Nth root

In mathematics, a root of a number x is any number which, when repeatedly multiplied by itself, eventually yields x:

In terms of exponentiation, r is a root of x if

for some positive integer n. For example, 2 is a root of 16 since 2⁴ = 2 × 2 × 2 × 2 = 16.

The number n is called the degree of the root. A root of degree 2 is called a square root, a root of degree 3 is called a cube root, a root of degree 4 is called a fourth root, and so forth. In general, a root of degree n is called an nth root. Roots are usually written using the radical symbol , with denoting the square root, denoting the cube root, denoting the fourth root, and so on. An unresolved root, especially one using the radical symbol, is often referred to as a surd.

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

Roots are particularly important in the theory of infinite series, where 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. Much of Galois theory is concerned with determining which algebraic numbers can be expressed using roots, leading to the famous Abel-Ruffini theorem that a general polynomial of degree five or higher cannot be solved using roots alone.

Contents 1 History 2 Definition and notation 2.1 Square roots 2.2 Cube roots 3 Working with surds 4 Infinite series 5 Computing principal roots 6 Complex roots 6.1 Square roots 6.2 Roots of unity 6.3 Nth roots 7 Solving polynomials 8 See also 9 References 9.1 External links

History

The origin of the root symbol √ is largely speculative. Some sources tell that the symbol was first used by Arabs, the first known use was by Abū al-Hasan ibn Alī al-Qalasādī (1421-1486), and that it is taken from the Arabic letter ج, the first letter in the word (Jathir, [with the "th" pronounced like the "th" in the english word "the"] in Arabic means root).

But many, including Leonhard Euler,^[1] believe it originates from the letter r, the first letter of the Latin word radix which refers 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 term 'surd' traces back to al-Khwārizmī (c. 825) who referred to rational and irrational numbers as 'audible' and 'inaudible', respectively. This later lead to the Arabic "asamm" (deaf, dumb) for irrational number being translated as surdus ("deaf" or "mute") into Latin. Gherardo of Cremona (c. 1150), Fibonacci (1202) and then Robert Recorde (1551) used the term to refer to unresolved irrational roots.^[2]

Definition and notation

The nth root of a number x, where n is a positive integer, is a number r whose nth power is x:

Every positive real number x has a single positive nth root, which is written . For n equal to 2 this is called the 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, 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, see complex roots below. The nth root of 0 is 0.

For most numbers, an nth root is irrational. For example,

All nth roots of integers, or in fact of any algebraic number, are however algebraic.

For the extension of powers and roots to indices that are not positive integers, see exponentiation.

The character codes for the radical symbols are:

Read	Character	Unicode	ASCII	URL	HTML (others)
Square root	√	U+221A	√	%E2%88%9A	√
Cube root	∛	U+221B	∛	%E2%88%9B
Fourth root	∜	U+221C	∜	%E2%88%9C

Square roots

The square root of a number x is that number r which, when squared, becomes 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:

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:

Every real number x has exactly one real cube root, written . For example,

Every real number has two additional complex cube roots (see complex roots below).

Working with surds

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

When simplifying an expression using the exponent form as in x^1/3 normally makes it easier to cancel out powers and roots.

When there is a denominator involving surds it may be 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:

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

If negative roots are allowed or when taking the nth roots of complex numbers problems can occur. For instance:

when taking the principal value of the roots. See failure of power and logarithm identities in the exponentiation article for more details.

Infinite series

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

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 or rational number. For instance, the fifth root of 34 is

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

until the desired precision is reached.

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

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

The error in the approximation is only about 0.03 %.

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

Complex roots

Every complex number 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 −4 are 2i and − 2i, and the square roots of i are

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:

Unfortunately, because the complex numbers cannot be categorized as positive and negative, there is no consistent way to choose one of the two square roots as principal. For example, the formula

introduces a branch cut in the complex plane along the positive real axis if , and along the negative real axis if .

Roots of unity

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

where

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

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

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

As with square roots, the formula above cannot be applied consistently to the entire complex plane, but instead leads to a branch cut at the points where θ / n suddenly “jumps”.

Solving polynomials

It was once conjectured that all roots of polynomials could be expressed in terms of radicals and elementary operations; however, the Abel-Ruffini theorem asserts that this is not true in general. For example, the solutions of the equation

cannot be expressed in terms of radicals.

For solving any equation of the nth degree, see Root-finding algorithm.

See also

Look up radical in Wiktionary, the free dictionary.

Look up surd in Wiktionary, the free dictionary.

• Nth root algorithm
• Shifting nth-root algorithm
• Irrational number
• Algebraic number
• Nested radical
• Twelfth root of two
• Super-root

References

External links

• principal nth root calculator reduces any number to principal nth root, shows simplest radical form