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Dalam matematika, sebuah akar ke-n dari bilangan x adalah bilangan r yang jika dipangkatkan n, menghasilkan x:

r^{n}=x,

dimana n adalah bilangan bulat positif, kadang-kadang disebut derajat dari akar. Akar derajat 2 disebut akar kuadrat dan akar derajat 3, sebuah akar pangkat tiga. Akar tingkat yang lebih tinggi dirujuk dengan menggunakan bilangan urut, seperti pada akar keempat, akar kedua puluh, dll. Perhitungan akar ke- $n$ adalah ekstraksi akar.

Misalnya, 3 adalah akar kuadrat dari 9, karena 3² = 9, dan 3 juga merupakan akar kuadrat dari 9, karena (−3)² = 9.

Setiap bilangan bukan nol yang dianggap sebagai bilangan kompleks memiliki $n$ akar ke- $n$ yang berbeda, termasuk real (paling banyak dua). Akar ke- $n$ dari 0 adalah nol untuk semua bilangan bulat positif $n$ , setelah $0 n = 0$ . Khususnya, jika $n$ genap dan $x$ adalah bilangan real positif, satunya adalah negatif, dan yang lainnya (ketika $n > 2$ ) bilangan kompleks non-real; jika $n$ genap dan $x$ adalah bilangan real negatif, tidak ada satupun akar ke- $n$ yang merupakan real. Jika $n$ ganjil dan $x$ real, satu akar $n$ adalah real dan bertanda sama sebagai $x$ , sedangkan akar lainnya ( $n - 1$ ) bukanlah real. Akhirnya, jika $x$ bukanlah real, maka tidak ada akar ke- $n$ yang merupakan real.

Akar bilangan real biasanya ditulis menggunakan simbol radikal atau radix ${\sqrt {{~^{~}}^{~}\!\!}}$ , dengan ${\sqrt {x}}$ menunjukkan akar kuadrat positif dari $x$ jika $x$ adalah positif; untuk akar tinggi, ${\sqrt[{n}]{x}}$ menunjukkan akar ke- $n$ yang sebenarnya jika $n$ adalah ganjil, dan akar ke-n positif jika $n$ adalah genap dan $x$ adalah positif. Dalam kasus lain, simbol tidak umum digunakan sebagai ambigu. Dalam ekspresi ${\sqrt[{n}]{x}}$ , bilangan bulat n disebut indeks dan $x$ disebut radikan .

Ketika kompleks akar ke- $n$ dipertimbangkan, seringkali berguna untuk memilih salah satu akar, yang disebut akar utama, sebagai nilai utama. Pilihan umum adalah memilih akar ke- $n$ utama dari $x$ sebagai akar ke- $n$ , dengan bagian real terbesar, dan, jika ada dua (untuk $x$ real dan negatif), yang memiliki bagian imajiner positif. Ini membuat akar ke- $n$ sebagai fungsi real dan positif untuk $x$ real dan positif, dan adalah kontinu diseluruh bidang kompleks, kecuali untuk nilai $x$ real dan negatif.

Kesulitan dengan pilihan ini adalah, untuk bilangan real negatif dan indeks ganjil, akar ke- $n$ utama yang bukan asli. Misalnya, $-8$ memiliki tiga akar pangkat tiga, $-2$ , $1+i{\sqrt {3}}$ dan $1-i{\sqrt {3}}.$ Akar pangkat tiga sebenarnya adalah $-2$ dan akar pangkat tiga utama adalah $1+i{\sqrt {3}}.$

Akar yang tidak terselesaikan, terutama yang menggunakan simbol radikal, kadang-kadang disebut sebagai surd^[1] atau "radikal".^[2] Setiap ekspresi yang mengandung radikal, apakah itu akar kuadrat, akar pangkat tiga, atau akar yang lebih tinggi, disebut ekspresi radikal, dan jika tidak mengandung fungsi transendental atau bilangan transendental disebut ekspresi aljabar.

Akar juga didefinisikan sebagai kasus khusus dari eksponensial, dimana eksponen adalah pecahan:

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

Akar digunakan untuk menentukan radius konvergensi dari deret pangkat dengan uji akar. Akar ke- $n$ dari 1 disebut akar satuan dan memainkan peran mendasar dalam berbagai bidang matematika, seperti teori bilangan, teori persamaan, dan transformasi Fourier.

History

Вавилонская табличка (около 1800—1600 г. до н. э.) с вычислением

{\sqrt {2}}\approx 1+24/60+51/60^{2}+10/60^{3}

=1{,}41421296\dots

Первые задачи, связанные с извлечением квадратного корня, обнаружены в трудах вавилонских математиков (о достижениях древнего Египта в этом отношении ничего не известно). Среди таких задач^[3]:

Применение теоремы Пифагора для нахождения стороны прямоугольного треугольника по известным двум другим сторонам.
Нахождение стороны квадрата, площадь которого задана.
Решение квадратных уравнений.

Вавилонские математики (II тысячелетие до н. э.) разработали для извлечения квадратного корня особый численный метод. Начальное приближение для ${\sqrt {a}}$ рассчитывалось исходя из ближайшего к корню (в меньшую сторону) натурального числа $n$ . Представив подкоренное выражение в виде: $a=n^{2}+r$ , получаем: $x_{0}=n+{\frac {r}{2n}}$ , затем применялся итеративный процесс уточнения, соответствующий методу Ньютона^[4]:

x_{n+1}={\frac {1}{2}}~\left(x_{n}+{\frac {a}{x_{n}}}\right)\

Итерации в этом методе очень быстро сходятся. Для ${\sqrt {5}}$ , например, $a=5;\;n=2;\;r=1;\ x_{0}={\frac {9}{4}}=2{,}25,$ и мы получаем последовательность приближений:

x_{1}={\frac {161}{72}}=2{,}23611;\;x_{2}={\frac {51841}{23184}}=2{,}2360679779

В заключительном значении верны все цифры, кроме последней.

Аналогичные задачи и методы встречаются в древнекитайской «Математике в девяти книгах»^[5]. Древние греки сделали важное открытие: ${\sqrt {2}}$ — иррациональное число. Детальное исследование, выполненное Теэтетом Афинским (IV век до н. э.), показало, что если корень из натурального числа не извлекается нацело, то его значение иррационально^[6].

Греки сформулировали проблему удвоения куба, которая сводилась к построению кубического корня с помощью циркуля и линейки. Проблема оказалась неразрешимой. Численные алгоритмы извлечения кубического корня опубликовали Герон (в трактате «Метрика», I век н. э.) и индийский математик Ариабхата I (V век)^[7].

Алгоритмы извлечения корней любой степени из целого числа, разработанные индийскими и исламскими математиками, были усовершенствованы в средневековой Европе. Николай Орем (XIV век) впервые истолковал^[8] корень $n$ -й степени как возведение в степень ${\frac {1}{n}}$ .

После появления формулы Кардано (XVI век) началось применение в математике мнимых чисел, понимаемых как квадратные корни из отрицательных чисел^[9]. Основы техники работы с комплексными числами разработал в XVI веке Рафаэль Бомбелли, который также предложил оригинальный метод вычисления корней (с помощью цепных дробей). Открытие формулы Муавра (1707) показало, что извлечение корня любой степени из комплексного числа всегда возможно и не приводит к новому типу чисел^[10].

Комплексные корни произвольной степени в начале XIX века глубоко исследовал Гаусс, хотя первые результаты принадлежат Эйлеру^[11]. Чрезвычайно важным открытием (Галуа) стало доказательство того факта, что не все алгебраические числа (корни многочленов) могут быть получены из натуральных с помощью четырёх действий арифметики и извлечения корня^[12].

Definition and notation

The four 4th roots of −1,
none of which are 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^1/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, ${\sqrt[{5}]{-2}}=-1.148698354\ldots$ 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, that is, expressions of the form ${\sqrt[{n}]{i}},$ in which $n$ and $i$ are integer numerals and the whole expression denotes an irrational number.^[13] Quadratic irrational numbers, that is, irrational numbers of the form ${\sqrt {i}},$ are also known as "quadratic surds".

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 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 $-1$ .

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 $x^{1/n}$ , makes it easier to manipulate powers and roots. If $a$ is a non-negative real number,

{\sqrt[{n}]{a^{m}}}=(a^{m})^{1/n}=a^{m/n}=(a^{1/n})^{m}=({\sqrt[{n}]{a}})^{m}.

Every non-negative number has exactly one non-negative real nth root, and so the rules for operations with surds involving non-negative radicands $a$ and $b$ are straightforward within the real numbers:

{\begin{aligned}{\sqrt[{n}]{ab}}&={\sqrt[{n}]{a}}{\sqrt[{n}]{b}}\\{\sqrt[{n}]{\frac {a}{b}}}&={\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^[14]

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\times 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.^[15]^[16] 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}}

Let $r=p/q$ , with $p$ and $q$ coprime and positive integers. Then ${\sqrt[{n}]{r}}={\sqrt[{n}]{p}}/{\sqrt[{n}]{q}}$ is rational if and only if both ${\sqrt[{n}]{p}}$ and ${\sqrt[{n}]{q}}$ are integers, which means that both $p$ and $q$ are nth powers of some integer.

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

Using Newton's method

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

x_{k+1}={\frac {n-1}{n}}x_{k}+{\frac {A}{nx_{k}^{n-1}}}

until the desired precision is reached. For example, to find the fifth root of 34, we plug in n = 5, A = 34 and x₀ = 2 (initial guess). The first 5 iterations are, approximately:
x₀ = 2
x₁ = 2.025
x₂ = 2.024397817
x₃ = 2.024397458
x₄ = 2.024397458
The approximation x₄ is accurate to 25 decimal places.

Newton's method can be modified to produce various generalized continued fraction for the nth root. 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 }}}}}}}}}}}}.

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 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 $y\leq c$ .
- 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                   10⁰·1·0⁰·1² + 10¹·2·0¹·1¹     ≤      1   <   10⁰·1·0⁰·2²   + 10¹·2·0¹·2¹         x = 1
         01                      y = 10⁰·1·0⁰·1²   + 10¹·2·0¹·1²   =  1 +    0   =     1
         00 52                10⁰·1·1⁰·2² + 10¹·2·1¹·2¹     ≤     52   <   10⁰·1·1⁰·3²   + 10¹·2·1¹·3¹         x = 2
         00 44                   y = 10⁰·1·1⁰·2²   + 10¹·2·1¹·2¹   =  4 +   40   =    44
            08 27             10⁰·1·12⁰·3² + 10¹·2·12¹·3¹   ≤    827   <   10⁰·1·12⁰·4²  + 10¹·2·12¹·4¹        x = 3
            07 29                y = 10⁰·1·12⁰·3²  + 10¹·2·12¹·3¹  =  9 +  720   =   729
               98 56          10⁰·1·123⁰·4² + 10¹·2·123¹·4¹ ≤   9856   <   10⁰·1·123⁰·5² + 10¹·2·123¹·5¹       x = 4
               98 56             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 to the nearest hundredth.

        1   6.  1   2   4
 3  /
  \/  004 192.000 000 000

      004                      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     x = 1
      001                         y = 10⁰·1·0⁰·1³   + 10¹·3·0¹·1²   + 10²·3·0²·1¹   =   1 +      0 +          0   =          1
      003 192                  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     x = 6
      003 096                     y = 10⁰·1·1⁰·6³   + 10¹·3·1¹·6²   + 10²·3·1²·6¹   = 216 +  1,080 +      1,800   =      3,096
          096 000              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    x = 1
          077 281                 y = 10⁰·1·16⁰·1³  + 10¹·3·16¹·1²  + 10²·3·16²·1¹  =   1 +    480 +     76,800   =     77,281
          018 719 000          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   x = 2
              015 571 928         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  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  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, 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.^[17]

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^{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^{i\theta /2}

which introduces a branch cut in the complex plane along the positive real axis with the condition $0 \leq θ < 2 π$ , 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

Wakilan geometris dari akar ke-2 hingga ke-6 bilangan kompleks

z

, dalam bentuk polar

re iφ

dimana

r = | z |

dan

φ = arg z

. If

z

adalah real,

φ = 0

atau

π

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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, ω², ... ωⁿ⁻¹ 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+bi then $r={\sqrt {a^{2}+b^{2}}}$ . 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)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 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)

Proof of irrationality for non-perfect nth power x

Assume that ${\sqrt[{n}]{x}}$ is rational. That is, it can be reduced to a fraction ${\frac {a}{b}}$ , where $a$ and $b$ are integers without a common factor.

This means that $x={\frac {a^{n}}{b^{n}}}$ .

Since x is an integer, $a^{n}$ and $b^{n}$ must share a common factor if $b\neq 1$ . This means that if $b\neq 1$ , ${\frac {a^{n}}{b^{n}}}$ is not in simplest form. Thus b should equal 1.

Since $1^{n}=1$ and ${\frac {n}{1}}=n$ , ${\frac {a^{n}}{b^{n}}}=a^{n}$ .

This means that $x=a^{n}$ and thus, ${\sqrt[{n}]{x}}=a$ . This implies that ${\sqrt[{n}]{x}}$ is an integer. Since x is not a perfect nth power, this is impossible. Thus ${\sqrt[{n}]{x}}$ is irrational.

References

^ Bansal, R.K. (2006). New Approach to CBSE Mathematics IX. Laxmi Publications. hlm. 25. ISBN 978-81-318-0013-3.
^ Silver, Howard A. (1986). Algebra and trigonometry . Englewood Cliffs, NJ: Prentice-Hall. ISBN 978-0-13-021270-2.
^ История математики 1970—1972, Том I, С. 42—46.
^ История математики 1970—1972, Том I, С. 47.
^ История математики 1970—1972, Том I, С. 169—171.
^ Башмакова И. Г. (1979). Становление алгебры (из истории математических идей). Новое в жизни, науке, технике. Математика, кибернетика, № 9. М.: Знание. hlm. 23.
^ Abhishek Parakh. (2007). "Ariabhata's root extraction methods" (PDF) (42.2) (edisi ke-Indian Journal of History of Science): 149—161. Diarsipkan dari versi asli (PDF) tanggal 2010-06-09.
^ История математики 1970—1972, Том I, С. 275—276.
^ История математики 1970—1972, Том I, С. 296—298.
^ История математики 1970—1972, Том III, С. 56—59.
^ История математики 1970—1972, Том III, С. 62.
^ Колмогоров А. Н., Юшкевич А. П. (ред.). (1978). Математика XIX века. Математическая логика, алгебра, теория чисел, теория вероятностей. I. М.: Наука. hlm. 58—66.
^ "Earliest Known Uses of Some of the Words of Mathematics". Mathematics Pages by Jeff Miller. Diakses tanggal 2008-11-30.
^ McKeague, Charles P. (2011). Elementary algebra. hlm. 470. ISBN 978-0-8400-6421-9.
^ B.F. Caviness, R.J. Fateman, "Simplification of Radical Expressions", Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, p. 329.
^ Richard Zippel, "Simplification of Expressions Involving Radicals", Journal of Symbolic Computation 1:189–210 (1985) DOI:10.1016/S0747-7171(85)80014-6.
^ Wantzel, M. L. (1837), "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas", Journal de Mathématiques Pures et Appliquées, 1 (2): 366–372 .

External links

Templat:Hyperoperations

[1] Bansal, R.K. (2006). New Approach to CBSE Mathematics IX. Laxmi Publications. hlm. 25. ISBN 978-81-318-0013-3.

[silver-2] Silver, Howard A. (1986). Algebra and trigonometry . Englewood Cliffs, NJ: Prentice-Hall. ISBN 978-0-13-021270-2.

[FOOTNOTEИстория_математики1970—1972Том_I,_С._42—46-3] История математики 1970—1972, Том I, С. 42—46.

[FOOTNOTEИстория_математики1970—1972Том_I,_С._47-4] История математики 1970—1972, Том I, С. 47.

[FOOTNOTEИстория_математики1970—1972Том_I,_С._169—171-5] История математики 1970—1972, Том I, С. 169—171.

[6] Башмакова И. Г. (1979). Становление алгебры (из истории математических идей). Новое в жизни, науке, технике. Математика, кибернетика, № 9. М.: Знание. hlm. 23.

[7] Abhishek Parakh. (2007). "Ariabhata's root extraction methods" (PDF) (42.2) (edisi ke-Indian Journal of History of Science): 149—161. Diarsipkan dari versi asli (PDF) tanggal 2010-06-09.

[FOOTNOTEИстория_математики1970—1972Том_I,_С._275—276-8] История математики 1970—1972, Том I, С. 275—276.

[FOOTNOTEИстория_математики1970—1972Том_I,_С._296—298-9] История математики 1970—1972, Том I, С. 296—298.

[FOOTNOTEИстория_математики1970—1972Том_III,_С._56—59-10] История математики 1970—1972, Том III, С. 56—59.

[FOOTNOTEИстория_математики1970—1972Том_III,_С._62-11] История математики 1970—1972, Том III, С. 62.

[12] Колмогоров А. Н., Юшкевич А. П. (ред.). (1978). Математика XIX века. Математическая логика, алгебра, теория чисел, теория вероятностей. I. М.: Наука. hlm. 58—66.

[13] "Earliest Known Uses of Some of the Words of Mathematics". Mathematics Pages by Jeff Miller. Diakses tanggal 2008-11-30.

[14] McKeague, Charles P. (2011). Elementary algebra. hlm. 470. ISBN 978-0-8400-6421-9.

[15] B.F. Caviness, R.J. Fateman, "Simplification of Radical Expressions", Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, p. 329.

[16] Richard Zippel, "Simplification of Expressions Involving Radicals", Journal of Symbolic Computation 1:189–210 (1985) DOI:10.1016/S0747-7171(85)80014-6.

[17] Wantzel, M. L. (1837), "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas", Journal de Mathématiques Pures et Appliquées, 1 (2): 366–372 .

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Pengguna:Klasüo/bak pasir

Daftar isi

History

Definition and notation

Square roots

Cube roots

Identities and properties

Simplified form of a radical expression

Infinite series

Computing principal roots

Using Newton's method

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

Proof of irrationality for non-perfect nth power x

See also

References

External links