TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a radicand, also known as an nth root, of a
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ca ...
''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'': :$r^n = x,$ where ''n'' is a
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
, sometimes called the ''degree'' of the root. A root of degree 2 is called a ''
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
'' and a root of degree 3, a ''
cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. ...
''. Roots of higher degree are referred by using ordinal numbers, as in ''fourth root'', ''twentieth root'', etc. The computation of an th root is a root extraction. For example, 3 is a square root of 9, since 3 = 9, and −3 is also a square root of 9, since (−3) = 9. Any non-zero number considered as a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
has different complex th roots, including the real ones (at most two). The th root of 0 is zero for all
positive integer In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s , since . In particular, if is even and is a positive real number, one of its th roots is real and positive, one is negative, and the others (when ) are non-real
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s; if is even and is a negative real number, none of the th roots is real. If is odd and is real, one th root is real and has the same sign as , while the other () roots are not real. Finally, if is not real, then none of its th roots are real. Roots of real numbers are usually written using the radical symbol or ''radix'' $\sqrt$, with $\sqrt$ denoting the positive square root of if is positive; for higher roots,
principal value In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
. The common choice is to choose the principal th root of as the th root with the greatest real part, and when there are two (for real and negative), the one with a positive
imaginary part In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
. This makes the th root a function that is real and positive for real and positive, and is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
in the whole
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
, except for values of that are real and negative. A difficulty with this choice is that, for a negative real number and an odd index, the principal th root is not the real one. For example, $-8$ has three cube roots, $-2$, $1 + i\sqrt$ and $1 - i\sqrt.$ The real cube root is $-2$ and the principal cube root is $1 + i\sqrt.$ An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd or a radical. Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a ''radical expression'', and if it contains no transcendental functions or
transcendental numbers In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
it is called an ''
algebraic expression In mathematics, an algebraic expression is an expression built up from integer constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For ...
''. Roots can also be defined as special cases of exponentiation, where the
exponent Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
is a
fraction A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
: :
Roots are used for determining the
radius of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
of a
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
with the
root test In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity :\limsup_\sqrt where a_n are the terms of the series, and states that the series converges absolutely if ...
. The th roots of 1 are called
roots of unity In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important ...
and play a fundamental role in various areas of mathematics, such as
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, theory of equations, and
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
.

# History

An archaic term for the operation of taking ''n''th roots is ''radication''.

# Definition and notation

An ''n''th root of a number ''x'', where ''n'' is a positive integer, is any of the ''n'' real or complex numbers ''r'' whose ''n''th power is ''x'': :$r^n = x.$ Every positive
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
''x'' has a single positive ''n''th root, called the principal ''n''th root, which is written
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
, has ''n'' different complex number ''n''th roots. (In the case ''x'' is real, this count includes any real ''n''th roots.) The only complex root of 0 is 0. The ''n''th roots of almost all numbers (all integers except the ''n''th powers, and all rationals except the quotients of two ''n''th powers) are
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
. For example, :$\sqrt = 1.414213562\ldots$ All ''n''th roots of rational numbers are
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the ...
s, and all ''n''th roots of integers are
algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficient ...
s. The term "surd" traces back to al-Khwārizmī (c. 825), who referred to rational and irrational numbers as ''audible'' and ''inaudible'', respectively. This later led to the Arabic word "" (''asamm'', meaning "deaf" or "dumb") for ''irrational number'' being translated into Latin as ''surdus'' (meaning "deaf" or "mute"). Gerard of Cremona (c. 1150),
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western ...
(1202), and then
Robert Recorde Robert Recorde () was an Anglo-Welsh physician and mathematician. He invented the equals sign (=) and also introduced the pre-existing plus sign (+) to English speakers in 1557. Biography Born around 1512, Robert Recorde was the second and last ...
(1551) all used the term to refer to ''unresolved irrational roots'', that is, expressions of the form in which $n$ and $i$ are integer numerals and the whole expression denotes an irrational number. Quadratic irrational numbers, that is, irrational numbers of the form $\sqrt,$ are also known as "quadratic surds".

## Square roots

A square root of a number ''x'' is a number ''r'' which, when squared, becomes ''x'': :$r^2 = x.$ 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 the principal square root, and is denoted with a radical sign: :$\sqrt = 5.$ Since the square of every real number is nonnegative, negative numbers do not have real square roots. However, for every negative real number there are two imaginary square roots. For example, the square roots of −25 are 5''i'' and −5''i'', where '' i'' represents a number whose square is .

## Cube roots

A cube root of a number ''x'' is a number ''r'' whose
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
is ''x'': :$r^3 = x.$ Every real number ''x'' has exactly one real cube root, written
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
cube roots.

# Identities and properties

Expressing the degree of an ''n''th root in its exponent form, as in $x^$, makes it easier to manipulate powers and roots. If $a$ is a non-negative real number, : Every non-negative number has exactly one non-negative real ''n''th root, and so the rules for operations with surds involving non-negative radicands $a$ and $b$ are straightforward within the real numbers: : Subtleties can occur when taking the ''n''th roots of negative or
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s. For instance: :$\sqrt\times\sqrt \neq \sqrt = 1,\quad$ but, rather, $\quad\sqrt\times\sqrt = i \times i = i^2 = -1.$ Since the rule strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.

# Simplified form of a radical expression

A non-nested radical expression is said to be in simplified form if # There is no factor of the radicand that can be written as a power greater than or equal to the index. # There are no fractions under the radical sign. # There are no radicals in the denominator. For example, to write the radical expression $\sqrt$ in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it: :$\sqrt = \sqrt = \sqrt \cdot \sqrt = 4 \sqrt$ Next, there is a fraction under the radical sign, which we change as follows: :$4 \sqrt = \frac$ Finally, we remove the radical from the denominator as follows: :$\frac = \frac \cdot \frac = \frac = \frac\sqrt$ 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. For instance using the factorization of the sum of two cubes: :$\frac = \frac = \frac .$ Simplifying radical expressions involving nested radicals can be quite difficult. It is not obvious for instance that: :$\sqrt = 1 + \sqrt$ The above can be derived through: :$\sqrt = \sqrt = \sqrt = \sqrt = 1 + \sqrt$ Let $r=p/q$, with and coprime and positive integers. Then

# Infinite series

The radical or root may be represented by the
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
: :$\left(1+x\right)^\frac = \sum_^\infty \fracx^n$ with $, x, <1$. This expression can be derived from the
binomial series In mathematics, the binomial series is a generalization of the polynomial that comes from a binomial formula expression like (1+x)^n for a nonnegative integer n. Specifically, the binomial series is the Taylor series for the function f(x)=(1+ ...
.

# Computing principal roots

## Using Newton's method

The th root of a number can be computed with
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a rea ...
, which starts with an initial guess and then iterates using the
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:$x_ = x_k-\frac$ until the desired precision is reached. For computational efficiency, the recurrence relation is commonly rewritten :$x_ = \frac\,x_k+\frac\,\frac 1.$ This allows to have only one exponentiation, and to compute once for all the first factor of each term. For example, to find the fifth root of 34, we plug in and (initial guess). The first 5 iterations are, approximately:

(All correct digits shown.) The approximation is accurate to 25 decimal places and is good for 51. Newton's method can be modified to produce various generalized continued fractions for the ''n''th root. For example, :

## Digit-by-digit calculation of principal roots of decimal (base 10) numbers

Building on the digit-by-digit calculation of a square root, it can be seen that the formula used there, $x\left(20p + x\right) \le c$, or $x^2 + 20xp \le c$, follows a pattern involving Pascal's triangle. For the ''n''th root of a number $P\left(n,i\right)$ is defined as the value of element $i$ in row $n$ of Pascal's Triangle such that $P\left(4,1\right) = 4$, we can rewrite the expression as $\sum_^10^i P\left(n,i\right)p^i x^$. For convenience, call the result of this expression $y$. Using this more general 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 the long 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 $10^n$ and add the digits from the next group. This will be the current value ''c''. # Find ''p'' and ''x'', as follows: #* Let $p$ be the part of the root found so far, ignoring any decimal point. (For the first step, $p = 0$). #* Determine the greatest digit $x$ such that $y \le c$. #* Place the digit $x$ as the next digit of the root, i.e., above the group of digits you just brought down. Thus the next ''p'' will be the old ''p'' times 10 plus ''x''. # Subtract $y$ from $c$ 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.

### Examples

Find the square root of 152.2756. 1 2. 3 4 / \/ 01 52.27 56 01 10·1·0·1 + 10·2·0·1 ≤ 1 < 10·1·0·2 + 10·2·0·2 x = 1 01 y = 10·1·0·1 + 10·2·0·1 = 1 + 0 = 1 00 52 10·1·1·2 + 10·2·1·2 ≤ 52 < 10·1·1·3 + 10·2·1·3 x = 2 00 44 y = 10·1·1·2 + 10·2·1·2 = 4 + 40 = 44 08 27 10·1·12·3 + 10·2·12·3 ≤ 827 < 10·1·12·4 + 10·2·12·4 x = 3 07 29 y = 10·1·12·3 + 10·2·12·3 = 9 + 720 = 729 98 56 10·1·123·4 + 10·2·123·4 ≤ 9856 < 10·1·123·5 + 10·2·123·5 x = 4 98 56 y = 10·1·123·4 + 10·2·123·4 = 16 + 9840 = 9856 00 00 Algorithm terminates: Answer is 12.34 Find the cube root of 4192 to the nearest hundredth. 1 6. 1 2 4 3 / \/ 004 192.000 000 000 004 10·1·0·1 + 10·3·0·1 + 10·3·0·1 ≤ 4 < 10·1·0·2 + 10·3·0·2 + 10·3·0·2 x = 1 001 y = 10·1·0·1 + 10·3·0·1 + 10·3·0·1 = 1 + 0 + 0 = 1 003 192 10·1·1·6 + 10·3·1·6 + 10·3·1·6 ≤ 3192 < 10·1·1·7 + 10·3·1·7 + 10·3·1·7 x = 6 003 096 y = 10·1·1·6 + 10·3·1·6 + 10·3·1·6 = 216 + 1,080 + 1,800 = 3,096 096 000 10·1·16·1 + 10·3·16·1 + 10·3·16·1 ≤ 96000 < 10·1·16·2 + 10·3·16·2 + 10·3·16·2 x = 1 077 281 y = 10·1·16·1 + 10·3·16·1 + 10·3·16·1 = 1 + 480 + 76,800 = 77,281 018 719 000 10·1·161·2 + 10·3·161·2 + 10·3·161·2 ≤ 18719000 < 10·1·161·3 + 10·3·161·3 + 10·3·161·3 x = 2 015 571 928 y = 10·1·161·2 + 10·3·161·2 + 10·3·161·2 = 8 + 19,320 + 15,552,600 = 15,571,928 003 147 072 000 10·1·1612·4 + 10·3·1612·4 + 10·3·1612·4 ≤ 3147072000 < 10·1·1612·5 + 10·3·1612·5 + 10·3·1612·5 x = 4 The desired precision is achieved: The cube root of 4192 is about 16.12

## Logarithmic calculation

The principal ''n''th root of a positive number can be computed using
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
s. Starting from the equation that defines ''r'' as an ''n''th root of ''x'', namely $r^n=x,$ with ''x'' positive and therefore its principal root ''r'' also positive, one takes logarithms of both sides (any base of the logarithm will do) to obtain :$n \log_b r = \log_b x \quad \quad \text \quad \quad \log_b r = \frac.$ The root ''r'' is recovered from this by taking the
antilog In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
: :$r = b^.$ (Note: That formula shows ''b'' raised to the power of the result of the division, not ''b'' multiplied by the result of the division.) For the case in which ''x'' is negative and ''n'' is odd, there is one real root ''r'' which is also negative. This can be found by first multiplying both sides of the defining equation by −1 to obtain $, r, ^n = , x, ,$ then proceeding as before to find , ''r'', , and using .

# Geometric constructibility

The
ancient Greek mathematicians Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek math ...
knew how to use 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 1837
Pierre Wantzel Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge. In a paper from 1837, Wantzel prov ...
proved that an ''n''th root of a given length cannot be constructed if ''n'' is not a power of 2..

# Complex roots

Every
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
other than 0 has ''n'' different ''n''th roots.

## Square roots

The two square roots of a complex number are always negatives of each other. For example, the square roots of are and , and the square roots of are :$\tfrac\left(1 + i\right) \quad\text\quad -\tfrac\left(1 + i\right).$ If we express a complex number in
polar form In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, then the square root can be obtained by taking the square root of the radius and halving the angle: :$\sqrt = \pm\sqrt \cdot e^.$ A ''principal'' root of a complex number may be chosen in various ways, for example :$\sqrt = \sqrt \cdot e^$ which introduces a
branch cut In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point ...
in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
along the positive real axis with the condition , or along the negative real axis with . Using the first(last) branch cut the principal square root $\scriptstyle \sqrt z$ maps $\scriptstyle z$ to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed in mathematical software like
Matlab MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
or
Scilab Scilab is a free and open-source, cross-platform numerical computational package and a high-level, numerically oriented programming language. It can be used for signal processing, statistical analysis, image enhancement, fluid dynamics simu ...
.

## Roots of unity

The number 1 has ''n'' different ''n''th roots in the complex plane, namely :$1,\;\omega,\;\omega^2,\;\ldots,\;\omega^,$ where :$\omega = e^\frac = \cos\left\left(\frac\right\right) + i\sin\left\left(\frac\right\right)$ These roots are evenly spaced around the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
in the complex plane, at angles which are multiples of $2\pi/n$. For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, $i$, −1, and $-i$.

## ''n''th roots

Every complex number has ''n'' different ''n''th roots in the complex plane. These are :$\eta,\;\eta\omega,\;\eta\omega^2,\;\ldots,\;\eta\omega^,$ where ''η'' is a single ''n''th root, and 1, ''ω'', ''ω'', ... ''ω'' are the ''n''th roots of unity. For example, the four different fourth roots of 2 are : In
polar form In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, a single ''n''th root may be found by the formula : Here ''r'' is the magnitude (the modulus, also called the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), a ...
) of the number whose root is to be taken; if the number can be written as ''a+bi'' then $r=\sqrt$. Also, $\theta$ 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 $\cos \theta = a/r,$ $\sin \theta = b/r,$ and $\tan \theta = b/a.$ Thus finding ''n''th roots in the complex plane can be segmented into two steps. First, the magnitude of all the ''n''th roots is the ''n''th 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 the ''n''th roots is $\theta / n$, where $\theta$ is the angle defined in the same way for the number whose root is being taken. Furthermore, all ''n'' of the ''n''th roots are at equally spaced angles from each other. If ''n'' is even, a complex number's ''n''th roots, of which there are an even number, come in
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (o ...
pairs, so that if a number ''r''1 is one of the ''n''th roots then ''r''2 = –''r''1 is another. This is because raising the latter's coefficient –1 to the ''n''th power for even ''n'' yields 1: that is, (–''r''1) = (–1) × ''r''1 = ''r''1. As with square roots, the formula above does not define a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
over the entire complex plane, but instead has a
branch cut In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point ...
at points where ''θ'' / ''n'' is discontinuous.

# Solving polynomials

It was once
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...
d that all
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s could be solved algebraically (that is, that all roots of a
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
could be expressed in terms of a finite number of radicals and elementary operations). However, while this is true for third degree polynomials ( cubics) and fourth degree polynomials ( quartics), the Abel–Ruffini theorem (1824) shows that this is not true in general when the degree is 5 or greater. For example, the solutions of the equation :$x^5 = x + 1$ cannot be expressed in terms of radicals. (''cf.''
quintic equation In algebra, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a ...
)

# Proof of irrationality for non-perfect ''n''th power ''x''

Assume that

* Shifting nth root algorithm *
Geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as th ...
*
Twelfth root of two The twelfth root of two or \sqrt 2/math> (or equivalently 2^) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio ( musical interval) of a ...
* Super-root