Inmathematics, annth root of anumber is the numberr which, when multiplied by itselfn times, yields x: Thepositive integern is called theindex ordegree, and the numberx of which the root is taken is theradicand. A root of degree 2 is called asquare root and a root of degree 3, acube root. Roots of higher degree are referred by usingordinal numbers, as infourth root,twentieth root, etc. The computation of annth root is aroot extraction.
Thenth root ofx is written as using theradical symbol. The square root is usually written as, with the degree omitted. Taking thenth root of a number, for fixed, is theinverse of raising a number to thenth power,[1] and can be written as afractional exponent:
For a positive real numberx, denotes the positive square root ofx and denotes the positive realnth root. For example,3 is a square root of9, since32 = 9, and−3 is also a square root of9, since(−3)2 = 9.[2] A negative real number−x has no real-valued square roots, but whenx is treated as a complex number it has twoimaginary square roots, and, wherei is theimaginary unit.
In general, any non-zerocomplex number hasn distinct complex-valuednth roots, equally distributed around a complex circle of constantabsolute value. (Thenth root of0 is zero withmultiplicityn, and this circle degenerates to a point.) Extracting thenth roots of a complex numberx can thus be taken to be amultivalued function. By convention theprincipal value of this function, called theprincipal root and denoted, is taken to be thenth root with the greatest real part and in the special case whenx is a negative real number, the one with a positiveimaginary part. The principal root of a positive real number is thus also a positive real number. As afunction, the principal root iscontinuous in the wholecomplex plane, except along the negative real axis. Thenth roots of 1 are calledroots of unity and play a fundamental role in various areas of mathematics, such asnumber theory,theory of equations, andFourier transform.
An unresolved root, especially one using the radical symbol, is sometimes referred to as asurd[3] or aradical.[4] Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called aradical expression, and if it contains notranscendental functions ortranscendental numbers it is called analgebraic expression.
The Babylonians, as early as 1800 BCE, demonstrated numerical approximations of irrational quantities such as thesquare root of 2 on clay tablets, with an accuracy analogous to six decimal places, as in the tabletYBC 7289.[5] Cuneiform tablets fromLarsa include tables of square and cube roots of integers.[6] The first to prove the irrationality of √2 was most likely thePythagoreanHippasus.[7]Plato in hisTheaetetus, then describes howTheodorus of Cyrene (c. 400 BC) proved the irrationality of,, etc. up to.[8] In the first century AD,Heron of Alexandria devised an iterative method tocompute the square root, which is actually a special case of the more generalNewton's method.[9]
Theancient Greek mathematicians knew how touse compass and straightedge to construct a length equal to the square root of a given length, when an auxiliary line of unit length is given. In 1837Pierre Wantzel proved that annth root of a given length cannot be constructed ifn is not a power of 2.[22]
The three 3rd roots of −1, one of which is a negative real
Annth root of a numberx, wheren is a positive integer, is any of then real or complex numbersr whosenth power isx:
Every positivereal numberx has a single positiventh root, called theprincipalnth root, which is written.[23] Forn equal to 2 this is called the principal square root and then is omitted. Thenth root can also be represented usingexponentiation asx1/n.[2]
For even values ofn, positive numbers also have a negativenth root, while negative numbers do not have a realnth root. For odd values ofn, every negative numberx has a real negativenth root.[23] For example, −2 has a real 5th root, but −2 does not have any real 6th roots.
Every non-zero numberx, real orcomplex, hasn different complex numbernth roots.[24] (In the casex is real, this count includes any realnth roots.) The only complex root of 0 is 0.
Thenth roots of almost all numbers (all integers except thenth powers, and all rationals except the quotients of twonth powers) areirrational.[25] For example,
Asquare root of a numberx is a numberr which, whensquared, becomesx:
Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as theprincipal square root,[2] and is denoted with a radical sign:
Since the square of every real number is nonnegative, negative numbers do not have real square roots.[26] However, for every negative real number there are twoimaginary square roots. For example, the square roots of −25 are 5i and −5i, wherei represents a number whose square is−1.
Expressing the degree of annth root in its exponent form, as in, makes it easier to manipulate powers and roots. If is anon-negative real number,[26]
Every non-negative number has exactly one non-negative realnth root, and so the rules for operations with surds involving non-negative radicands and are straightforward within the real numbers:[26]
Subtleties can occur when taking thenth roots of negative orcomplex numbers. For instance:
but, rather,
Since the rule strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.[29]
Anon-nested radical expression is said to be insimplified form if no factor of the radicand can be written as a power greater than or equal to the index; there are no fractions inside the radical sign; and there are no radicals in the denominator.[30]
For example, to write the radical expression in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:
Next, there is a fraction under the radical sign, which we change as follows:
Finally, we remove the radical from the denominator as follows:
When there is a denominator involving surds it is always possible to find a factor to multiply both numerator and denominator by to simplify the expression.[31][32] For instance using thefactorization of the sum of two cubes:
Simplifying radical expressions involvingnested radicals can be quite difficult. In particular, denesting is not always possible, and when possible, it may involve advancedGalois theory. Moreover, when complete denesting is impossible, there is no generalcanonical form such that the equality of two numbers can be tested by simply looking at their canonical expressions.
For example, it is not obvious that
The above can be derived through:
Let, withp andq coprime and positive integers. Then is rational if and only if both and are integers, which means that bothp andq arenth powers of some integer.
Thenth root of a positive real numberA can be computed withNewton's method, which starts with an initial guessx0, which is also a positive real number, and then iterates using therecurrence relation[35]
until the desired precision is reached. For computational efficiency, the recurrence relation can be rewritten[35]
This allows the relation to only have oneexponentiation, which is computed once for each iteration. Thenth root ofx can then be defined as thelimit of ask approaches infinity.
For example, to find the fifth root of 34, we plug inn = 5,A = 34 andx0 = 2 (initial guess). The first 5 iterations are, approximately:
A suitable initial guess for Newton's method may need to be identified using thebisection method ormethod of false position.[36] For large values ofn and higher requirements for precision, a more rapid algorithm than Newton's method for finding thenth root is to use a truncatedTaylor series with aPadé approximant.[37]
The technique of François Viète, published c. 1600, can be used to perform digit-by-digit calculation of principal roots of decimal (base 10) numbers.[38] This method is based on thebinomial theorem and is essentially an inverse algorithm solvingwhere, thebinomial coefficient, is thekth entry on thenth row ofPascal's triangle.
To compute the root of a numberC, choose a series of approximationsthat satisfy, where the difference between and is the next digit in the approximation. Thedecimal fraction is chosen to be the largest number with a singlesignificant digit that satisfiesthen per the binomial theoremThe term is just a multiple of theith remainder,.
Using this expression, any positive principal root can be computed, digit-by-digit, as follows.
Write the original number in decimal form. The numbers are written similar to thelong division algorithm, and, as in long division, the root will be written on the line above. Now separate the digits into groups of digits equating to the root being taken, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the radicand. One digit of the root will appear above each group of digits of the original number.
Beginning with the left-most group of digits, do the following procedure for each group:
Starting on the left, bring down the most significant (leftmost) group of digits not yet used (if all the digits have been used, write "0" the number of times required to make a group) and write them to the right of the remainder from the previous step (on the first step, there will be no remainder). In other words, multiply the remainder by and add the digits from the next group. This will be thecurrent valuec.
Findp andx, as follows:
Let be thepart of the root found so far, ignoring any decimal point. (For the first step, and).
Determine the greatest digit such that.
Place the digit as the next digit of the root, i.e., above the group of digits you just brought down. Thus the nextp will be the oldp times 10 plusx.
Subtract from to form a new remainder.
If the remainder is zero and there are no more digits to bring down, then the algorithm has terminated. Otherwise go back to step 1 for another iteration.
The principalnth root of a positive number can be computed usinglogarithms. Starting from the equation that definesr as annth root ofx, namely withx positive and therefore its principal rootr also positive, one takes logarithms of both sides (anybase of the logarithm will do) to obtain
The rootr is recovered from this by taking theantilog:[39]
(Note: That formula showsb raised to the power of the result of the division, notb multiplied by the result of the division.)
For the case in whichx is negative andn is odd, there is one real rootr which is also negative. This can be found by first multiplying both sides of the defining equation by −1 to obtain then proceeding as before to find |r|, and usingr = −|r|.
The two square roots of a complex number are always negatives of each other.[40] For example, the square roots of−4 are2i and−2i, and the square roots ofi are
If we express a complex number inpolar form, then the square root can be obtained by taking the square root of the radius and halving the angle:[41]
Aprincipal root of a complex number may be chosen in various ways, for example
Using the first(last) branch cut the principal square root maps to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed inmathematical software likeMatlab orScilab.
The number 1 hasn differentnth roots in the complex plane,[24] namely
where
These roots are evenly spaced around theunit circle in the complex plane, at angles which are multiples of. For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1,, −1, and. As a result of this symmetry, the sum of thenth roots of unity equals zero.[42]
Geometric representation of the 2nd to 6th roots of a complex numberz, in polar formreiφ wherer = |z | andφ = argz. Ifz is real,φ = 0 orπ. Principal roots are shown in black.
Every complex number hasn differentnth roots in the complex plane.[24] These are
whereη is a singlenth root, and 1, ω, ω2, ... ωn−1 are thenth roots of unity. Thus, since they are all just multiplied by the same scalarη, the sum of thenth roots equals zero.[42] For example, the four different fourth roots of 2 are
Herer is the magnitude (the modulus, also called theabsolute value) of the number whose root is to be taken; if the number can be written as then. The is the angle formed as one pivots on the origin counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the properties that
Thus findingnth roots in the complex plane can be segmented into two steps. First, the magnitude of all thenth roots is thenth root of the magnitude of the original number. Second, the angle between the positive horizontal axis and a ray from the origin to one of thenth roots is, where is the angle defined in the same way for the number whose root is being taken. Furthermore, alln of thenth roots are at equally spaced angles from each other, as proven by thenth root theorem[44]
Ifn is even, a complex number'snth roots, of which there are an even number, come inadditive inverse pairs, so that if a numberr1 is one of thenth roots thenr2 = −r1 is another. This is because raising the latter's coefficient −1 to thenth power for evenn yields 1: that is, (−r1)n = (−1)n ×r1n =r1n.
As with square roots, the formula above does not define acontinuous function over the entire complex plane, but instead has abranch cut at points whereθ / n is discontinuous.
Aroot of apolynomial is a number such that. Annth root of a number is by definition a root of the polynomial.Algebraic numbers are the numbers that are polynomial roots.
During the two next centuries, a considerable effort was devoted to the question of whether every algebraic number can be expressed in terms of radicals. In 1824, the proof of theAbel–Ruffini theorem showed that there is nogeneral formula for the degree 5.[46] This did not completely exclude the possibility of expressing polynomial roots in terms of radicals with formulas depending on each specific polynomial. For example, thequintic polynomial
In spite of this obstacle,Demoivre's theorem demonstrates that annth root of a number can always be extracted, even for aquintic function.[43] The two following results, proved in the 19th century resolve fundamental problems on polynomial roots that were set in the 17th century. Thefundamental theorem of algebra asserts that every polynomial hascomplex roots.[19] There are numbers, calledtranscendental numbers that are not polynomial roots. The numberπ is an example of such a transcendental number.[48]
It may be unclear why any number hasn roots rather than just a primary root. To demonstrate this, for the principal roota of the variablex taken to thenth power, the following polynomial relation holds:This polynomial can befactored as follows:[49]Thus, the polynomial is zero forx equal toa, or for anyx that solves the equation:[50]By the fundamental theorem of algebra, this series has roots, for a combined total of.
As an example, let and, then find the cube roots of 1[51]
Thus the first root is, and the other two roots can be derived using thequadratic equation with[52]
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^Caviness, B. F.; Fateman, R. J."Simplification of Radical Expressions"(PDF).Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation. p. 329.
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^Chen, S.-G.; Hsieh, P. Y. (1989). "Fast computation of the Nth root".Computers & Mathematics with Applications.17 (10):1423–1427.doi:10.1016/0898-1221(89)90024-2.
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