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symbols, equations and theories section, pure mathematics
==Symbols, Equations, and Theories==
==Symbols, Equations, and Theories==
Pure mathematics is the study of mathematics for its own sake, motivated for reasons other than application. It exhibits a trend towards increasing generality and abstraction.
Mathematics is expressed with [[symbol]]s. Some of the most commonly used are the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These are symbols used to express our intuitive notion of [[quantity]]. Other symbol used in elementary mathematics are the equality ( = ), addition ( + ), subtraction ( - ), multiplication ( x ), less than ( < ), greater than ( > ), etc. More advanced branches of mathematics have their own symbols.
An [[equation]] is a mathematical statement that asserts the equality of two expressions. Some equations, like (x + 2 = 5), express the equality of two quantities. Other equations, called [[differential equation]]s, express the equality of two functions.
A mathematical theory is expressed as a set of sentences, called [[axiom]]s. These axioms should be self consistent, that is, they must not contradict with each other. From these axioms, new results can be derived adhering strictly to mathematical logic. These derived results are called [[theorem]s. It is important to note that, according to the [[Godel's Incompleteness Theorems]], it is impossible to state a self consistent set of axioms from which the whole mathematics can be derived.
==Pure and Applied Mathematics==
==Pure and Applied Mathematics==
====Pure mathematics====
====Pure mathematics====
Pure mathematics is the study of mathematics for its own sake, motivated for reasons other than application. It exhibits a trend towards increasing generality and abstraction.
==Branches of Mathematics==
==Branches of Mathematics==