Centuries ago, negative numbers weren’t taken seriously. How could you have less than none of anything? When the set of all negative whole numbers was finally joined together with the set of natural numbers, the result became known as the set of integers. That set is symbolized Z, like this:
Z= {..., −3, −2, −1, 0, 1, 2, 3, ...}
Negative numbers
What is a negative number, exactly? That question is deeper than it seems at first thought. We can start to answer it by creating situations where negative numbers are really useful.
In the United States, most nonscientific people use the Fahrenheit temperature scale, where 32 degrees represents the freezing point of water. Scientists, and people outside the United States, use the Celsius temperature scale, where 0 degrees represents the freezing point of water.
In either system, temperatures often get colder than 0 degrees. Then people start calling tem-peratures negative.
Here’s another real-life situation where negative numbers come in handy. These days, nearly everyone has a credit card. When you first get the card, it has a balance of 0. That means you haven’t put any money in the bank that gave you the card, but you don’t owe the bank any money, either. What if you buy some items at the local department store, “charging”
up a balance of $49? How much money is in the account now? If you think of it as the bank’s The Integers 45
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
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.”
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 with a scheme similar to the one we used to build up the set of natural numbers.
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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= ∅
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
If we remember the basic multiplication sign rules, we can see that
−100= −5× 5 × 2 × 2
This same technique can be applied to any negative nonprime number smaller than −3 to show that it’s composite! We have to be sure that “negative primes” are allowed in the mathematical system we’re deal-ing with. Accorddeal-ing to the traditional definition, all the primes are natural numbers larger than 1, so this trick won’t work.
Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. A. The solutions in the appendix may not represent the only way a problem can be figured out. If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it!
1. If the number 0 is the set containing nothing, then what number does nothing represent?
2. The number 3 is odd. If a number n is divisible by 3 without a remainder, does that meann must be odd?
3. When an odd number is multiplied by 3, is the result always odd? If so, demonstrate why. If not, show a counterexample (a situation where an odd number is multiplied by 3 to get an even number).
Figure 3-6 This drawing, and careful choice of words, can help you avoid confusion when comparing the values of integers.
Practice Exercises 49
50 Natural Numbers and Integers
4. Find out whether or not 901 is a prime number.
5. Break down 1,081 into a product of primes.
6. Break down 841 into a product of primes. What interesting property does this number have?
7. Break down 2,197 into a product of primes. What interesting property does this number have?
8. Are any negative integers composite if we insist on using the traditional definition of a prime number?
9. Can you think of a good reason why the natural numbers 0 and 1 are not defined as prime? Here’s a hint: It should never be possible for a number to be both prime and composite.
10. Show how the natural numbers can be paired off one-to-one with the integers. Here’s a hint: Use Fig. 3-5 with its pattern of dashed, arrowed guidelines to create an “implied one-ended list” of the integers that captures them all.
Let’s take a close look at the processes, also called operations, known as addition and subtrac-tion. Much of this material will seem like a review of arithmetic to you, but you’ll need to know it “forward and backward” to work with the algebra to come later.