Mathematics (often shortened to maths) is the study of the different ways in which both real-world and abstract values can be manipulated, and how they relate to each other. Opinions differ widely on whether mathematics is an art, a science, or something entirely separate. The ability to describe the properties of both physical and theoretical entities and the relationships between them in mathematical terms, however, is vital to the pursuance of many fields of endeavour, including science, engineering, medicine, commerce, and education. Mathematics has been used since prehistoric times, when early man used basic mathematical principles to count both physical objects like sheep and cattle, and more abstract quantities like time. Elementary arithmetic (the ability to add, subtract, multiply and divide) would have been vital to the pursuance of trade, even in a relatively primitive society. Early methods of recording numbers included the use of knotted ropes by the Incas to store numerical data. Written numerals are thought to have first been used by the Egyptians, who developed the modern decimal system and introduced the concept of a number to represent zero.
Early applications of mathematics included solving problems related to commerce, the measurement of land, and the study of the heavens (astronomy). As time went by, however, man became fascinated with the study of mathematics for its own sake, which led to the pursuit of more abstract mathematical theories. The development of civilisation meant that mathematics also found many practical uses, including civil engineering projects and military campaign logistics. The foundations for the study of both pure and applied mathematics were thus set in place at a very early stage of man's development. Many discoveries attributable to the study of pure mathematics have found practical applications, while the application of mathematical solutions to real-world problems has often inspired new fields of mathematical study (the development of digital computers and electronic communication systems, for example, gave rise to the branch of mathematics known as information theory).
During the last three centuries or so, the expression of mathematical ideas has evolved from a rather laborious process of writing expressions, formulae and theorems out in plain language to a more efficient and much faster form of symbolic mathematical notation. The relative ease with which mathematicians can express their ideas using such notation comes at a price. Because the information is represented in an extremely compressed format, it is not intuitively understood by students when they first encounter it. This perhaps explains why many students shy away from or have difficulty with mathematics, despite the fact that it is a vital element of just about all branches of science and engineering, as well as being essential for a career in commerce or industry.
These pages will attempt to provide a source of reference for students taking technology-based courses of study at level 3 or higher, and will (initially at least) only cover topics that the author has encountered in relation to such courses. At the time of writing, this is likely to include number systems, algebra, geometry, trigonometry and basic calculus. The approach taken will be fairly generic, while hopefully providing sufficient insight into each topic to be useful to students studying in specialist areas, such as electronics, telecommunications or computing.