The mathematics that we teach and learn today includes concepts and ideas that once were pondered only by the most brilliant men and women of ancient, and not so ancient, times. Numbers such as 1,000, for example, or two, or zero, were at one time considered very abstract ideas. There was a time when a quantity more than two or three was simply called “many.” Yet we have grown up learning all about quantities and how to manipulate them. We teach even young children the concept of fractions as we ask them to share, or divide, their candy between them. Today, in many ways, what used to be stimulating thought for only the privileged few is now considered child’s play.
Yet scholars, philosophers, scientists, and writers of the past have spent lifetimes devising ways to explain these concepts to benefit merchants, kings, and countries. The idea that two items of different weight could fall to the Earth at the same rate was, in its time, controversial. Creating calculations that pointed to the fact that the Earth revolved around the Sun was heresy. Mathematicians have, in fact, been beheaded by kings, imprisoned by churches, and murdered by angry mobs for their knowledge. Times have changed, thankfully. It is fair to say we have come a long way.
This book is designed to help you come even further in your understanding of algebra. To start with, there is a lot of algebra that you already know. The Additive Identity Property, the Commutative Property of Multiplication, the Multiplicative Property of Equality, and the Zero Product Theorem are already concepts that, while you might not know them by name, are in your personal database of mathematical knowledge. This book will help you identify, and make a connection with, the algebra that you already do know, and it will give you the opportunity to discover new ideas and concepts that you are about to learn.