Inmathematics, acube root of a numberx is a numbery that has the given number as itsthird power; that is The number of cube roots of a number depends on thenumber system that is considered.
Everyreal numberx has exactly one real cube root that is denoted and called thereal cube root ofx or simplythe cube root ofx in contexts wherecomplex numbers are not considered. For example, the real cube roots of8 and−8 are respectively2 and−2. The real cube root of aninteger or of arational number is generally not a rational number, neither aconstructible number.
Every nonzero real orcomplex number has exactly three cube roots that are complex numbers. If the number is real, one of the cube roots is real and the two other are nonrealcomplex conjugate numbers. Otherwise, the three cube roots are all nonreal. For example, the real cube root of8 is2 and the other cube roots of8 are and. The three cube roots of−27i are and The number zero has a unique cube root, which is zero itself.
The cube root is amultivalued function. Theprincipal cube root is itsprincipal value, that is a unique cube root that has been chosen once for all. The principal cube root is the cube root with the largestreal part. In the case of negative real numbers, the largest real part is shared by the two nonreal cube roots, and the principal cube root is the one with positive imaginary part. So, for negative real numbers,the real cube root is not the principal cube root. Forpositive real numbers, the principal cube root is the real cube root.
Ify is any cube root of the complex numberx, the other cube roots are and
In analgebraically closed field ofcharacteristic different from three, every nonzero element has exactly three cube roots, which can be obtained from any of them by multiplying it by eitherroot of the polynomial In an algebraically closed field of characteristic three, every element has exactly one cube root.
In other number systems or otheralgebraic structures, a number or element may have more than three cube roots. For example, in thequaternions, a real number has infinitely many cube roots.
The cube roots of a numberx are the numbersy which satisfy the equation
For any real numberx, there is exactly one real numbery such that. Indeed, thecube function is increasing, so it does not give the same result for two different inputs, and covers all real numbers. In other words, it is abijection or one-to-one correspondence.
That is, one can definethe cube root of a real number as its unique cube root that is also real. With this definition, the cube root of anegative number is a negative number.
However this definition may be confusing when real numbers are considered as specific complex numbers, since, in this casethe cube root is generally defined as the principal cube root, and the principal cube root of a negative real number is not real.
Ifx andy are allowed to becomplex, then there are three solutions (ifx is non-zero) and sox has three cube roots. A real number has one real cube root and two further cube roots which form acomplex conjugate pair. For instance, the cube roots of1 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.
For complex numbers, the principal cube root is usually defined as the cube root that has the greatestreal part, or, equivalently, the cube root whoseargument has the leastabsolute value. It is related to the principal value of thenatural logarithm by the formula
If we writex as
wherer is a non-negative real number and lies in the range
then the principal complex cube root is
This means that inpolar 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 −2, but rather
This difficulty can also be solved by considering the cube root as amultivalued function: if we write the original complex numberx in three equivalent forms, namely
The principal complex cube roots of these three forms are then respectively
Unlessx = 0, these three complex numbers are distinct, even though the three representations ofx were equivalent. For example, may then be calculated to be −2,, or.
This is related with the concept ofmonodromy: if one follows bycontinuity the functioncube root along a closed path around zero, after a turn the value of the cube root is multiplied (or divided) by
Cube roots arise in the problem of finding an angle whose measure is one third that of a given angle (angle trisection) and in the problem of finding the edge of a cube whose volume is twice that of a cube with a given edge (doubling the cube). In 1837Pierre Wantzel proved that neither of these can be done with acompass-and-straightedge construction.
Newton's method is aniterative method that can be used to calculate the cube root. For realfloating-point numbers this method reduces to the following iterative algorithm to produce successively better approximations of the cube root ofa:
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:
Thisconverges 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 that1/3a 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 ofx0 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 thisgeneralized continued fraction, based on thenth root method:
Ifx is a good first approximation to the cube root ofa and, then:
The second equation combines each pair of fractions from the first into a single fraction, thus doubling the speed of convergence.
Cubic equations, which arepolynomial equations of the third degree (meaning the highest power of the unknown is 3) can always be solved for their three solutions in terms of cube roots and square roots (although simpler expressions only in terms of square roots exist for all three solutions, if at least one of them is arational number). If two of the solutions are complex numbers, then all three solution expressions involve the real cube root of a real number, while if all three solutions are real numbers then they may be expressed in terms of thecomplex cube root of a complex number.
Quartic equations can also be solved in terms of cube roots and square roots.
The calculation of cube roots can be traced back toBabylonian mathematicians from as early as 1800 BCE.[2] In the fourth century BCEPlato posed the problem ofdoubling the cube, which required acompass-and-straightedge construction of the edge of acube with twice the volume of a given cube; this required the construction, now known to be impossible, of the length.
A method for extracting cube roots appears inThe Nine Chapters on the Mathematical Art, aChinese mathematical text compiled around the second century BCE and commented on byLiu Hui in the third century CE.[3] TheGreek mathematicianHero of Alexandria devised a method for calculating cube roots in the first century CE. His formula is again mentioned by Eutokios in a commentary onArchimedes.[4] In 499 CEAryabhata, amathematician-astronomer from the classical age ofIndian mathematics andIndian astronomy, gave a method for finding the cube root of numbers having many digits in theAryabhatiya (section 2.5).[5]
