Methods of computing square rootsMethods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted , , or ) of a real number. Arithmetically, it means given , a procedure for finding a number which when multiplied by itself, yields ; algebraically, it means a procedure for finding the non-negative root of the equation ; geometrically, it means given two line segments, a procedure for constructing their geometric mean. Every real number except zero has two square roots.
Cube (algebra)In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example 23 = 8 or (x + 1)3. The cube is also the number multiplied by its square: n3 = n × n2 = n × n × n. The cube function is the function x ↦ x3 (often denoted y = x3) that maps a number to its cube. It is an odd function, as (−n)3 = −(n3).
TetrationIn mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though and the left-exponent xb are common. Under the definition as repeated exponentiation, means , where n copies of a are iterated via exponentiation, right-to-left, i.e. the application of exponentiation times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation. It would be read as "the nth tetration of a".
Generalized continued fractionIn complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values. A generalized continued fraction is an expression of the form where the an (n > 0) are the partial numerators, the bn are the partial denominators, and the leading term b0 is called the integer part of the continued fraction.
Elementary arithmeticElementary arithmetic is a branch of mathematics involving basic numerical operations, namely addition, subtraction, multiplication, and division. Due to its low level of abstraction, broad range of application, and being the foundation of all mathematics, elementary arithmetic is generally the first critical branch of mathematics to be taught in schools. Numerical digit Symbols called digits are used to represent the value of numbers in a numeral system. The most commonly used digits are the Arabic numerals (0 to 9).
Radical symbolIn mathematics, the radical symbol, radical sign, root symbol, radix, or surd is a symbol for the square root or higher-order root of a number. The square root of a number x is written as while the nth root of x is written as It is also used for other meanings in more advanced mathematics, such as the radical of an ideal. In linguistics, the symbol is used to denote a root word. Each positive real number has two square roots, one positive and the other negative. The square root symbol refers to the principal square root, which is the positive one.
AM-GM InequalityIn mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same (in which case they are both that number). The simplest non-trivial case – i.e., with more than one variable – for two non-negative numbers x and y, is the statement that with equality if and only if x = y.
Resolvent (Galois theory)In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group G is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial p and has, roughly speaking, a rational root if and only if the Galois group of p is included in G. More exactly, if the Galois group is included in G, then the resolvent has a rational root, and the converse is true if the rational root is a simple root. Resolvents were introduced by Joseph Louis Lagrange and systematically used by Évariste Galois.
Mathematical fallacyIn mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof. For example, the reason why validity fails may be attributed to a division by zero that is hidden by algebraic notation.
Septic equationIn algebra, a septic equation is an equation of the form where a ≠ 0. A septic function is a function of the form where a ≠ 0. In other words, it is a polynomial of degree seven. If a = 0, then f is a sextic function (b ≠ 0), quintic function (b = 0, c ≠ 0), etc. The equation may be obtained from the function by setting f(x) = 0. The coefficients a, b, c, d, e, f, g, h may be either integers, rational numbers, real numbers, complex numbers or, more generally, members of any field.
