Mathematics for the liberal arts (2013)

Part II. TWO PILLARS OF MATHEMATICS

Chapter 5. NUMBER THEORY

God made integers, all else is the work of man.

LEOPOLD KRONECKER (1823–1891)

Now we turn to our second major topic in detail, the theory of numbers. While calculus studies continuously changing processes, number theory deals primarily with a discrete set, the integers. The integers are both familiar and mysterious. People have been fascinated by the integers since the beginning of history. We will investigate properties of the integers, and prove some of these properties.

5.1 What Is Number Theory?

What are numbers? In different times and places, numbers have meant different things. Here are some examples.

Counting numbers: 1, 2, 3,…

Whole numbers: 0,1, 2,…

Integers: 0, ±1, ±2,…

Rational numbers: numbers of the form , where a and b are integers and b ≠ 0

Real numbers: the numbers on a number line, e.g., the x-axis

Complex numbers: numbers of the form a + bi, where a and b are real numbers, and i² = – 1

Although people study all these numbers, the term number theory has come to mean the study of the integers. So that will be the focus of this chapter.