Algebra Know-It-ALL

The whole numbers starting with 0 and counting upward are usually called the natural num-

bers. Some mathematicians don’t call 0 a natural number. It’s a little like the dispute among

astronomers over whether Pluto should be called a planet, or whether empty space should be

called a part of nature. In this book, we’ll call 0 a natural number.

How Natural Numbers are Made

In Chap. 2, you saw how we can build sets from nothing. A similar scheme can be used to gen-

erate the natural numbers. From the natural numbers, we can create fractions, square roots,

and all the other kinds of numbers.

The starting point: 0

In order to build anything, we need a foundation. The natural numbers start from 0 and

increase one by one. Zero is an excellent place to begin the number-building process. If you

think of the natural numbers as evenly spaced points along a straight ray or half-line, 0 is at

the very beginning. The empty set is a good way to define 0. So let’s agree that the number 0 is

the set containing no elements, and illustrate it as a point at the left-hand end of an infinitely

long half-line (Fig. 3-1).

How 1 is defined

We can define the number 1 as the set containing the number 0, so it has one element.

That makes it different from 0, but doesn’t require that we invent anything new. Because

the number 0 is the null set, the number 1 can also be imagined as the set containing the

null set (Fig. 3-2). We now have three different ways we can write the number 1 in terms of

other things we’ve already defined:

1 = {0}

1 = {∅}

1 = {{ }}

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CHAPTER

3

Natural Numbers and Integers

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