Template:Short description Template:About Template:Cs1 config Template:More citations needed In mathematics, an Template:Mvarth root of a number Template:Mvar is a number Template:Mvar which, when raised to the power of Template:Mvar, yields Template:Mvar: <math display="block">r^n = \underbrace{r \times r \times \dotsb \times r}_{n\text{ factors}} = x.</math>
The positive integer Template:Mvar is called the index or degree, and the number Template:Mvar of which the root is taken is the radicand. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an Template:Mvarth root is a root extraction.
For example, Template:Math is a square root of Template:Math, since Template:Math, and Template:Math is also a square root of Template:Math, since Template:Math.
The Template:Mvarth root of Template:Mvar is written as <math>\sqrt[n]{x}</math> using the radical symbol <math>\sqrt{\phantom x}</math>. The square root is usually written as Template:Tmath, with the degree omitted. Taking the Template:Mvarth root of a number, for fixed Template:Tmath, is the inverse of raising a number to the Template:Mvarth power,<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> and can be written as a fractional exponent:
<math display="block">\sqrt[n]{x} = x^{1/n}.</math>
For a positive real number Template:Mvar, <math>\sqrt{x}</math> denotes the positive square root of Template:Mvar and <math>\sqrt[n]{x}</math> denotes the positive real Template:Mvarth root. A negative real number Template:Math has no real-valued square roots, but when Template:Mvar is treated as a complex number it has two imaginary square roots, Template:Tmath and Template:Tmath, where Template:Mvar is the imaginary unit.
In general, any non-zero complex number has Template:Mvar distinct complex-valued Template:Mvarth roots, equally distributed around a complex circle of constant absolute value. (The Template:Mvarth root of Template:Math is zero with multiplicity Template:Mvar, and this circle degenerates to a point.) Extracting the Template:Mvarth roots of a complex number Template:Mvar can thus be taken to be a multivalued function. By convention the principal value of this function, called the principal root and denoted Template:Tmath, is taken to be the Template:Mvarth root with the greatest real part and in the special case when Template:Mvar is a negative real number, the one with a positive imaginary part. The principal root of a positive real number is thus also a positive real number. As a function, the principal root is continuous in the whole complex plane, except along the negative real axis.
An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd<ref>Template:Cite book</ref> or a radical.<ref name=silver>Template:Cite book</ref> 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 it is called an algebraic expression.
Template:Arithmetic operations
Roots are used for determining the radius of convergence of a power series with the root test. The Template:Mvarth roots of 1 are called roots of unity and play a fundamental role in various areas of mathematics, such as number theory, theory of equations, and Fourier transform.
HistoryEdit
Template:Main article An archaic term for the operation of taking nth roots is radication.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Definition and notationEdit
none of which are real
one of which is a negative real
An Template:Mvarth root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x:
<math display="block">r^n = x.</math>
Every positive real number x has a single positive nth root, called the principal nth root, which is written <math>\sqrt[n]{x}</math>. For n equal to 2 this is called the principal square root and the n is omitted. The nth root can also be represented using exponentiation as x1/n.
For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nth root. For odd values of n, every negative number x has a real negative nth root. For example, −2 has a real 5th root, <math>\sqrt[5]{-2} = -1.148698354\ldots</math> but −2 does not have any real 6th roots.
Every non-zero number x, real or complex, has n different complex number nth roots. (In the case x is real, this count includes any real nth roots.) The only complex root of 0 is 0.
The nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two nth powers) are irrational. For example,
<math display="block">\sqrt{2} = 1.414213562\ldots</math>
All nth roots of rational numbers are algebraic numbers, and all nth roots of integers are algebraic integers.
The term "surd" traces back to Al-Khwarizmi (Template:Circa), who referred to rational and irrational numbers as audible and inaudible, respectively. This later led to the Arabic word {{#invoke:Lang|lang}} ({{#invoke:Lang|lang}}, meaning "deaf" or "dumb") for irrational number being translated into Latin as {{#invoke:Lang|lang}} (meaning "deaf" or "mute"). Gerard of Cremona (Template:Circa), Fibonacci (1202), and then Robert Recorde (1551) all used the term to refer to unresolved irrational roots, that is, expressions of the form <math>\sqrt[n]{r}</math>, in which <math>n</math> and <math>r</math> are integer numerals and the whole expression denotes an irrational number.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Irrational numbers of the form <math>\pm\sqrt{a},</math> where <math>a</math> is rational, are called pure quadratic surds; irrational numbers of the form <math>a \pm\sqrt{b}</math>, where <math>a</math> and <math>b</math> are rational, are called mixed quadratic surds.<ref>Template:Cite book</ref>
Square rootsEdit
A square root of a number x is a number r which, when squared, becomes x:
<math display="block">r^2 = x.</math>
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:
<math display="block">\sqrt{25} = 5.</math>
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 5i and −5i, where i represents a number whose square is Template:Math.
Cube rootsEdit
A cube root of a number x is a number r whose cube is x:
<math display="block">r^3 = x.</math>
Every real number x has exactly one real cube root, written <math>\sqrt[3]{x}</math>. For example,
<math display="block">\begin{align} \sqrt[3]{8} &= 2\\ \sqrt[3]{-8} &= -2. \end{align}</math>
Every real number has two additional complex cube roots.
Identities and propertiesEdit
Expressing the degree of an nth root in its exponent form, as in <math>x^{1/n}</math>, makes it easier to manipulate powers and roots. If <math>a</math> is a non-negative real number,
<math display="block">\sqrt[n]{a^m} = (a^m)^{1/n} = a^{m/n} = (a^{1/n})^m = (\sqrt[n]a)^m.</math>
Every non-negative number has exactly one non-negative real nth root, and so the rules for operations with surds involving non-negative radicands <math>a</math> and <math>b</math> are straightforward within the real numbers:
<math display="block">\begin{align}
\sqrt[n]{ab} &= \sqrt[n]{a} \sqrt[n]{b} \\
\sqrt[n]{\frac{a}{b}} &= \frac{\sqrt[n]{a}}{\sqrt[n]{b}}
\end{align}</math>
Subtleties can occur when taking the nth roots of negative or complex numbers. For instance:
<math display="block">\sqrt{-1}\times\sqrt{-1} \neq \sqrt{-1 \times -1} = 1,\quad</math>
but, rather,
<math display="block">\quad\sqrt{-1}\times\sqrt{-1} = i \times i = i^2 = -1.</math>
Since the rule <math>\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab} </math> strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above.
Simplified form of a radical expressionEdit
A non-nested radical expression is said to be in simplified 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.<ref>Template:Cite book</ref>
For example, to write the radical expression <math>\textstyle \sqrt{32/5}</math> in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:
<math display="block"> \sqrt{\frac{32}{5}} = \sqrt{\frac{16 \cdot 2}{5}} = \sqrt{16} \cdot \sqrt{\frac{2}{5}} = 4 \sqrt{\frac{2}{5}} </math>
Next, there is a fraction under the radical sign, which we change as follows:
<math display="block">4 \sqrt{\frac{2}{5}} = \frac{4 \sqrt{2}}{\sqrt{5}}</math>
Finally, we remove the radical from the denominator as follows:
<math display="block">\frac{4 \sqrt{2}}{\sqrt{5}} = \frac{4 \sqrt{2}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{4 \sqrt{10}}{5} = \frac{4}{5}\sqrt{10}</math>
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.<ref>Template:Cite conference</ref><ref>Template:Cite journal</ref> For instance using the factorization of the sum of two cubes:
<math display="block">
\frac{1}{\sqrt[3]{a} + \sqrt[3]{b}} =
\frac{\sqrt[3]{a^2} - \sqrt[3]{ab} + \sqrt[3]{b^2}}{\left(\sqrt[3]{a} + \sqrt[3]{b}\right)\left(\sqrt[3]{a^2} - \sqrt[3]{ab} + \sqrt[3]{b^2}\right)} =
\frac{\sqrt[3]{a^2} - \sqrt[3]{ab} + \sqrt[3]{b^2}}{a + b} .
</math>
Simplifying radical expressions involving nested radicals can be quite difficult. In particular, denesting is not always possible, and when possible, it may involve advanced Galois theory. Moreover, when complete denesting is impossible, there is no general canonical 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
<math display="block">\sqrt{3 + 2\sqrt{2}} = 1 + \sqrt{2}.</math>
The above can be derived through:
<math display="block">\sqrt{3 + 2\sqrt{2}} = \sqrt{1 + 2\sqrt{2} + 2} = \sqrt{1^2 + 2\sqrt{2} + \sqrt{2}^2} = \sqrt{\left(1 + \sqrt{2}\right)^2} = 1 + \sqrt{2}</math>
Let <math>r=p/q</math>, with Template:Mvar and Template:Mvar coprime and positive integers. Then <math>\sqrt[n]r = \sqrt[n]{p}/\sqrt[n]{q}</math> is rational if and only if both <math>\sqrt[n]{p}</math> and <math>\sqrt[n]{q}</math> are integers, which means that both Template:Mvar and Template:Mvar are nth powers of some integer.
Infinite seriesEdit
The radical or root may be represented by the infinite series:
<math display="block">(1+x)^\frac{s}{t} = \sum_{n=0}^\infty \frac{\prod_{k=0}^{n-1} (s-kt)}{n!t^n}x^n</math>
with <math>|x|<1</math>. This expression can be derived from the binomial series.
Computing principal rootsEdit
Using Newton's methodEdit
The Template:Mvarth root of a number Template:Math can be computed with Newton's method, which starts with an initial guess Template:Math and then iterates using the recurrence relation
<math display="block">x_{k+1} = x_k-\frac{x_k^n-A}{nx_k^{n-1}}</math>
until the desired precision is reached. For computational efficiency, the recurrence relation is commonly rewritten
<math display="block">x_{k+1} = \frac{n-1}{n}\,x_k+\frac{A}{n}\,\frac 1{x_k^{n-1}}.</math>
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 Template:Math and Template:Math (initial guess). The first 5 iterations are, approximately:
Template:Block indent Template:Block indent Template:Block indent Template:Block indent Template:Block indent Template:Block indent
(All correct digits shown.)
The approximation Template:Math is accurate to 25 decimal places and Template:Math is good for 51.
Newton's method can be modified to produce various generalized continued fractions for the nth root. For example,
<math display="block">
\sqrt[n]{z} = \sqrt[n]{x^n+y} = x+\cfrac{y} {nx^{n-1}+\cfrac{(n-1)y} {2x+\cfrac{(n+1)y} {3nx^{n-1}+\cfrac{(2n-1)y} {2x+\cfrac{(2n+1)y} {5nx^{n-1}+\cfrac{(3n-1)y} {2x+\ddots}}}}}}.
</math>
Digit-by-digit calculation of principal roots of decimal (base 10) numbersEdit
Building on the digit-by-digit calculation of a square root, it can be seen that the formula used there, <math>x(20p + x) \le c</math>, or <math>x^2 + 20xp \le c</math>, follows a pattern involving Pascal's triangle. For the nth root of a number <math>P(n,i)</math> is defined as the value of element <math>i</math> in row <math>n</math> of Pascal's Triangle such that <math>P(4,1) = 4</math>, we can rewrite the expression as <math>\sum_{i=0}^{n-1}10^i P(n,i)p^i x^{n-i}</math>. For convenience, call the result of this expression <math>y</math>. 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 <math>10^n</math> and add the digits from the next group. This will be the current value c.
- Find p and x, as follows:
- Let <math>p</math> be the part of the root found so far, ignoring any decimal point. (For the first step, <math>p = 0</math> and <math>0^0 = 1</math>).
- Determine the greatest digit <math>x</math> such that <math>y \le c</math>.
- Place the digit <math>x</math> 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 <math>y</math> from <math>c</math> 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.
ExamplesEdit
Find the square root of 152.2756.
1 2. 3 4
/
\/ 01 52.27 56 (Results) (Explanations)
01 x = 1 100·1·00·12 + 101·2·01·11 ≤ 1 < 100·1·00·22 + 101·2·01·21
01 y = 1 y = 100·1·00·12 + 101·2·01·11 = 1 + 0 = 1
00 52 x = 2 100·1·10·22 + 101·2·11·21 ≤ 52 < 100·1·10·32 + 101·2·11·31
00 44 y = 44 y = 100·1·10·22 + 101·2·11·21 = 4 + 40 = 44
08 27 x = 3 100·1·120·32 + 101·2·121·31 ≤ 827 < 100·1·120·42 + 101·2·121·41
07 29 y = 729 y = 100·1·120·32 + 101·2·121·31 = 9 + 720 = 729
98 56 x = 4 100·1·1230·42 + 101·2·1231·41 ≤ 9856 < 100·1·1230·52 + 101·2·1231·51
98 56 y = 9856 y = 100·1·1230·42 + 101·2·1231·41 = 16 + 9840 = 9856
00 00
Algorithm terminates: Answer is 12.34
Find the cube root of 4192 truncated to the nearest thousandth.
1 6. 1 2 4
3 /
\/ 004 192.000 000 000 (Results) (Explanations)
004 x = 1 100·1·00·13 + 101·3·01·12 + 102·3·02·11 ≤ 4 < 100·1·00·23 + 101·3·01·22 + 102·3·02·21
001 y = 1 y = 100·1·00·13 + 101·3·01·12 + 102·3·02·11 = 1 + 0 + 0 = 1
003 192 x = 6 100·1·10·63 + 101·3·11·62 + 102·3·12·61 ≤ 3192 < 100·1·10·73 + 101·3·11·72 + 102·3·12·71
003 096 y = 3096 y = 100·1·10·63 + 101·3·11·62 + 102·3·12·61 = 216 + 1,080 + 1,800 = 3,096
096 000 x = 1 100·1·160·13 + 101·3·161·12 + 102·3·162·11 ≤ 96000 < 100·1·160·23 + 101·3·161·22 + 102·3·162·21
077 281 y = 77281 y = 100·1·160·13 + 101·3·161·12 + 102·3·162·11 = 1 + 480 + 76,800 = 77,281
018 719 000 x = 2 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 ≤ 18719000 < 100·1·1610·33 + 101·3·1611·32 + 102·3·1612·31
015 571 928 y = 15571928 y = 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 = 8 + 19,320 + 15,552,600 = 15,571,928
003 147 072 000 x = 4 100·1·16120·43 + 101·3·16121·42 + 102·3·16122·41 ≤ 3147072000 < 100·1·16120·53 + 101·3·16121·52 + 102·3·16122·51
The desired precision is achieved. The cube root of 4192 is 16.124...
Logarithmic calculationEdit
The principal nth root of a positive number can be computed using logarithms. Starting from the equation that defines r as an nth root of x, namely <math>r^n=x,</math> 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
<math display="block">n \log_b r = \log_b x \quad \quad \text{hence} \quad \quad \log_b r = \frac{\log_b x}{n}.</math>
The root r is recovered from this by taking the antilog:
<math display="block">r = b^{\frac{1}{n}\log_b x}.</math>
(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 <math>|r|^n = |x|,</math> then proceeding as before to find |r|, and using Template:Nowrap.
Geometric constructibilityEdit
The ancient Greek mathematicians 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 proved that an nth root of a given length cannot be constructed if n is not a power of 2.<ref>Template:Citation</ref>
Complex rootsEdit
Every complex number other than 0 has n different nth roots.
Square rootsEdit
The two square roots of a complex number are always negatives of each other. For example, the square roots of Template:Math are Template:Math and Template:Math, and the square roots of Template:Math are
<math display="block">\tfrac{1}{\sqrt{2}}(1 + i) \quad\text{and}\quad -\tfrac{1}{\sqrt{2}}(1 + i).</math>
If we express a complex number in polar form, then the square root can be obtained by taking the square root of the radius and halving the angle:
<math display="block">\sqrt{re^{i\theta}} = \pm\sqrt{r} \cdot e^{i\theta/2}.</math>
A principal root of a complex number may be chosen in various ways, for example
<math display="block">\sqrt{re^{i\theta}} = \sqrt{r} \cdot e^{i\theta/2}</math>
which introduces a branch cut in the complex plane along the positive real axis with the condition Template:Math, or along the negative real axis with Template:Math.
Using the first(last) branch cut the principal square root <math>\scriptstyle \sqrt z</math> maps <math>\scriptstyle z</math> to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed in mathematical software like Matlab or Scilab.
Roots of unityEdit
The number 1 has n different nth roots in the complex plane, namely
<math display="block">1,\;\omega,\;\omega^2,\;\ldots,\;\omega^{n-1},</math>
where
<math display="block">\omega = e^\frac{2\pi i}{n} = \cos\left(\frac{2\pi}{n}\right) + i\sin\left(\frac{2\pi}{n}\right).</math>
These roots are evenly spaced around the unit circle in the complex plane, at angles which are multiples of <math>2\pi/n</math>. For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, <math>i</math>, −1, and <math>-i</math>.
nth rootsEdit
Template:Visualisation complex number roots.svg Every complex number has n different nth roots in the complex plane. These are
<math display="block">\eta,\;\eta\omega,\;\eta\omega^2,\;\ldots,\;\eta\omega^{n-1},</math>
where η is a single nth root, and 1, ω, ω2, ... ωn−1 are the nth roots of unity. For example, the four different fourth roots of 2 are
<math display="block">\sqrt[4]{2},\quad i\sqrt[4]{2},\quad -\sqrt[4]{2},\quad\text{and}\quad -i\sqrt[4]{2}.</math>
In polar form, a single nth root may be found by the formula
<math display="block">\sqrt[n]{re^{i\theta}} = \sqrt[n]{r} \cdot e^{i\theta/n}.</math>
Here r is the magnitude (the modulus, also called the absolute value) of the number whose root is to be taken; if the number can be written as a+bi then <math>r=\sqrt{a^2+b^2}</math>. Also, <math>\theta</math> 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 <math>\cos \theta = a/r,</math> <math> \sin \theta = b/r,</math> and <math> \tan \theta = b/a.</math>
Thus finding nth roots in the complex plane can be segmented into two steps. First, the magnitude of all the nth roots is the nth 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 nth roots is <math>\theta / n</math>, where <math>\theta</math> is the angle defined in the same way for the number whose root is being taken. Furthermore, all n of the nth roots are at equally spaced angles from each other.
If n is even, a complex number's nth roots, of which there are an even number, come in additive inverse pairs, so that if a number r1 is one of the nth roots then r2 = −r1 is another. This is because raising the latter's coefficient −1 to the nth power for even n yields 1: that is, (−r1)n = (−1)n × r1n = r1n.
As with square roots, the formula above does not define a continuous function over the entire complex plane, but instead has a branch cut at points where θ / n is discontinuous.
Solving polynomialsEdit
It was once conjectured that all polynomial equations could be solved algebraically (that is, that all roots of a polynomial 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
<math display="block">x^5 = x + 1</math>
cannot be expressed in terms of radicals. (cf. quintic equation)
Proof of irrationality for non-perfect nth power xEdit
Assume that <math>\sqrt[n]{x}</math> is rational. That is, it can be reduced to a fraction <math>\frac{a}{b}</math>, where Template:Mvar and Template:Mvar are integers without a common factor.
This means that <math>x = \frac{a^n}{b^n}</math>.
Since x is an integer, <math>a^n</math>and <math>b^n</math>must share a common factor if <math>b \neq 1</math>. This means that if <math>b \neq 1</math>, <math>\frac{a^n}{b^n}</math> is not in simplest form. Thus b should equal 1.
Since <math>1^n = 1</math> and <math>\frac{n}{1} = n</math>, <math>\frac{a^n}{b^n} = a^n</math>.
This means that <math>x = a^n</math> and thus, <math>\sqrt[n]{x} = a</math>. This implies that <math>\sqrt[n]{x}</math> is an integer. Since Template:Mvar is not a perfect Template:Mvarth power, this is impossible. Thus <math>\sqrt[n]{x}</math> is irrational.