Starting at the beginning: The natural numbers

Abstract

In this text, we will review the material you have learnt in high school and in elementary calculus classes, but as rigorously as possible. To do so we will have to begin at the very basics - indeed, we will go back to the concept of numbers and what their properties are. Of course, you have dealt with numbers for over ten years and you know how to manipulate the rules of algebra to simplify any expression involving numbers, but we will now turn to a more fundamental issue, which is: why do the rules of algebra work at all? For instance, why is it true that a(b+c) is equal to ab + ac for any three numbers a, b, c? This is not an arbitrary choice of rule; it can be proven from more primitive, and more fundamental, properties of the number system.