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In printed mathematics, the norm is to set variables and constants in an italic typeface. [20] For example, a general quadratic function is conventionally written as ax 2 + bx + c, where a, b and c are parameters (also called constants, because they are constant functions), while x is the variable of the function.
In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers) [5] and providing an output (which may also be a number). [5] A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variable. [6]
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications.
This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a ...
v. limits of functions of a continuous variable. continuous and discontinuous functions; vi. derivatives and integrals; vii. additional theorems in the differential and integral calculus; viii. the convergence of infinite series and infinite integrals; ix. the logarithmic, exponential and circular functions of a real variable
An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high. [122]
Behnke–Stein theorem (several complex variables) Behrend's theorem (number theory) Bell's theorem (quantum mechanics) Beltrami's theorem (Riemannian geometry) Belyi's theorem (algebraic geometry) Bendixson–Dulac theorem (dynamical systems) Berge's theorem (graph theory) Berger–Kazdan comparison theorem (Riemannian geometry)
A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist.