Concepts inAnalysis in the Computable Number Field

Computable number

In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. Equivalent definitions can be given using ¿-recursive functions, Turing machines or ¿-calculus as the formal representation of algorithms.  more from Wikipedia

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions. These theories are often studied in the context of real numbers, complex numbers, and real and complex functions. Analysis may be conventionally distinguished from geometry.  more from Wikipedia

Field (mathematics)

In abstract algebra, a field is a ring whose nonzero elements form a commutative group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic number fields, p-adic fields, and so forth.  more from Wikipedia

Constructive analysis

In mathematics, constructive analysis is mathematical analysis done according to the principles of constructive mathematics. This contrasts with classical analysis, which (in this context) simply means analysis done according to the (ordinary) principles of classical mathematics. Generally speaking, constructive analysis can reproduce theorems of classical analysis, but only in application to separable spaces; also, some theorems may need to be approached by approximations.  more from Wikipedia

Function of a real variable

In mathematics, a function of a real variable is a mathematical function whose domain is the real line. More loosely, a function of a real variable is sometimes taken to mean any function whose domain is a subset of the real line. Functions of a real variable were the classical object of study in mathematical analysis, specifically real analysis.  more from Wikipedia

Real analysis

Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.  more from Wikipedia

Constructive proof

In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem) which proves the validity of a proposition without considering an example.  more from Wikipedia

Countable set

In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor. The elements of a countable set can be counted one at a time¿although the counting may never finish, every element of the set will eventually be associated with a natural number. Some authors use countable set to mean a set with the same cardinality as the set of natural numbers.  more from Wikipedia