Mathematics
Mathematics is the science of structure and change. Mathematics is important to the other sciences because it provides rigourous methods for developing models of complex phenomena. Such phenomena include the spread of computer viruses on a network, the growth of tumors, the risk associated with certain contracts traded on the stock market, and the formation of turbulence around an aircraft. Mathematics provides a kind of "quality control" for the development of trustworthy theories and equations which are important to people in most modern technical discplines such as engineering and economics.
Contents
Overview

The ten digits making up the base ten number system. 
Modern mathematics is built around a system of axioms, which is a name given to "the rules of the game." Mathematicians then use various methods of formal proof to extend the axioms to come up with surprising and elegant results. Such methods include proof by induction, and proof by contradiction, for example.
Mathematical Subject Classification
There are numerous categories and subcategories of mathematics, as shown by the American Mathematical Society's Mathematics Subject Classification scheme.
A common way of classifying mathematics is into Pure Maths, and Applied Maths. Pure Maths is maths which is studied in order to make mathematics more stable and powerful, and a knowledge of Pure Maths is required to understand the foundations of Applied Maths. Pure Maths is often considered to be divided into the areas of Higher Algebra, Analysis, and Topology.
Applied Maths consists of taking the techniques from Pure Maths and using them to develop models of "the real world." Applied Maths is sometimes considered to be divided into the areas of Dynamical Systems, Approximation Techniques, and Probability & Statistics. There are also various Applied Mathematical disciplines which use a combination of these areas, but focus on a particular type of application. Examples include Mathematical Physics and Mathematical Biology.
Arithmetic
{mainArithmetic} Arithmetic is a branch of mathematics and their basic properties under the operations of addition, subtraction, multiplication and division and exponentsas well as percents radicals fractions and decimals
In general, more basic properties of the integers belong to arithmetic while deeper or more difficult results belong to number theory, but the boundary is not extremely clear. For instance, modular arithmetic might be considered part of arithmetic as well as part of number theory.
One of the earlier arithmetic devices was the abacus.
according to Wikipedia Arithmetic comes from the Greek word arithmos, "number" and tiké [téchne], "art")
Algebra
{mainalgebra} In mathematics, algebra can denote many things. As a subject, it generally denotes the study of calculations on some set. In high school, this can the study of examining, manipulating, and solving equations, inequalities, and other mathematical expressions. Algebra revolves around the concept of the variable, an unknown quantity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with algebraic manipulation. Algebra can be used to solve different types of equations, but algebra is also many other things Modern algebra (or "higher", or "abstract" algebra) deals (in part) with generalisations of the normal operations seen arithmetic and high school algebra. Groups, rings, fields, modules, and vector spaces are common objects of study in higher algebra. Algebra Involving Equation Algebra can be used to solve equations as simple as 3x=9 but in some cases so complex that mathematicians have not figured how to solve the particular equation yet. As if to add to the confusion, "algebra" is the name for a certain kind of structure in modern algebra.
Modern algebra also arguably contains the field of number theory, which has important applications in computer science. (It is commonly claimed that the NSA is the largest employer in the USA of mathematicians, due to the applications of number theory to cryptanalysis.) However, number theory concerns itself with a specific structure (the ring ), whereas algebra in general deals with general classes of structure. Furthermore, number theory interacts more specifically with certain areas of mathematics (e.g., analysis) than does algebra in general. Indeed, number theory is traditionally divided into different branches, the most prominent of which are algebraic number theory and analytic number theory.
History of Mathematics
Mathematics was noted by the earliest humans. Over time, as humans evolved, the complexity of mathematics also evolved. There was an astounding discovery on how the numbers correlated with each other, as well as in nature, so well, as they created the concept of numbers. Many cultures throughout the world contributed to the development of mathematics in historical times, from China and India to the Middle East and Greece.
Modern Mathematics began in Europe during the Renaissance, after various Arabic texts were translated into European languages during the 12th and 13th centuries. Islamic cultures in the Middle East had preserved various ancient Greek and Hindu texts, and had furthermore extended these old results into new areas. The popularity of the printing press combined with the increasing need for navigational accuracy as European powers began colonizing other parts of the globe initiated a huge mathematical boom, which has continued to this day.
"God created the integers. All the rest is the work of man."
Leopold Kronecker