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Suppose p is an integer and n is a positive integer. From the above facts, you can see that whenever you multiply p by n, you add p to itself (n− 1) times. On the number line, you move away from the “number reflector” (n− 1) times by a distance equal to the abso-lute value of p. If your starting integer p is above the “number reflector,” you move up; if your starting integer p is below the “number reflector,” you move down. Figure 5-1 illustrates how this works for 2 × 3 and −2 × 5. The “number reflector” is shown as a horizontal, dashed line.
What if n is negative instead of positive? To multiply p by n in this situation, first take the additive inverse (negative) of your starting integer p, and then move away from the “number reflector” |n|− 1 times by a distance equal to the absolute value of p. Figure 5-2 shows how this works for 2 × (−3) and −2× (−5).
When you multiply two quantities, you get a product. In a multiplication problem, the first quantity (the one to be multiplied) is sometimes called the multiplicand, and the second quantity (the one you are multiplying by) is sometimes called the multiplier. More often, they are both called factors.
To divide, move in
Now imagine the multiplication process in reverse. For examples, look at Figs. 5-1 and 5-2 and think backward!
In Fig. 5-1, suppose you start with 6 and you want to divide it by 3. You reduce your distance from the “number reflector” by a factor of 3 but stay on the same side, so you end up 66 Multiplication and Division
−2 is multiplied by 5. To avoid clutter, only the even-integer points are shown on the number line.
at the point representing 2. If you start with −10 and divide by 5, you cut your distance from the “number reflector” by a factor of 5 and stay on the same side, so you end up at −2.
Dividing by a negative number is trickier. In Fig. 5-2, suppose you start with −6 and divide it by −3. You jump to the other side of the “number reflector” and then reduce your distance from it by a factor of 3, finishing at 2. If you start with 10 and divide by −5, you jump to the other side of the “number reflector” and then cut your distance from it by a factor of 5, finishing at −2.
In these situations, the integers divide each other “cleanly” without remainders. Remain-ders occur when one nonzero integer divides another integer and the result is not an integer.
“Messy division” produces fractions, which we’ll study in Chap. 6. An example is the division of 5 by 3. Another example is the division of −7 by −12.
You can add, subtract, or multiply any two integers and always end up with another inte-ger. In math jargon, the operations of addition, subtraction, and multiplication are closed over the set of integers. When you do a division problem with integers, you don’t always get another integer. Therefore, the operation of division is not closed over the set of integers.
When you divide a quantity by another quantity, you get a quotient. Sometimes it is called aratio. In a division problem, the first number is sometimes called the dividend, and the sec-ond number is sometimes called the divisor.
Division by 0
What happens if you try to divide an integer by 0? You don’t get any integer, or any other known type of number. Look at this problem “inside-out.” What must you multiply 0 by if Moving Out and In 67 even-integer points are shown on the number line.
you want to get, say, 3? How many times must you add 0 to itself to get anything but 0? No integer can do this trick. In fact, no known number solves this problem.
Most mathematicians will tell you that division by 0 is “not defined.” That’s what my 7th-grade math teacher kept saying, and I pestered her about it. I would ask, “Why not?” or retort, “Let’s define it, then!” She would repeat herself, “Division by 0 is not defined.” I did not take her seriously, so I started trying to make division by 0 work. I never came up with a well-defined way to do it. But I came pretty close, and had a lot of fun trying.
Are you confused?
Have you heard that dividing a positive integer by 0 gives you “infinity”? If not, you probably will some day. Be skeptical! The first thing you must do to figure out if it’s really true is to define “infinity.” That’s not easy. No meaningful, enduring definition of “infinity” produced by mathematicians has ever had anything to do with division by 0.
Again, look at the problem “inside-out.” If you want to multiply 0 by any positive integer n, you must add 0 to itself (n− 1) times. No matter how large you make n, you always get 0 when you add it to itself (n− 1) times. Why should adding 0 to itself forever make any difference? It’s tempting to suppose that it might, but that doesn’t prove that it will. In mathematics, we need proof before we can claim something is true!
Manipulating equations
Whenever we add or subtract a certain quantity to or from both sides of the equation, we still have a valid equation. The same is true if we multiply both sides of an equation by a certain quantity, or divide either side by a certain quantity other than 0.
The quantity you add, subtract, or multiply by can be a number, a variable, or a compli-cated expression, as long as it is the same for the left-hand side of the equation as for the right-hand side. If you divide both sides of an equation by anything, it’s best to stick to nonzero numbers. If you divide both sides of an equation by a variable or an expression containing a variable, you can get into trouble, as you’ll see in Chap. 11.
Keep these rules in mind. That way, you won’t get confused later on when we do some-thing like divide both sides of an equation by 999, or multiply both sides by (a+b). The fact that we can do these things makes solving equations and proving various facts far easier than they would be otherwise.
Here’s a challenge!
In terms of the number line and displacements, show what happens when you multiply the integer −1 over and over, endlessly, by −2.
Solution
Figure 5-3 illustrates this process. Because the multiplier is negative, we jump to the opposite side of the “number reflector” each time we multiply. Then the result becomes a new multiplicand. Because the 68 Multiplication and Division
absolute value of the multiplier is 2, we double our distance from the “number reflector” with each jump.
Expressed as equations, we have
−1× (−2)= 2
2 × (−2)= −4
−4× (−2)= 8
8 × (−2)= −16
−16× (−2)= 32
↓ and so on, forever
Moving Out and In 69
Start at -1 2 8
- 4
-16
To 32
Figure 5-3 Here is what happens when we start at
−1 and multiply by −2 over and over.
We jump back and forth across the
“number reflector,” doubling our distance from it with each jump. To avoid clutter, only the even-integer points are shown on the number line.
Identity, Grouping, and Signs
Let’s review how signs work in multiplication and division. Then we’ll proceed to the three major laws that govern the interplay between multiplication, division, addition, and subtraction.
Notation for multiplication
When you want to multiply two numbers, you can use the familiar “times sign” and put the numerals for the factors on either side. This symbol (×) looks like a tilted cross or a letter “x.”
Another symbol you’ll often see is the small, elevated dot (·). When a number is multiplied by a variable, or when a variable is multiplied by another variable, you’ll see their symbols run together without any space between. Parentheses are placed around complicated expressions when they are multiplied by each other.
• When you see 3 × 7, it means 3 times 7.
In this book, we’ll use the forward slash (/) to indicate division. In arithmetic, you some-times see the dash with two dots (÷), but that’s rarely used in algebra. When expressions are complicated, the dividend (the number you want to divide) can be placed on top of a long horizontal line, and the divisor (the number you divide by) is placed underneath. As with multiplication, parentheses are placed around complicated expressions when they are divided by each other.
You can multiply or divide any integer by 1, and it won’t change the value. Because multiply-ing or dividmultiply-ing by 1 always gives you the same number again, 1 is called the multiplicative identity element. (For some reason I’ve never heard it called the “divisive identity element,” but technically this term is okay.) For any integer a
a1=a 70 Multiplication and Division
and
a/1=a The sign-changing element
When you multiply or divide any integer by −1, you reverse the sign but do not change the absolute value. A positive integer becomes negative, and a negative integer becomes positive, but the distance from 0 on the integer line stays the same. For any integer a
a(−1)= −a and
a/(−1)= −a Conversely,
−a(−1)=a and
−a/(−1)=a
Note that a(−1) here means a times −1, not a minus 1. Those parentheses are important! The inte-ger−1 can be called the multiplicative sign-changing element or the divisive sign-changing element.
Parentheses in simple products and quotients Look at these expressions:
3× (−5)=a and
15/(−3)=b If you don’t write the parentheses, the above expressions are
3× −5=a and
15/−3=b
Expressions like these might be clear enough to you, but they would confuse some people.
Don’t be afraid to add parentheses to an expression if you think they will prevent ambiguity.
Just be sure that for every opening parenthesis you put in, you include a corresponding closing parenthesis later in the expression.
Identity, Grouping, and Signs 71
Signs in multiplication and division
Here is a set of rules for multiplication and division by any integer except −1, 0, or 1. Remem-ber that “more” always means more positive, or moving upward on the integer line, and “less”
always means more negative, or moving down. Study these rules carefully. If any of them confuse you, “plug in” some actual numbers and test them.
• When you multiply a positive integer by 2 or more, the result stays positive and the absolute value increases (it gets farther from 0).
• When you divide a positive integer by 2 or more, the result stays positive and the abso-lute value decreases (it gets closer to 0).
• When you multiply a negative integer by 2 or more, the result stays negative and the absolute value increases (it gets farther from 0).
• When you divide a negative integer by 2 or more, the result stays negative and the absolute value decreases (it gets closer to 0).
• When you multiply a positive integer by −2 or less, the result becomes negative and the absolute value increases (it gets farther from 0).
• When you divide a positive integer by −2 or less, the result becomes negative and the absolute value decreases (it gets closer to 0).
• When you multiply a negative integer by −2 or less, the result becomes positive and the absolute value increases (it gets farther from 0).
• When you divide a negative integer by −2 or less, the result becomes positive and the absolute value decreases (it gets closer to 0).
“Homogenize” these!
You would do well to memorize (or, as my dad would always say, “homogenize”) two general facts. For any two integers a and b
a× (−b)= −ab= −(ab) and
a/(−b)= −a/b= −(a/b)
Are you confused?
Suppose you see an expression with addition, subtraction, multiplication, and division all mixed up, but with no parentheses. Here’s an example.
2+ 48/4 × 6 − 2 × 5 + 12/2 × 2 − 5
You’ll get an answer that depends on which operations you do first. Do not approach a mixed-operation problem like this by simply grinding out the arithmetic from left to right. You must use certain rules of precedence. In this order:
• Group all the multiplications
• Do all the multiplications from left to right 72 Multiplication and Division
• Group all the divisions
• Do all the divisions from left to right
• Convert all the subtractions to negative additions
• Do the additions from left to right
When you use these rules correctly, the above problem simplifies as shown in Table 5-1. As you read down-ward, each expression is equal to the one above it.
Here’s a challenge!
Start with the integer 5, multiply by −4, then divide that result by −2, then multiply that result by 8, then divide that result by −20, and finally divide that result by −4. What do you end up with?
Solution
We can break this down step-by-step, paying careful attention to signs and using parentheses when we need them:
5 × (−4)= −20
−20/(−2)= 10
10 × 8 = 80
80/(−20)= −4
−4/(−4)= 1