Integers

Abstract

Before You Get Started. You have used numbers like 1,2,3,34,101, etc., ever since you learned how to count. And a little later you also met numbers like 0,–11,–,40. You know that these numbers—they are called integers—come equipped with two operations, “plus” and “times.” You know some of the properties of these operations: for example, 3+5 = 5+3 and, more generally, m+n = n+m. Another example: 3 ∙ 5 ∙ 7 is the same whatever the order of multiplying, or, more abstractly, (k ∙ m) ∙ n = k ∙ (m ∙ n). List seven similar examples of properties of the integers, things you know are correct. Are there some of these that can be derived from others? If so, does that make some features on your list more fundamental than others? In this chapter we organize information about the integers, making clear what follows from what.We begin by writing down a list of properties of the integers that your previous experience will tell you ought to be considered to be true, things you always believed anyway. We call these properties axioms. Axioms are statements that form the starting point of a mathematical discussion; items that are assumed (by an agreement between author and reader) without question or deeper analysis. Once the axioms are settled, we then explore how much can be logically deduced from them. A mathematical theory is rich if a great deal can be deduced from a few primitive (and intuitively acceptable) axioms. If you open a mathematics book in the library, you usually will not see a list of axioms on the first page, but they are present implicitly: the author is assuming knowledge of more basic mathematics that rests on axioms known to the reader. In short, we have to start somewhere. The axioms in one course may in fact be theorems in a deeper course whose axioms are more primitive. The list of axioms is simply a clearly stated starting point.