Rational function in the context of Polynomial function

In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x − 2)(x + 2) is a polynomial factorization of x − 4.

Factorization is not usually considered meaningful within number systems possessing division, such as the real or complex numbers, since any ${\displaystyle x}$ can be trivially written as ${\displaystyle (xy)\times (1/y)}$ whenever ${\displaystyle y}$ is not zero. However, a meaningful factorization for a rational number or a rational function can be obtained by writing it in lowest terms and separately factoring its numerator and denominator.

In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.

The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field.

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ${\displaystyle \infty }$ for infinity. With the Riemann model, the point ${\displaystyle \infty }$ is near to very large numbers, just as the point ${\displaystyle 0}$ is near to very small numbers.

The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as ${\displaystyle 1/0=\infty }$ well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.

In mathematics, an elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are polynomial functions, rational functions, the trigonometric functions, the exponential and logarithm functions, the n-th root, and the inverse trigonometric functions, as well as those functions obtained by addition, multiplication, division, and composition of these. Some functions which are encountered by beginners are not elementary, such as the absolute value function and piecewise-defined functions. More generally, in modern mathematics, elementary functions comprise the set of functions previously enumerated, all algebraic functions (not often encountered by beginners), and all functions obtained by roots of a polynomial whose coefficients are elementary.

This list of elementary functions was originally set forth by Joseph Liouville in 1833. A key property is that all elementary functions have derivatives of any order, which are also elementary, and can be algorithmically computed by applying the differentiation rules (or the rules for implicit differentiation in the case of roots). The Taylor series of an elementary function converges in a neighborhood of every point of its domain. More generally, they are global analytic functions, defined (possibly with multiple values, such as the elementary function ${\displaystyle {\sqrt {z}}}$ or ${\displaystyle \log z}$) for every complex argument, except at isolated points. In contrast, antiderivatives of elementary functions need not be elementary and is difficult to decide whether a specific elementary function has an elementary antiderivative.