Mathematics: The Study of Quantity, Structure, Space, and Change

Abstract

Mathematics provides the last letter of the acronym STEM for science, technology, engineering and mathematics. However, it forms the basis of all of the STEM components. Mathematics is the most precise language for science, the most important tool of engineering and is broadly used in technology from deciphering the genetic code to solar energy generation.Most of the mathematics that is needed to understand science, technology and engineering can actually be taught in middle school. The chapter contains a detailed section on numbers starting with the numbers that we use to count and their symbolic representation by different cultures. This basic number system is extended to negative numbers and the system of integers, to fractions or rational numbers and to numbers that cannot be written as fractions (irrationals). The section on numbers ends with binary numbers that are the basis for modern computation and explains how computers “crunch” numbers.The arithmetic of numbers leads then in a natural way to the algebra of solving equations for numbers that are at first not known. The examples presented are engineering problems such as designing a garden sprinkler system or an electrical circuit.This knowledge of the arithmetic and algebra of numbers is used to introduce the geometry of Euclid, starting with his axioms and discussing some famous rules (or theorems, such as that of Pythagoras). Applications include the spherical geometry of the earth, the distance to the horizon and the timing of sunsets on beaches and mountains. Finally these geometrical methods are used to explain how geometry problems are implemented on and solved by computers, how computer graphics is accomplished and how the global positioning system (GPS) works.The mathematics presented in this first chapter is sufficient to understand all topics and to solve explicitly all problems presented in Chaps. 2, 3, and 4. In addition explanations are given how to solve these same problems by using the computer software MATHEMATICA in elementary ways. The last Chap.5 deals with advanced problems. There, additional mathematics, generally known as differential and integral calculus is covered in a pragmatic way and at the high school level.