Algebra Know-It-ALL

natural numbers as being attached to a ray that stands straight up above the “number reflec-
tor,” and their negatives as being attached to a ray that dangles straight down.
This is a fine way to imagine the integers, but in mathematical terms, it is a little “impure.”
In order to define the negative numbers this way, we have to come up with new gimmicks
that we did not need to define the natural numbers. A pure mathematician would demand
some way to define all the integers, positive, negative, and 0, using only the idea of a set and
nothing else. We can define the entire set of integers in the same way as we defined the set of
natural numbers. Put your “abstract thinking cap” on again (if it isn’t glued to your head by
now), and keep in mind that what you’re about to read does not represent the only way the set
of integers can be defined in a “pure” way.

Building the integers

The natural numbers have a clear starting point, which is 0. But the integers go on forever in
two directions. At least, that’s the impression you’ll get if you look at Fig. 3-4. How can you
start moving along a line that goes in two directions, and cover every point on it? You have to
pick one direction or the other, right?
Wrong! In the real world that might be true, but in the “mathematical cosmos” we have
powers that ordinary mortals lack.
Take a look at Fig. 3-5. Instead of hopping from 0 to 1, and then from 1 to 2, and then
from 2 to 3, always moving in the same direction, suppose you hop alternately back and forth.
Start at 0, then move up one unit to 1. Then go down two units to −1. Then go up three units

0

1

2

{0}

{0, 1}

3

• 2

{0, 1, - 1 }

• 1

{0, 1, -1,2}

{0,1, 1-,22,- }

• 3 {0, 1, - 1 ,2 2,- , 3 }

Start here

Proceed

forever!

Figure 3-5 Here’s a way to generate the set of integers

with a scheme similar to the one we used

to build up the set of natural numbers.

The Integers 47