Algebra Know-It-ALL

Building new numbers

Let’s make up a rule that we can use to generate new natural numbers, one after another.

Suppose we’ve built a certain natural number. Call it n. If we want to create the next higher

natural number, n+ 1, we can take all the natural numbers up to and including n, make them

into elements of a set, and then call that set the new number. A mathematician would write

it like this:

n+ 1 = {0, 1, 2, 3, ..., n}

If we have any particular natural number p, we can define it on the basis of all the natural

numbers less than itself, like this:

p= {0, 1, 2, 3, 4, ..., p − 1}

If that’s a little too abstract for you, look at Fig. 3-3. This drawing shows the first six natural

numbers (0, 1, 2, 3, 4, and 5) as they are built up and assigned to a vertical stack of points that

ascends upward as far as we care to go.

This is a nifty scheme! A natural number p is a set containing p elements. In theory, those

elements could be anything, such as apples, stars, atoms, or people. But it’s convenient to use

all the natural numbers less than p as those elements. That allows us to build the set of natural

numbers, one on top of the other, like stacking coins. Just as each coin in the stack rests on all

the coins below itself, every element p in the set of natural numbers “rests on” all the natural

numbers “below” itself. Once this process is defined, it sets in motion a mathematical “chain

reaction” that never ends.

36 Natural Numbers and Integers

Start here

and go on forever

Figure 3-1 The number 0 can be defined as the null set. We

can show how it starts to generate natural numbers

by placing it at the beginning of an endless string

of points.

{ }

(^01) Whole numbers
yet to be defined
Figure 3-2 The number 1 is the set containing the null
set, which is the same as the set containing the
number 0. It has one element.