Algebra Know-It-ALL

46 Natural Numbers and Integers

account, they have a claim to $49 of your money. If you think of it as your account, you’re

$49 dollars in debt. You have, in a sense, negative $49. If you go to another store and charge

$10 more, you’ll end up with negative $59. In theory, there is no limit to how large negatively

your account, in dollars, can become. (In practice, the bank will put a limit on it.)

Negative whole numbers are denoted by putting a minus sign in front of a natural num-

ber. The exception is 0, where a negative sign doesn’t change the meaning. “Negative 0” is the

same thing as “positive 0” in ordinary mathematics. In the credit-card situation just described,

you start out with $0 and then go to −$49, then to −$59. The same thing can happen with

temperature. If it was 0 degrees yesterday afternoon and then the temperature fell by

10 degrees overnight, it was −10 degrees in the morning.

A “number reflector”

We’ve already shown how the natural numbers can be generated from sets. How can we add

the negative natural numbers to the “normal” or positive ones, making sure to include 0 so we

get the entire set of integers?

We can take two natural-number rays (or half-lines), put minus signs in front of all the

numbers on one of the rays, and then stick the rays together end-to-end so “positive 0” and

“negative 0” are on top of each other. Figure 3-4 shows how this works. You might think of the

0

1

Proceed forever!

2

{0}

{0, 1}

3 {0, 1, 2}

Proceed forever!

{0}

{0, 1}

{0, 1, 2}

1

2

3

“Number reflector” –

Figure 3-4 The negative numbers can be built up from the positive ones

by inventing an imaginary “number reflector” that reverses the

“sense” of every natural number and gives it a “twin.”