Wikiblog Cultuland [CZ]

In mathematics, an nth root of a number x, where n is usually assumed to be a positive integer, is a number r which, when raised to the power n yields x:

r^{n}=x,

where n is the degree of the root. 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.

For example:

3 is a square root of 9, since 3 = 9.
−3 is also a square root of 9, since (−3) = 9.

Any non-zero number, considered as complex number, has n different "complex roots of degree n" (nth roots), including those with zero imaginary part, i.e. any real roots. The root of 0 is zero for all degrees n, since 0 = 0. In particular, if n is even and x is a positive real number, one of its nth roots is positive, one is negative, and the rest (when n > 2) are complex but not real; if n is even and x is a negative real, none of the nth roots is real. If n is odd and x is real, one nth root is real and has the same sign as x, while the other (n − 1) roots are not real. Finally, if x is not real, then none of its nth roots is real.

Roots are usually written using the radical symbol or radix with

{\sqrt {x}}

denoting the principal square root of

x

,

{\sqrt[{3}]{x}}

denoting the principal cube root,

{\sqrt[{4}]{x}}

denoting the principal fourth root, and so on. In the expression

{\sqrt[{n}]{x}}

, n is called the index,

{\sqrt {\;}}

is the radical sign or radix, and

x

is called the radicand. Since the radical symbol denotes a function, it is defined to return only one result for a given argument

x

, which is called the principal nth root of

x

. Conventionally, a real root, preferably non-negative, if there is one, is designated as the principal nth root.

A complementary definition of principal root (though not formally defined or universally accepted) is to say that it is always the complex root that has the least value of the argument among all roots; here “argument” is bound to

[0,2\pi )

and means the counterclockwise angle in radian between the positive real axis and the line joining the complex number to the origin.

For example:

$-8$ has three cube roots: $-2$ , $1+i{\sqrt {3}}$ and $1-i{\sqrt {3}}$ with arguments $\pi,\;{\tfrac {1}{3}}\pi,\;{\tfrac {5}{3}}\pi,$ respectively. Of these, $1+i{\sqrt {3}}$ has the least argument and hence in some contexts is considered the principal cube root, while in other contexts $-2$ is said to be the principal cube root because it is the only real one.
$16$ has four fourth roots: $2,\;2i,\;-2,$ and $-2i$ , having arguments $0,\;{\tfrac {1}{2}}\pi,\;\pi$ and ${\tfrac {3}{2}}\pi$ respectively. So $2$ is always considered the unique principal fourth root, because it is a positive real, which necessarily has the least argument possible: $0$ .

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 it is called an algebraic expression.

In calculus, roots are treated as special cases of exponentiation, where the exponent is a fraction:

{\sqrt[{n}]{x}}=x^{\frac {1}{n}}.

Roots are particularly important in the theory of infinite series; the root test determines the radius of convergence of a power series. Roots can also be defined for complex numbers, and the complex roots of 1 (the roots of unity) play an important role in higher mathematics. Galois theory can be used to determine which algebraic numbers can be expressed using roots and to prove the Abel–Ruffini theorem, which states that a general polynomial equation of degree five or higher cannot be solved using roots alone; this result is also known as "the insolubility of the quintic".

History

An archaic term for the operation of taking nth roots is radication.

Definition and notation

The four 4th roots of −1,
none of which is real

The three 3rd roots of −1,
one of which is a negative real

An nth 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:

r^{n}=x.

Every positive real number x has a single positive nth root, called the principal nth root, which is written

{\sqrt[{n}]{x}}

. 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 x.

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,

{\sqrt[{5}]{-2}}=-1.148698354\ldots

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,

{\sqrt {2}}=1.414213562\ldots

All nth roots of integers are algebraic numbers.

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 "أصم‎" (asamm, meaning "deaf" or "dumb") for irrational number being translated into Latin as "surdus" (meaning "deaf" or "mute"). Gerard of Cremona (c. 1150), Fibonacci (1202), and then Robert Recorde (1551) all used the term to refer to unresolved irrational roots.

Square roots

The graph

y=\pm {\sqrt {x}}

.

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 {25}}=5.

Since the square of every real number is a positive real number, negative numbers do not have real square roots. However, every negative number has two imaginary square roots. For example, the square roots of −25 are 5i and −5i, where i represents a square root of −1.

Cube roots

The graph

y={\sqrt[{3}]{x}}

.

A cube root of a number x is a number r whose cube is x:

r^{3}=x.

Every real number x has exactly one real cube root, written

{\sqrt[{3}]{x}}

. For example,

{\sqrt[{3}]{8}}=2

and

{\sqrt[{3}]{-8}}=-2.

Every real number has two additional complex cube roots.

Identities and properties

Expressing the degree of an nth root in its exponent form, as in

x^{\frac {1}{n}}

, makes it easier to manipulate powers and roots.

{\sqrt[{n}]{a^{m}}}\equiv \left(a^{m}\right)^{\frac {1}{n}}\equiv a^{\frac {m}{n}}.

Every positive real number has exactly one positive real nth root, and such the rules for operations with surds involving positive radicands

a,\;b

are straightforward within the real numbers:

{\begin{aligned}{\sqrt[{n}]{ab}}&\equiv {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}\\{\sqrt[{n}]{\frac {a}{b}}}&\equiv {\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}}\end{aligned}}

Subtleties can occur when taking the nth roots of negative or complex numbers. For instance:

{\sqrt {-1}}\times {\sqrt {-1}}\neq {\sqrt {-1\times -1}}=1,\quad

but rather

\quad {\sqrt {-1}}\times {\sqrt {-1}}=i\times i=i^{2}=-1.

Since the rule

{\sqrt[{n}]{a}}\times {\sqrt[{n}]{b}}={\sqrt[{n}]{ab}}

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 {\tfrac {32}{5}}}

in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:

{\sqrt {\tfrac {32}{5}}}={\sqrt {\tfrac {16\cdot 2}{5}}}=4{\sqrt {\tfrac {2}{5}}}

Next, there is a fraction under the radical sign, which we change as follows:

4{\sqrt {\tfrac {2}{5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}

Finally, we remove the radical from the denominator as follows:

{\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}}

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 {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}}\,.

Simplifying radical expressions involving nested radicals can be quite difficult. It is not obvious for instance that:

{\sqrt {3+2{\sqrt {2}}}}=1+{\sqrt {2}}

The above can be derived through:

{\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}}

Infinite series

The radical or root may be represented by the infinite series:

(1+x)^{\frac {s}{t}}=\sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(s-kt)}{n!t^{n}}}x^{n}

with

|x|<1

. This expression can be derived from the binomial series.

Computing principal roots

The nth root of an integer k is only an integer if k is the product of nth powers of integers. In all other cases the nth root of an integer is an irrational number. For instance, the fifth root of

{\sqrt[{5}]{248832}}={\sqrt[{5}]{3^{5}\cdot 2^{5}\cdot 2^{5}}}=12

and the fifth root of 34 is

{\sqrt[{5}]{34}}={\sqrt[{5}]{2\cdot 17}}=2.024397458\ldots,

where here the dots signify not only that the decimal expression does not end after a finite number of digits, but also that the digits never enter a repeating pattern, because the number is irrational.

Since for positive real numbers

a

and

b

the equality

\;{\sqrt[{n}]{a/b}}={\sqrt[{n}]{a}}/{\sqrt[{n}]{b}}\;

holds, the above property can be extended to positive rational numbers. Let

r=p/q

, with

p

and

q

coprime and positive integers, be a rational number, then

r

has a rational nth root, if both positive integers

p

and

q

have an integer nth root, i.e.,

\;r=p\cdot {\tfrac {1}{q}}\;

is the product of nth powers of rational numbers. If one or both nth roots of

p

or

q

are irrational, the quotient is irrational, too.

nth root algorithm

The nth root of a number A can be computed by the nth root algorithm, a special case of Newton's method. Start with an initial guess x₀ and then iterate using the recurrence relation

x_{k+1}={\frac {1}{n}}\left({(n-1)x_{k}+{\frac {A}{x_{k}^{n-1}}}}\right)

until the desired precision is reached.

Depending on the application, it may be enough to use only the first Newton approximant:

{\sqrt[{n}]{x^{n}+y}}\approx x+{\frac {y}{nx^{n-1}}}.

For example, to find the fifth root of 34, note that 2 = 32 and thus take x = 2, n = 5 and y = 2 in the above formula. This yields

{\sqrt[{5}]{34}}={\sqrt[{5}]{32+2}}\approx 2+{\frac {2}{5\cdot 16}}=2.025.

The error in the approximation is only about 0.03%.

Newton's method can be modified to produce a generalized continued fraction for the nth root which can be modified in various ways as described in that article. For example:

{\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 }}}}}}}}}}}};

{\sqrt[{n}]{z}}=x+{\cfrac {2x\cdot y}{n(2z-y)-y-{\cfrac {(1^{2}n^{2}-1)y^{2}}{3n(2z-y)-{\cfrac {(2^{2}n^{2}-1)y^{2}}{5n(2z-y)-{\cfrac {(3^{2}n^{2}-1)y^{2}}{7n(2z-y)-\ddots }}}}}}}}.

In the case of the fifth root of 34 above (after dividing out selected common factors):

{\sqrt[{5}]{34}}=2+{\cfrac {1}{40+{\cfrac {4}{4+{\cfrac {6}{120+{\cfrac {9}{4+{\cfrac {11}{200+{\cfrac {14}{4+\ddots }}}}}}}}}}}}=2+{\cfrac {4\cdot 1}{165-1-{\cfrac {4\cdot 6}{495-{\cfrac {9\cdot 11}{825-{\cfrac {14\cdot 16}{1155-\ddots }}}}}}}}.

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

Pascal's Triangle showing

P(4,1)=4

.

Building on the digit-by-digit calculation of a square root, it can be seen that the formula used there,

x(20p+x)\leq c

, or

x^{2}+20xp\leq c

, follows a pattern involving Pascal's triangle. For the nth root of a number

P(n,i)

is defined as the value of element

i

in row

n

of Pascal's Triangle such that

P(4,1)=4

, we can rewrite the expression as

\sum _{i=0}^{n-1}10^{i}P(n,i)p^{i}x^{n-i}

. 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 square. 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 $y\leq c$ .
- 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 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

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{hence}}\quad \quad \log _{b}r={\frac {\log _{b}x}{n}}.

The root r is recovered from this by taking the antilog:

r=b^{{\frac {1}{n}}\log _{b}x}.

(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 r = −|r|.

Geometric constructibility

The ancient Greek mathematicians knew how to use compass and straightedge to construct a length equal to the square root of a given length. In 1837 Pierre Wantzel proved that an nth root of a given length cannot be constructed if n is not a power of 2.

Complex roots

Every complex number other than 0 has n different nth roots.

Square roots

The square roots of i

The two square roots of a complex number are always negatives of each other. For example, the square roots of

-4

are

2 i

and

-2 i

, and the square roots of

i

are

{\tfrac {1}{\sqrt {2}}}(1+i)\quad {\text{and}}\quad -{\tfrac {1}{\sqrt {2}}}(1+i).

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:

{\sqrt {re^{i\theta }}}=\pm {\sqrt {r}}\cdot e^{\frac {i\theta }{2}}.

A principal root of a complex number may be chosen in various ways, for example

{\sqrt {re^{i\theta }}}={\sqrt {r}}\cdot e^{\frac {i\theta }{2}}

which introduces a branch cut in the complex plane along the positive real axis with the condition

0 \leq θ < 2π

, or along the negative real axis with

-π < θ \leq π

.

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 or Scilab.

Roots of unity

The three 3rd roots of 1

The number 1 has n different nth roots in the complex plane, namely

1,\;\omega,\;\omega ^{2},\;\ldots,\;\omega ^{n-1},

where

\omega =e^{\frac {2\pi i}{n}}=\cos \left({\frac {2\pi }{n}}\right)+i\sin \left({\frac {2\pi }{n}}\right)

These roots are evenly spaced around the unit circle 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

.

nth roots

Visualisation of the square to sixth roots of a complex number z, in polar form re where φ = arg z and r = |z | – if z is real, φ = 0 or

π

. Principal roots are in black.

Every complex number has n different nth roots in the complex plane. These are

\eta,\;\eta \omega,\;\eta \omega ^{2},\;\ldots,\;\eta \omega ^{n-1},

where η is a single nth root, and 1, ω, ω, ... ω are the nth roots of unity. For example, the four different fourth roots of 2 are

{\sqrt[{4}]{2}},\quad i{\sqrt[{4}]{2}},\quad -{\sqrt[{4}]{2}},\quad {\text{and}}\quad -i{\sqrt[{4}]{2}}.

In polar form, a single nth root may be found by the formula

{\sqrt[{n}]{re^{i\theta }}}={\sqrt[{n}]{r}}\cdot e^{i\theta /n}.

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+bithen

r={\sqrt {a^{2}+b^{2}}}

. 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 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

\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 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) = (–1) × r₁= r₁.

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 polynomials

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

x^{5}=x+1

cannot be expressed in terms of radicals. (cf. quintic equation)

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Wikiblog Cultuland [CZ]

nth root (radication)

nth root (radication)

Definition

History

Definition and notation

Square roots

Cube roots

Identities and properties

Simplified form of a radical expression

Infinite series

Computing principal roots

nth root algorithm

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

Examples

Logarithmic calculation

Geometric constructibility

Complex roots

Square roots

Roots of unity

nth roots

Solving polynomials

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