Algebra Know-It-ALL

48 Natural Numbers and Integers

to 2, down four units to −2, up five units to 3, down six units to −3, and so on. Keep hop-

ping alternately up and down, making your hop one unit longer every time. In Fig. 3-5, the

integers themselves are shown to the left side of the vertical line, and their equivalents, built

up as sets of previously defined integers, are shown on the right side. Pick any integer, positive

or negative, as big or small as you want. You’ll eventually reach it if you make enough hops.

The next time you are at a party with a bunch of mathematics lovers and somebody asks

you, “What is the number −2, really?” you can say, “Well, that can be debated. But if you like,

we can define it as the set containing 0, 1, −1, and 2.” That should get you a raised eyebrow.

If you want to bring down the house, you can go to an old-fashioned chalk blackboard (every

good mathematics party has one, right?) and scribble out the following to make your point:

0 =∅

1 = {0} = {∅}

−1= {0, 1} = {∅, {∅}}

2 = {0, 1, −1} = {∅, {∅}, {∅,{∅}}}

−2= {0, 1, −1, 2} = {∅, {∅}, {∅, {∅}},{∅, {∅}, {∅, {∅}}}}

↓

and so on, forever

Are you confused?

The integers can get confusing when you compare values. If you draw a number line and represent the

integers as points on it, such as is done in Figs. 3-4 or 3-5, what does it mean if one number is “larger” or

“smaller” than another? How about the expressions “less than” or “greater than”?

A mathematician will tell you that the integers get smaller as you move downward in Figs. 3-4 or 3-5, and

larger as you go upward. For example, −5 is smaller (or less) than −2, and any negative integer is smaller

(or less) than any natural number. Conversely, −2 is larger (or greater) than −5, and any natural number

is larger (or greater) than any negative integer. But that can begin to seem strange if you think about it

awhile. How can −158 be “smaller” than −12? If you find yourself in debt by $158, isn’t it a bigger problem

than if you are in debt by $12?

In the literal sense, −158 is indeed smaller (or less) than −12, just as −158° is colder than −12°. In fact,

the integer −158 is less than −12 or −32 or −157. But −158 is larger negatively than −12 or −32 or −157.

To avoid confusion when comparing numbers, the best policy is to be careful with your choice of words.

Figure 3-6 should clear up any lingering uncertainty you might have about this.

Here’s a challenge!

If we allow all negatives of primes (i.e., −2, −3, −5, −7, −11, −13, −17, −19, ...) to be called prime, does

that make all the nonprime negative numbers composite?

Solution

Let’s keep the traditional definition of composite number: a product of two or more primes. Now imagine

that we have some positive composite number. It is therefore a product of primes that are all positive. If we

make one of those primes negative, we get the negative of that composite number. For example:

100 = 5 × 5 × 2 × 2