Nth root
In mathematics, an th root of a number is a number which, when raised to the power of, yields : The positive integer is called the index or degree, and the number 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 th root is a root extraction.
The th root of is written as using the radical symbol. The square root is usually written as, with the degree omitted. Taking the th root of a number, for fixed, is the inverse of raising a number to the th power, and can be written as a fractional exponent:
For a positive real number, denotes the positive square root of and denotes the positive real th root. For example, is a square root of, since, and is also a square root of, since. A negative real number has no real-valued square roots, but when is treated as a complex number it has two imaginary square roots, and, where is the imaginary unit.
In general, any non-zero complex number has distinct complex-valued th roots, equally distributed around a complex circle of constant absolute value. Extracting the th roots of a complex number can thus be taken to be a multivalued function. By convention the principal value of this function, called the principal root and denoted, is taken to be the th root with the greatest real part and in the special case when 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. The th 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.
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.
History
The Babylonians, as early as 1800 BCE, demonstrated numerical approximations of irrational quantities such as the square root of 2 on clay tablets, with an accuracy analogous to six decimal places, as in the tablet YBC 7289. Cuneiform tablets from Larsa include tables of square and cube roots of integers. The first to prove the irrationality of √2 was most likely the Pythagorean Hippasus. Plato in his Theaetetus, then describes how Theodorus of Cyrene proved the irrationality of,, etc. up to. In the first century AD, Heron of Alexandria devised an iterative method to compute the square root, which is actually a special case of the more general Newton's method.The term surd traces back to Al-Khwarizmi, who referred to rational and irrational numbers as "audible" and "inaudible", respectively. This later led to the Arabic word أصم for "irrational number" being translated into Latin as surdus. Gerard of Cremona, Fibonacci, and then Robert Recorde all used the term to refer to "unresolved irrational roots", that is, expressions of the form, in which and are integer numerals and the whole expression denotes an irrational number. Irrational numbers of the form where is rational, are called "pure quadratic surds"; irrational numbers of the form , where and are rational, are called mixed quadratic surds. An archaic term from the late 15th century for the operation of taking nth roots is radication, and an unresolved root is a radical.
In the fourteenth century, Jamshid al-Kashi used an iterative technique now called the Ruffini-Horner method to extract nth roots for an arbitrary n. This technique has been used since antiquity to determine square roots, then by China and Kushyar ibn Labban during the tenth century to determine cube roots. In 1665, Isaac Newton discovered the general binomial theorem, which can convert an nth root into an infinite series. Based on approach developed by François Viète, Newton devised an iterative method for solving a non-linear function of the form, which can be used to extract an nth root. This technique was further refined by Joseph Raphson and became known as the Newton–Raphson method. In 1690, Michel Rolle introduced the notation for the nth root of the value a.
In 1629, Albert Girard proposed the fundamental theorem of algebra, but failed to produce a proof. This theorem states that every single-variable polynomial of degree n has n roots. Further, a polynomial with complex coefficients has at least one complex root. Equivalently, the theorem states that the field of complex numbers is algebraically closed. Among the notable mathematicians who worked on a proof during the 18th and 19th centuries were d'Alembert, Gauss, Bolzano, and Weierstrass, with Gauss usually being credited with the first correct proof. A consequence of this proof is that any nth root of a real or complex number will be on the complex plane.
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.
Definition and notation
An th 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:Every positive real number x has a single positive nth root, called the principal nth root, which is written. 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, but −2 does not have any real 6th roots.
Every non-zero number x, real or complex, has n different complex number nth roots. The only complex root of 0 is 0.
The nth roots of almost all numbers are irrational. For example,
All nth roots of rational numbers are algebraic numbers, and all nth roots of integers are algebraic integers.
Square roots
A square root of a number x is a 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 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.
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.
Identities and properties
Expressing the degree of an nth root in its exponent form, as in, makes it easier to manipulate powers and roots. If is a non-negative real number,Every non-negative number has exactly one non-negative real nth root, and so the rules for operations with surds involving non-negative radicands and are straightforward within the real numbers:
Subtleties can occur when taking the nth roots of negative or complex numbers. For instance:
but, rather,
Since the rule 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 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.For example, to write the radical expression in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it:
Next, there is a fraction under the radical sign, which we change as follows:
Finally, we remove the radical from the denominator as follows:
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:
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
The above can be derived through:
Let, with and coprime and positive integers. Then is rational if and only if both and are integers, which means that both and are nth powers of some integer.
Infinite series
The radical or root may be represented by the generalized binomial theorem:with. This expression can be derived from the binomial series. For the nth root, this becomes
For numbers, choose a value such that
then per above, solve for
As an example, for and, choose
Nth roots are used to check for convergence of a power series with the root test.