression nested inside a radical. ... In mathematics, the nth roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n. ... 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...

References


  Results from FactBites:
 
PlanetMath: cube root (140 words)
The cube root operation is distributive for multiplication and division, but not for addition and subtraction.
The cube root is a special case of the general nth root.
This is version 8 of cube root, born on 2001-11-10, modified 2006-06-21.
Cube root - Wikipedia, the free encyclopedia (595 words)
For example, the cube root of 8 is 2, because 2 × 2 × 2 = 8.
The cube root operation is associative with exponentiation and distributive with multiplication and division, but is not associative or distributive with addition or subtraction.
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.
  More results at FactBites »

 

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