Nth root

Origin of the root symbol

The origin of the root symbol √ is largely speculative. Some sources imply that the symbol was first used by Arab mathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī (1421–1486). Legend has it that it was taken from the Arabic letter "ج‎" (ǧīm, /dʒim/), which is the first letter in the Arabic word "جذر‎" (jadhir, meaning "root"; /ˈdʒɑːðir/). However, many scholars, including Leonhard Euler, believe it originates from the letter "r", the first letter of the Latin word "radix" (meaning "root"), referring to the same mathematical operation. The symbol was first seen in print without the vinculum (the horizontal "bar" over the numbers inside the radical symbol) in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician.

The Unicode and HTML character codes for the radical symbols are:

Etymology of "surd"

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.

Definition and notation

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 x n . 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, − 2 5 = − 1.148698354 … but −2 does not have any real 6th roots.

Every non-zero number x, real or complex, has n different complex number nth roots including any positive or negative roots. They are all distinct except in the case of x = 0, all of whose nth roots equal 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,

2 = 1.414213562 …

All nth roots of integers, are algebraic numbers.

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:

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

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 x 3 . For example,

8 3 = 2 and − 8 3 = − 2.

Every real number has two additional complex cube roots.

Identities and properties

Every positive real number has a positive nth root and the rules for operations with such surds are straightforward:

a b n ≡ a n b n a b n ≡ a n b n

Using the exponent form as in x 1 / n normally makes it easier to cancel out powers and roots.

a m n

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

− 1 × − 1 = − 1 ≠ − 1 × − 1

when taking the principal value of the roots.

Simplified form of a radical expression

A non-nested radical expression is said to be in simplified form if

1. There is no factor of the radicand that can be written as a power greater than or equal to the index.
2. There are no fractions under the radical sign.
3. There are no radicals in the denominator.

For example, to write the radical expression 32 5 in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:

32 5 = 16 ⋅ 2 5 = 4 2 5

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

4 2 5 = 4 2 5

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

4 2 5 = 4 2 5 ⋅ 5 5 = 4 10 5 = 4 5 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:

1 a 3 + b 3 = a 2 3 − a b 3 + b 2 3 ( a 3 + b 3 ) ( a 2 3 − a b 3 + b 2 3 ) = a 2 3 − a b 3 + b 2 3 a + b .

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

3 + 2 2 = 1 + 2

The above can be derived through:

3 + 2 2 = 1 + 2 2 + 2 = 1 2 + 2 2 + 2 2 = ( 1 + 2 ) 2 = 1 + 2

Infinite series

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

( 1 + x ) s t = ∑ n = 0 ∞ ∏ k = 0 n − 1 ( s − k t ) 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 is not always an integer, and if it is not an integer then it is not a rational number. For instance, the fifth root of 34 is

34 5 = 2.024397458 … ,

where the dots signify that the decimal expression does not end after any finite number of digits. Since in this example the digits after the decimal never enter a repeating pattern, the number is irrational.

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 = 1 n ( ( n − 1 ) x k + A x k n − 1 )

until the desired precision is reached.

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

x n + y n ≈ x + y n x n − 1 .

For example, to find the fifth root of 34, note that 2⁵ = 32 and thus take x = 2, n = 5 and y = 2 in the above formula. This yields

34 5 = 32 + 2 5 ≈ 2 + 2 5 ⋅ 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:

z n = x n + y n = x + y n x n − 1 + ( n − 1 ) y 2 x + ( n + 1 ) y 3 n x n − 1 + ( 2 n − 1 ) y 2 x + ( 2 n + 1 ) y 5 n x n − 1 + ( 3 n − 1 ) y 2 x + ⋱ ; z n = x + 2 x ⋅ y n ( 2 z − y ) − y − ( 1 2 n 2 − 1 ) y 2 3 n ( 2 z − y ) − ( 2 2 n 2 − 1 ) y 2 5 n ( 2 z − y ) − ( 3 2 n 2 − 1 ) y 2 7 n ( 2 z − y ) − ⋱ .

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

34 5 = 2 + 1 40 + 4 4 + 6 120 + 9 4 + 11 200 + 14 4 + ⋱ = 2 + 4 ⋅ 1 165 − 1 − 4 ⋅ 6 495 − 9 ⋅ 11 825 − 14 ⋅ 16 1155 − ⋱ .

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 ( 20 p + x ) ≤ c , or x 2 + 20 x p ≤ 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 ∑ 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:

1. 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.
2. Find p and x, as follows:
3. Let p be the part of the root found so far, ignoring any decimal point. (For the first step, p = 0 ).
4. Determine the greatest digit x such that y ≤ c .
5. 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.
6. Subtract y from c to form a new remainder.
7. 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 computation

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; base 10 is used here) to obtain

n log 10 ⁡ r = log 10 ⁡ x hence log 10 ⁡ r = log 10 ⁡ x n .

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

r = 10 log 10 ⁡ x n .

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 two square roots of a complex number are always negatives of each other. For example, the square roots of −4 are 2i and −2i, and the square roots of i are

1 2 ( 1 + i ) and − 1 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:

r e i θ = ± r e i θ 2 .

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

r e i θ = r e i θ 2

which introduces a branch cut in the complex plane along the positive real axis with the condition 0 ≤ θ < 2π, or along the negative real axis with −π < θ ≤ π.

Using the first(last) branch cut the principal square root z maps 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 number 1 has n different nth roots in the complex plane, namely

1 , ω , ω 2 , … , ω n − 1 ,

where

ω = e 2 π i n = cos ⁡ ( 2 π n ) + i sin ⁡ ( 2 π n )

These roots are evenly spaced around the unit circle in the complex plane, at angles which are multiples of 2 π / 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

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

η , η ω , η ω 2 , … , η ω 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

2 4 , i 2 4 , − 2 4 , and − i 2 4 .

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

r e i θ n = r n e i θ / 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 = a 2 + b 2 . Also, θ 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 ⁡ θ = a / r , sin ⁡ θ = b / r , and tan ⁡ θ = 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 θ / n , where θ 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)