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In mathematics, an nth root of a number a is a number b such that bn=a. When referring to the nth root of a real number a it is assumed that what is desired is the principal nth root of the number, which is denoted ![sqrt[n]{a}](http://upload.wikimedia.org/math/9/a/2/9a2b6d33f3d62a1e8bd99c76f3cb79f5.png) using the radical symbol  The principal nth root of a real number a is the unique real number b which is an nth root of a and is of the same sign as a. Note that if n is even, negative numbers will not have a principal nth root. When n = 2, the nth root is called the square root, and when n = 3, the nth root is called the cube root. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
For other uses, see Number (disambiguation). ...
In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
Look up een, even in Wiktionary, the free dictionary. ...
A negative number is a number that is less than zero, such as −3. ...
In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...
Plot of y = In mathematics, the cube root of a number, denoted or x1/3, is the number a such that a3 = x. ...
Symbol The origin of the radical symbol  is largely speculative, 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. Look up Origin in Wiktionary, the free dictionary. ...
Leonhard Paul Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ...
Vintage German letter balance for home use Look up letter in Wiktionary, the free dictionary. ...
For other uses, see Latin (disambiguation). ...
The radix (Latin for root), also called base, is the number of various unique symbols (or digits or numerals) a positional numeral system uses to represent numbers. ...
In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ...
For other uses, see Print. ...
A vinculum is a horizontal line placed over a mathematical expression, used to indicate that it is to be considered a group. ...
Horizontal plane is used in radio to plot a antennas relative field strength (which directly affects a stations coverage area) on a polar graph. ...
Events January 21 - The Swiss Anabaptist Movement was born when Conrad Grebel, Felix Manz, George Blaurock, and about a dozen others baptized each other in the home of Manzs mother on Neustadt-Gasse, Zürich, breaking a thousand-year tradition of church-state union. ...
Fundamental operations Operations with radicals are given by the following formulas: In mathematics and in the sciences, a formula (plural: formulae, formulæ or formulas) is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ...
![sqrt[n]{ab} = sqrt[n]{a} sqrt[n]{b} qquad a ge 0, b ge 0](http://upload.wikimedia.org/math/0/c/d/0cdedfcbe5dfbe6983841724e3f74cbd.png) ![sqrt[n]{frac{a}{b}} = frac{sqrt[n]{a}}{sqrt[n]{b}} qquad a ge 0, b > 0](http://upload.wikimedia.org/math/5/a/f/5afc148312953575978540260c5c7af2.png) ![sqrt[n]{a^m} = left(sqrt[n]{a}right)^m = left(a^{frac{1}{n}}right)^m = a^{frac{m}{n}},](http://upload.wikimedia.org/math/3/b/c/3bc5c02ca1aaf68df2b3a6baa4067cbd.png) where a and b are positive. In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ...
For every non-zero complex number a, there are n different complex numbers b such that bn = a, so the symbol cannot be used unambiguously. The nth roots of unity are of particular importance. The term null vector can have two different meanings: null vector (vector space) null vector (Minkowski space) This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematics, the nth roots of unity, or de Moivre numbers, are all the complex numbers which yield 1 when raised to a given power n. ...
Once a number has been changed from radical form to exponentiated form, the rules of exponents still apply (even to fractional exponents), namely Exponentiation is a mathematical operation, written an, involving two numbers, the base a and the exponent n. ...
Look up rule in Wiktionary, the free dictionary. ...
For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ...
   For example: ![sqrt[3]{a^5}sqrt[5]{a^4} = a^frac{5}{3} a^frac{4}{5} = a^frac{25 + 12}{15} = a^frac{37}{15}](http://upload.wikimedia.org/math/4/b/6/4b6fc079d5f717b2035569fd276ac53a.png) ![frac{sqrt{a}}{sqrt[4]{a}} = a^frac{1}{2}a^frac{-1}{4}= a^frac{4 - 2}{8} = a^frac{2}{8} = a^frac{1}{4}](http://upload.wikimedia.org/math/f/6/4/f646eec09502624f37e1488878ffee8d.png) If you are going to do addition or subtraction, then you should notice that the following concept is important. 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ...
5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ...
For other uses, see Concept (disambiguation). ...
![sqrt[3]{a^5} = sqrt[3]{aaaaa} = sqrt[3]{a^3a^2} = asqrt[3]{a^2}](http://upload.wikimedia.org/math/8/1/d/81d7fc6b63d9b15b8b5b9b9aec8e3be5.png) If you understand how to simplify one radical expression, then addition and subtraction is simply a question of grouping "like terms". An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ...
This picture illustrates how the hours on a clock form a group under modular addition. ...
For example,
![sqrt[3]{a^5}+sqrt[3]{a^8}](http://upload.wikimedia.org/math/9/4/e/94e35bd06ecacb35ced4927663fad539.png) ![=sqrt[3]{a^3a^2}+sqrt[3]{a^6 a^2}](http://upload.wikimedia.org/math/8/7/6/8763df5a2178d6dde0decb3dace59436.png) ![=asqrt[3]{a^2}+a^2sqrt[3]{a^2}](http://upload.wikimedia.org/math/6/c/c/6cc68f1603421abf90aaab90c60b25bd.png) ![=({a+a^2})sqrt[3]{a^2}](http://upload.wikimedia.org/math/3/f/f/3ffa1ce6d0b2e79be80c29327622996c.png)
Working with surds | Surd | | al-Khowarizmi (c. 825) 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] | Often it is simpler to leave the nth roots of numbers "unresolved" (ie. with radicals visible). These unresolved expressions, called "surds", may then be manipulated into simpler forms or arranged to divide each other out. Notationally, the radical symbol () depicts surds, with the upper line above the expression called the vinculum. A cube root takes the form: In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...
Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ...
![sqrt[3]{a}](http://upload.wikimedia.org/math/7/f/5/7f552661c061361dbc630fb4077f0711.png) , which corresponds to  , when expressed using indices. All roots can remain in surd form. In mathematics, an index is a superscript or subscript to a symbol. ...
Basic techniques for working with surds arise from identities. Some basic examples include: In mathematics, an identity is an equality that remains true regardless of the values of any variables that appear within it. ...
 - The above can be combined with index reduction:
![sqrt[6]{a^6b^4} = sqrt[3cdot 2]{a^2a^2a^2b^2b^2} = sqrt[3]{a^3b^2} = asqrt[3]{b^2}](http://upload.wikimedia.org/math/b/0/e/b0e0bad6e8c3c0834ed47bb8fbacddd8.png) ![sqrt[n]{a^m b} = a^{frac{m}{n}}sqrt[n]{b}](http://upload.wikimedia.org/math/b/3/7/b372247805535b8f54dc58e199d59b32.png) The last of these may serve to rationalize the denominator of an expression, moving surds from the denominator to the numerator. It follows from the identity A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ...
A denominator is a name. ...
In algebra, a vulgar fraction consists of one integer divided by a non-zero integer. ...
 , which exemplifies a case of the difference of two squares. Variants for cube and other roots exist, as do more general formulae based on finite geometric series. Case analysis is one of the most general and applicable methods of analytical thinking, depending only on the division of a problem, decision or situation into a sufficient number of separate cases. ...
In mathematics, the difference of two squares refers to the identity a2 − b2 = (a + b)(a − b) from elementary algebra. ...
In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...
In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ...
Infinite series The radical or root may be represented by the infinite series: Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...
Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
 with  .
Finding all roots All the roots of any number, real or complex, may be found with a simple algorithm. The number should first be written in the form aeiφ (see Euler's formula). Then all the nth roots are given by: In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ...
Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...
![e^{(frac{varphi+2kpi}{n})i} times sqrt[n]{a}](http://upload.wikimedia.org/math/a/d/b/adb6eb95f204c635b43d16ebc4c65c31.png) for  , where represents the principal nth root of a.
Positive real numbers All the complex solutions of xn = a, or the nth roots of a, where a is a positive real number, are given by the simplified equation: ![e^{2pi i frac{k}{n}} times sqrt[n]{a}](http://upload.wikimedia.org/math/f/a/1/fa1fa6c467c099ca972905f159336a24.png) for , where represents the principal nth root of a.
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 In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. ...
In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...
It has been suggested that this article or section be merged into Elementary matrix transformations. ...
The Abel-Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. ...
 cannot be expressed in terms of radicals.
For solving any equation of the nth degree, see Root-finding algorithm. A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. ...
See also The principal nth root of a positive real number A, is the positive real solution of the equation (for integer n there are n distinct complex solutions to this equation if , but only one is positive). ...
The shifting nth-root algorithm is an algorithm for extracting the nth root of a positive real number which proceeds iteratively by shifting in n digits of the radicand, starting with the most significant, and produces one digit of the root on each iteration, in a manner similar to long...
In mathematics, an irrational number is any real number that is not a rational number — that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers. ...
In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...
In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ...
Plot of y = In mathematics, the cube root of a number, denoted or x1/3, is the number a such that a3 = x. ...
The Twelfth root of two is a quantity representing the frequency ratio between any two consecutive notes of a modern chromatic scale in equal temperament. ...
External links
References - ^ Leonhard Euler (1755). Institutiones calculi differentialis (in Latin).
- ^ Earliest Known Uses of Some of the Words of Mathematics. Mathematics Pages by Jeff Miller. Retrieved on 2007-04-20.
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