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239
CH 44  MULTIPLYING AND
DIVIDING SIGNED NUMBERS
 Division and Those Pesky Zeros
O
ne of the most important facts in all of mathematics is that the
denominator (bottom) of a fraction can NEVER be zero.
Sometimes this is phrased
“Never divide by zero.”
What’s the big deal?
Recall from somewhere in your past that
56  8 because 8  7 = 56
7
This is the result of
dividing by zero.
If you prefer, we can write the division problem and its “check” like
this:
8
7 56
which is checked by seeing that 8  7 = 56.
Let’s put zeros in fractions and see what happens.
Zero on the Top
How shall we interpret the division problem
0 = ???
7
What number times 7 yields an answer of 0? Well, 0 works; that is,
0  0 because 0  7 = 0.
7
So the answer is 0. Moreover, no other number besides 0 will work.
Ch 44  Multiplying and
Dividing Signed Numbers
240
Zero on the Bottom
Now let’s put a zero only on the bottom and see what happens:
9 = ???
0
Your first guess is probably 0; let’s check it out:
9 = 0 would be true only if 0  0 = 9, which it isn’t.
0
How about we try an answer of 9?
9 = 9 which is checked by seeing if 9  0 = 9. Sorry.
0
Could the answer be ? Nope;   0 = 0, not 9.
In fact, any number we conjure up as the answer will have to multiply
with 0 to make a product of 9. But this is impossible, since any number
times 0 is always 0, never 9. In short, no number in the whole world
will work as the answer to this division problem.
Zero on the Top AND the Bottom
Now for an even stranger problem with division and zeros:
0 = ???
0
We can try 0: 0 = 0 which seems reasonable, since 0  0 = 0.
0
Let’s try an answer of 5: 0 = 5 , which -- amazingly -- works out too.
0
Could  possibly work? 0 =  -- still true, since   0 is surely 0.
0
That’s three answers for this division problem: 0, 5, and .
Is there any end to this madness? Apparently not, since any number
we imagine will multiply with 0 to make a product of 0. In short,
every number in the whole world will work in this division problem.
Ch 44  Multiplying and
Dividing Signed Numbers
241
Summary:
1)
2)
3)
Zero on the top of a fraction is perfectly okay, as long as the
bottom is NOT zero. The answer to this kind of division
problem is always zero. For example, 0 = 0 .
7
There is no answer to the division problem 9 . Clearly, we
0
can never work a problem like this.
There are infinitely many answers to the division problem
0 . This may be a student’s dream come true, but in
0
mathematics we don’t want a division problem with billions
and billions of answers.
Each of the division problems with a zero in the denominator
leads to a major conundrum, so we summarize cases 2) and 3) by
stating that
DIVISION BY ZERO IS UNDEFINED!
More examples:
0 = 0
34
17 is undefined
0
“Black holes
are where
God divided
by zero.”
Steven Wright
0 is undefined
0
Note: The two division problems with zero in the denominator
may both be undefined, but they’re undefined for totally different
reasons. Your teacher may require that you understand this.
Ch 44  Multiplying and
Dividing Signed Numbers
242
Homework
1.
Evaluate each expression, and explain your conclusion:
a. 0
15
2.
3.
4.
5.
6.
b. 32
0
c. 0
0
Evaluate each expression:
a. 0
17
b. 0
9
c. 6  6
17  3
2
d. 3  8  1
100
e. 98
0
f. 44
0
g. 73  8
2 8
2
h. 7  40
23  23
i. 0
0
j.
2
k. 52  25
0  03
l. 4  53 2  10
3 9
9  9
10  10
0 = 0 because

a. 0 is the only number that when multiplied by  gives 0.
b. No number times  equals 0.
c. Every number times  equals 0.
0 is undefined because
0
a. no number times 0 equals 0.
b. every number times 0 equals 0.
c. any number divided by itself is 1.
7 is undefined because
0
a. 0 is the only number multiplied by 0 to get 7.
b. No number times 0 equals 7.
c. Every number times 0 equals 7.
a. The numerator of a fraction is 0. What can you conclude?
b. The denominator of a fraction is 0. What can you conclude?
Ch 44  Multiplying and
Dividing Signed Numbers
243
 Multiplying Signed Numbers
Recall that the result of multiplying two numbers is called the product
of the two numbers. So the product of 3 and 5 is 15. Some ways to
represent the product of 3 and 5 are
35
35
3(5)
(3)5
(3)(5)
(all of these products equal 15)
Also recall that the order in which you multiply a pair of numbers
makes no difference in the product: ab = ba for any numbers a and b.
[This is called the commutative property of multiplication.]
Positive Times Positive What should 6  4 be? It better be 24. In other
words, we see what we learned in elementary school:
A positive number times a positive number is positive.
Positive Times Negative This one’s not so obvious. We’ll determine the
answer by sneaking up on the problem while it’s not looking:
(7)(2)  14
since a positive times a positive is positive
(7)(1)  7
same rule, or, anything times 1 is itself
(7)(0)  0
anything times 0 is 0
(7)( 1)  ???
what should the product be?
What comes next in the sequence 14, 7, 0, ? Since this sequence of
numbers is decreasing by 7 at each step, the next number in the
sequence must be 7, and we see that (7)(1) = 7. It appears that
A positive number times a negative number is negative.
Negative Times Positive To calculate (3)(4), first use the commutative
propery to reverse the order of the factors: (3)(4) = (4)(3), which is
now a product of a positive with a negative. By the previous rule, we
know that the answer is 12. So (3)(4) = 12, and thus,
A negative number times a positive number is negative.
Ch 44  Multiplying and
Dividing Signed Numbers
244
Negative Times Negative Now for the most interesting situation, the
product of two negative numbers -- for example (5)(1).
( 5)(2)   10
since a negative times a positive is negative
( 5)(1)   5
same rule, or, anything times 1 is itself
( 5)(0)  0
anything times 0 is 0
( 5)( 1)  ???
what should the product be?
What comes next in the sequence of answers 10, 5, 0 ? This sequence
of numbers is growing by 5 at each step. The next number is
therefore 5, giving us the following logical pattern:
( 5)(2)   10 
( 5)(1)   5 
 The answers are growing by 5.
( 5)(0)  0


( 5)( 1) = 5
We’ve reached the inescapable conclusion that
A negative number times a negative number is positive!
See if you can deduce the two rules in the following box from the four
rules stated above.
Multiplying Signed Numbers :
If the signs are the same, the product is positive.
If the signs are different, the product is negative.
Ch 44  Multiplying and
Dividing Signed Numbers
245
Homework
7.
8.
Find the product:
a. (17)(3)
b. (4)(7)
c. 3(10)
d. (3)(4)
e. 7(2)
f. 2(7)
g. 1(8)
h. 1(9)
i. (1)(134)
j. (765)(0)
k. 3  4
l. 7  10
m. (18)(2)
n. 7(3)
o. 8(7)
p. 3  9
q. (2)(99)
r. (1)(7)
s. (7)(10)
t. (4)(5)
Find the product:
a. (2)(3)(4)
b. 7  6  3
c. (1)(2)(3)(4)
d. 4  4  4(1)(3)
e. 5(1)(1)(2)
f. 4(3)(2)(8)
g. (2)(1)(3)
h. (7)(6)(2)(1)
i. (1)(1)(1)(8)
j. (1)(3)(5)(7)
k. 3(4)(5)2
l. (2)(2)(1)(3)(5)0
 Dividing Signed Numbers
The secret to the division rules for signed numbers is the fact that
division is checked by multiplication. For example,
56  8 only because 8  7 = 56.
7
Positive Divided by Positive
6  3 , since (3)(2) = 6.
2
Therefore, a positive number divided by a positive number
is positive.
Ch 44  Multiplying and
Dividing Signed Numbers
246
Negative Divided by Positive
6   3 , because (3)(2) = 6.
2
Thus, a negative number divided by a positive number is
negative.
Positive Divided by Negative
6   3 . This can be checked by seeing that (3)(2) = 6.
2
Conclusion: a positive number divided by a negative
number is negative.
Negative Divided by Negative
6  3 , which is confirmed by the fact that (3)(2) = 6.
2
We see that a negative number divided by a negative
number is positive.
Dividing Signed Numbers :
If the signs are the same, the quotient is positive.
If the signs are different, the quotient is negative.
Notes:
1.
2.
Do these four rules for dividing signed numbers
remind you of anything? The rules for dividing
signed numbers are the same as the rules for
multiplying signed numbers.
10 = 10 =  10 because each of these three
2
2
2
division problems results in 5. In general, the
following three fractions are equal:
 a = a = a
b
b
b
Ch 44  Multiplying and
Dividing Signed Numbers
247
Homework
9.
Find the quotient (leave answers as simplified fractions):
a. 30
5
b.
60
10
c. 100
20
d. 14
2
e. 3
6
f.
18
4
g. 24
5
h. 34
24
j.
345
345
k. 10
10
l.
c. 3  3  9
7
d. 18  2
2  18
g. 12  15
3
3
h. 9  1
6 5
i.
10.
11.
1
9
120
70
Evaluate each expression:
10(2)
4
a. 5  1
1 2
b.
e. 1  2  3
2  4
f. 10  2
2 1
Evaluate each expression, leaving your answers as decimals
rounded to the tenths place:
a.
(2)( 3)
( 3)(4)
b.
1  2  3
2  3  4
c.
20  (16  4)
32  22
d.
1  ( 2)
1  4
e.
10  ( 2)
( 1  4)
f.
(1)(2)( 3)( 4)
12(2)( 1)
Ch 44  Multiplying and
Dividing Signed Numbers
248
Solutions
1.
a. 0 = 0 since 0  15 = 0, and 0 is the only number that accomplishes
15
this.
32 is undefined because any number times 0 is 0, never 32; thus NO
0
number works.
b.
0 is undefined because any number times 0 is 0; thus EVERY
0
number works.
c.
2.
a. 0
b. 0
c. 0
d. 0
e. Undefined
f. Undefined
i. Undefined
j. Undefined
g. Undefined
h. Undefined
k. Undefined
l. 0
3.
a.
b.
6.
a. You can’t conclude anything -- it depends on what’s on the bottom. If
4.
5.
b.
the bottom is a non-zero number (like 7), then 0 = 0 . If the bottom is
7
zero, then 0 is undefined.
0
b.
This time we can conclude that the fraction is undefined, since
division by 0 is undefined, no matter what’s on the top of the fraction.
28
9
21
20
7.
a. 51
g. 8
m. 36
s. 70
b.
h.
n.
t.
8.
a. 24
g. 6
b. 126
h. 84
c. 30
i. 134
o. 56
d. 12
j. 0
p. 27
e. 14
k. 12
q. 198
f. 14
l. 70
r. 7
c. 24
i. 8
d. 192
j. 105
e. 10
k. 120
f. 192
l. 0
Ch 44  Multiplying and
Dividing Signed Numbers
249
9.
e. 1
2
a. 6
b. 6
c. 5
d. 7
f.  9 =  4 1
2
2
g.  24 =  4 4
5
5
l. 12 = 1 5
7
7
h. 17 = 1 5
12
12
i.
g. 9
k. 1
10. a. 4
b. 5
c. 0
d. 1
e. 1
f. 7
11. a. 0.5
b. 0.7
c. 0
d. 1
e. 1.6
f. 1
Ch 44  Multiplying and
Dividing Signed Numbers
1
9
j. 1
h. 8
250
 Jean Piaget (1896-1980) Swiss cognitive psychologist
Ch 44  Multiplying and
Dividing Signed Numbers