Numbers are crucial to computing. In addition to using a computer to execute numeric computations, all types of information that we store and manage using a computer are ultimately stored as numbers. At the lowest level, computers store all information using just the digits 0 and 1. So to begin our exploration of computers, we need to first begin by exploring numbers.

First, let’s recall that numbers can be classified into all sorts of categories. There are natural numbers, negative numbers, rational numbers, irrational numbers, and many others that are important in mathematics. Only some of these are crucial to the understanding of computing. Let’s review the relevant category definitions briefly.

First, let’s define the general concept of a number: A number is a unit belonging to an abstract mathematical system and is subject to specified laws of succession, addition, and multiplication. That is, a number is a representation of a value, and certain arithmetic operations can be consistently applied to such values.

Now let’s separate numbers into categories. A natural number is the number 0 or any number obtained by repeatedly adding 1 to this number. Natural numbers are the ones we use in counting. A negative number is less than zero and is opposite in sign to a positive number. An integer is any of the natural numbers or any of the negatives of these numbers. A rational number is an integer or the quotient of two integers—that is, any value that can be expressed as a fraction.

In this chapter, we focus on natural numbers and the ways that they are represented in various number systems. As part of our discussion, we establish how all number systems relate to each other. In Chapter 3, we examine the computer representation of negative and rational numbers, as well as how we use numbers to represent other forms of data such as characters and images.

Some of the material in this chapter may already be familiar to you. Certainly some of the underlying ideas should be. You probably take for granted some basic principles of numbers and arithmetic because you’ve become so used to them. Part of our goal in this chapter is to remind you of those underlying principles and to show you that they apply to all number systems. Then the idea that a computer uses binary values—that is, 1s and 0s—to represent information should be less mysterious.