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.
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.
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