Download Lesson 22: More Operations with Negative Numbers

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Lesson 22  page 1
Lesson 22
More Operations with Negative Numbers
Now that we can add and subtract with negative numbers, we fill in the rest of the arithmetic operations: multiplying, dividing,
and raising negative numbers to a power.
Multiplying with Negative Numbers
If you think of multiplication as repeated addition, a problem like 3(4) means 4 + 4 + 4 = 12. To show
multiplication graphically we could line up three columns of four. The graphic model is helpful because it
makes it clear that multiplication is commutative, since 3 columns of 4 is the same as 4 rows of 3.
To show 3(–4), then, we could line up 3 columns with –4 in each column:
Adding –4 three times we get –4 + –4 +–4 = –12. And since multiplication is
commutative, this should be the same as the multiplication (–4)(3).
We now have:
4 • 3 = 12
3 • –4 = –12
–4 • 3 = –12
You can see from the way we did the above examples that the problems below will work the same way. (If you take a minute
to verify this yourself by drawing a picture as above, you are destined for mathematical greatness.)
4 • 3 = 12
4 • –3 = –12
–3 • 4 = –12
The only difficulty remaining is the problem (–3)(–4). I can’t make –3 columns, or –4 rows, those don’t make sense. It’s
probably some form of 12, but is it positive or negative?
A Negative Times A Negative
There are two ways to think about a double negative in grammar. In some languages or sentence constructions a double
negative intensifies the negative effect. “I will never, never go there again,” means that the person feels very strongly about
never going to that awful place again. Sometimes a double negative is a kind of reversal, the opposite of the opposite, that
turns around to a positive. “I do not disagree,” means the speaker agrees with you.
Let’s look at a simple
situation. Each number is
multiplied by –1. At first we
know what to do, and then
we hope our momentum will
carry us along:
It’s easy to see a pattern.
Multiplying a positive
number by –1 changes the
number to its opposite,
which is negative. The
products are moving along
the number line in the
positive direction.
To fill in the missing
numbers, we go with the
pattern.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 22  page 2
The pattern on the number line suggests that multiplying a number by –1 changes it to its opposite. If the number is positive
and multiplied by –1 it will become negative and if the number is negative and is multiplied by –1 it will become positive. The
product of two negative numbers is positive.
The Four Types of Multiplication Problems
SAME SIGN
DIFFERENT SIGNS
3 • 4 = 12
(–3) • (–4) = 12
3 • (–4) = –12
(–3) • 4 = –12
positive • positive
negative • negative
positive • negative
negative • positive
ANSWER is POSITIVE
ANSWER is NEGATIVE
Example: Multiply.
8•2
8 • –2
16
–8 • –2
–16
–8 • 2
16
–16
When we multiply strings of factors, remember that pairs of negatives turn positive. If there are three negatives,
(negative • negative) • negative =
(positive)
• negative = negative
When multiplying several numbers, you multiply them in order, two by two. Or you can figure out the sign of the answer first.
Then just multiply the numbers.
Example: Multiply.
(2)(–3)(5)
(–2)(–3)(5)
(–6)(5) = –30
(6)(5) = 30
(–2)(–3)(–5)
(2)(–3)(–5)
(6)(–5) = –30
(–6)(–5) = 30
Example: Write a multiplication to find the answer.
Travis owed $10 each to 5 different friends. How much did
he owe in all?
The submarine started at the surface and made 4 dives,
each 200 feet down. How far down was the submarine?
(–10)(5) = –50
4(–200) = –800
He owed $50.
It was 800 feet below the surface.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 22  page 3
Using the Distributive Property with Negative Numbers
Everything works exactly the same way it did before. Really.
Example: Simplify using the distributive property.
3( 5x 8)
3(2x 7)
3(2x ) –3(7)
6x
3( 5x )
3(4x 6)
15x
15x
–21
6x 21
Remember to use the efficient notation
and change +–21 to – 21.
3(4x
3(8)
3(4x )
24
24
6)
3( 6)
12x 18
Same here, change adding a negative
back to subtraction.
Here change the subtraction to addition
before you start, so you know it’s a –6.
Dividing with Negative Numbers
The related multiplication/division equations allow us to figure out the rules for division from the multiplication rules.
3 • 4 = 12
is related to the equation
12 ÷ 4 = 3
–3 • –4 = 12
is related to the equation
12 ÷ –4 = –3
3 • –4 = –12
is related to the equation
–12 ÷ –4 = 3
–3 • 4 = –12
is related to the equation
–12 ÷ 4 = –3
positive ÷ positive
POSITIVE
positive ÷ negative
NEGATIVE
negative ÷ negative
POSITIVE
negative ÷ positive
NEGATIVE
Just as in multiplication, if the signs are the same, the quotient is positive. If the signs are different, the quotient is negative.
Example: Divide.
8÷2
8 ÷ –2
4
–8 ÷ –2
–4
–8 ÷ 2
4
–4
Example: Write a division to find the answer.
The 25 negative team members were split evenly between 5
houses. How many in each house?
Jake wants to lose 16 pounds over the next 8 weeks. How
much should he lose per week?
–25 ÷ 5 = –5
–16 ÷ 8 = –2
5 negative team members in each house.
He should lose 2 lb per week.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 22  page 4
Negative Fractions
Since fraction bars are a kind of division symbol, the rules about
division with negatives help with dealing with negative fractions.
Think of the fraction 1/2 as the answer to the division
problem 1 ÷ 2. Then the fraction –(1/2) is the answer to the
division problem –1 ÷ 2 or to the division problem 1 ÷ –2.
In fraction notation, if the fraction is negative, we can write the
negative sign in front, in the numerator, or in the denominator.
Since negative ÷ negative = positive, when numerator and
denominator are both negative the fraction should be written as
positive.
Negative Fraction Notation
The negative sign can go in front of the fraction, or in
the numerator, or in the denominator (least preferred).
1
2
1
2
1
2
If both numerator and denominator are negative, the
fraction is positive.
1
2
1
2
Example: Simplify.
5 3
•
6 10
1
1
5
3
6
2
•
10
6
1
4
18 •
2
positive • negative = negative
5
3
18 5
•
1 3
30
negative • negative = positive
Exponents on Negative Numbers
Now that we know how to multiply with negative numbers, we can figure out what a negative number to a power should be.
Example: Write each expression as a multiplication, then find the product.
35
(3)(3)(3)(3)(3)
243
( 3)5
( 3)( 3)( 3)( 3)( 3)
243
(3)4
(3)(3)(3)(3)
81
( 3)4
( 3)( 3)( 3)( 3)
81
Notice the signs of the answers. If a negative number is raised to a power, the result may be either positive or negative. It
depends on how many negatives there are in the product. If there are an even number of negatives, they will form positives
in pairs, and the result will be positive. If there are odd numbers of negatives, the pairs will make positives and there will be
one more negative left over, making the result negative.
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 22  page 5
Exponentiation comes first in the order of operations, so unless the parentheses hold the negative sign onto the number, the
exponent doesn’t apply to the negative sign. It works like this:
( 3)4
( 3)( 3)( 3)( 3)
but
34
(3)(3)(3)(3)
Every math student I have ever met has found this confusing at first, so don’t get discouraged if you don’t remember it every
time right away. It seems a bit arbitrary and finicky at first, but there are good reasons for it when working with variables, so
invest the time to understand and remember the difference between the two expressions.
Example: Write each expression as a multiplication, then find the product.
( 5)2
52
(5)(5)
25
(52 )
( 5)( 5)
25
52
(5 • 5)
25
(5)(5)
25
A Lot of Rules
Negative numbers really altered the way we look at addition and subtraction, because a negative is opposite a positive in
terms of addition. But once we established how the new numbers behave in addition and subtraction, the rest of the
operations follow. Multiplication is repeated addition, and our understanding of the negatives from addition gave us the two
most important rules of multiplication:
The product of two numbers with the same sign is positive.
The product of two numbers with different signs is negative.
Everything else is a consequence of this. Understanding division comes from the related multiplication/division equations,
and the same rules apply there. Understanding exponents with negatives comes from writing them as multiplications, and
being finicky about notation and the order of operations.
Now that we can do all the arithmetic operations, the negatives are full-fledged real numbers at our disposal. Now we’ll start
using them in formulas and equations.

© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 22  page 6
Lesson 22: More Operations with Negative Numbers
Worksheet
Name _________________________________________
1. Fill in the answers to the problems and the sign of the answer in the chart:
SAME SIGN
DIFFERENT SIGNS
3 • 4 = 12
(–3) • (–4) = 12
3 • (–4) = –12
(–3) • 4 = –12
positive • positive
negative • negative
positive • negative
negative • positive
ANSWER is POSITIVE
ANSWER is NEGATIVE
2. Simplify.
7( 3)
2•5
35 ( 7)
6 3
1• 9
( 4)( 6)
54 ( 6)
32 ( 8)
3. Write the exponential expression as a multiplication, then evaluate.
24
( 2)4
24
( 2)5
25
\
25
4. Use the distributive property to simplify.
5(2x 1)
© 2010 Cheryl Wilcox
5( 2x 1)
5(2x 1)
Free Pre-Algebra
Lesson 22  page 7
5. Simplify
1
5
15
2
1
5
15
2
1
5
15
2
6. Mixed Practice
3 4
3 4
3( 4)
3
4
3
3 9
3( 9)
3 ( 9)
9
18
9
18 9
18 ( 9)
18( 9)
23
20
5
© 2010 Cheryl Wilcox
( 2)3
20
5
( 2)(3)
20
5
( 2)( 3)
20
5
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Lesson 22  page 8
Lesson 22: More Operations with Negative Numbers
Homework 22A
Name _____________________________________
1. Simplify.
2. Simplify.
a.
10
16
b.
8
2
c.
17
1
d.
a.
6 2•3
5 1
b.
2 7•2
2•9 3•6
c.
102
510
2a 5b
•
b2 6
g.
6 33
•
13 35
42
23
d.
|4 5|
2
e.
–9 4
2•5 5
189a 2
e.
819a
f.
43
3. Solve the equations.
a. 5x
h. Build four fractions equivalent to
2
.
5
b. 3(x
i. Change
8 22
1) 27
111
to a mixed number.
5
j. Change 17
2
to an improper fraction.
9
© 2010 Cheryl Wilcox
c.
2x
3
6
Free Pre-Algebra
Lesson 22  page 9
4. Simplify each expression, then compare with >, <, =.
5. Change any subtractions to additions, then find the sum.
a. 5
a. 6
3 8
8 2
b. 7
b.
5
8 3
c. 8 ( 7)
d. 9
c.
5
(–3)2
2
8 10 9
e. 8 ( 8)
6. Using the formula |a – b|, find the distance between
7. Simplify.
a. 10 and 3
a. 4 • 9
b. 10 and –3
b. ( 4)( 9)
c. 36 ( 4)
c. Sam on Floor 87 and Holly on Floor –22.
d. 36 ( 9)
e.
36
4
8. Write the expression as a multiplication, then evaluate.
9. Simplify.
a. 53
a. 6x
3x
b. ( 5)3
b. 6(x
3)
c. 53
c. 6(x
3)
d. 6(x
3) 2( 3x 9)
d. ( 5)4
© 2010 Cheryl Wilcox
Free Pre-Algebra
Lesson 22  page 10
Lesson 22: More Operations with Negative Numbers
Homework 22A Answers
1. Simplify.
a.
10
16
b.
8
2
2 •5
2 •2•2•2
2 •2•2
17
c.
1
d.
2. Simplify.
4
1
2
5
8
a.
6 2•3
5 1
4
b.
2 7•2
2•9 3•6
17
c.
2 • 3 • 17
102
510
2 • 3 • 5 • 17
189a 2
e.
819a
1
5
3 • 3 •3• 7 • a •a
3 • 3 • 7 • 13 • a
2a 5b
•
b2 6
2 •a 5• b
•
b• b 2 •3
5a
3b
6 33
•
g.
13 35
2 • 3 3 • 11
•
13 5 • 7
198
455
f.
h. Build four fractions equivalent to
2
5
i. Change
3a
13
4
10
6
15
43
42
2
d.
|4 5|
2
e.
–9 4
2•5 5
j. Change 17
a. 5x
2 14
18 18
| 1|
2
16
undefined
0
48
8
6
5
5
1
1
2
5
10 5
8 22
5x
8 22
5x 30
x
b. 3(x
1
5
3x
c.
2x
3
5x 8 8 22 8
5x / 5 30 / 5
6
1) 27
3 27
3x 24
x
2
to an improper fraction.
9
2 153 2 155
17
9
9
9
9
© 2010 Cheryl Wilcox
0
3. Solve the equations.
2
.
5
8 10
20 25
22
0
6
64 16
8
3
111
to a mixed number.
5
111 5 22r 1
6 6
6
3x 3 3 27 3
3x / 3 24 / 3
8
6
2x
3
2x
x
6
18
9
2x
•3 6•3
3
2x / 2 18 / 2
Free Pre-Algebra
Lesson 22  page 11
4. Simplify each expression, then compare with >, <, =.
5. Change any subtractions to additions, then find the sum.
a. 5
a. 6
3 8
5
5
8 2
b. 7
6
2
5
9
8 7 15
5
d. 9
8 10 9
(9 8) (10
5 < 5
5
2
c. 8 ( 7)
8 3
5
c.
0
7
5 < 5
b.
8 2
9) 1 1 2
e. 8 ( 8)
(–3)2
8 8 0
5 < 9
6. Using the formula |a – b|, find the distance between
7. Simplify.
a. 10 and 3
a. 4 • 9
10 3
7
36
36
b. ( 4)( 9)
b. 10 and –3
10 ( 3)
10 3
13
c. Sam on Floor 87 and Holly on Floor –22.
87 ( 22)
87 22
c. 36 ( 4)
9
d. 36 ( 9)
4
109 floors
e.
36
4
9
8. Write the expression as a multiplication, then evaluate.
9. Simplify.
a. 53
a. 6x
3x
(5)(5)(5) 125
9x
b. 6(x
b. ( 5)3
( 5)( 5)( 5)
125
c. 6(x
c. 53
(5)(5)(5)
d. ( 5)
d. 6(x
( 5)( 5)( 5)( 5) 625
6x
18
6x
18
3)
125
4
© 2010 Cheryl Wilcox
3)
3) 2( 3x 9)
6x
18
6x 18 0
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Lesson 22  page 12
Lesson 22: More Operations with Negative Numbers
Homework 22B
Name _____________________________________
1. Simplify.
2. Simplify.
a.
12
16
b.
9
3
b.
15
1
c.
c.
d.
e.
a.
260
910
f.
52
3
2
5
4
5(3 1)
32
12
e.
–6 4
5(8 6)
3. Solve the equations.
8 14
g.
•
15 3
a. 9
h. Build four fractions equivalent to
4x
3) 14
218
to a mixed number.
7
j. Change 23
4
to an improper fraction.
5
© 2010 Cheryl Wilcox
7
3
.
10
b. 2(x
i. Change
22 • 32
| 8 10 |
2
285x 3
7x 2 y
•
y 2 21x
6
d.
190x
2
6 2•3
2
c.
2x
7
6
Free Pre-Algebra
Lesson 22  page 13
4. Simplify each expression, then compare with >, <, =.
a.
5
5. Change any subtractions to additions, then find the sum.
a. 3 ( 5)
2
b. 3
b. 5 1
c.
5 1
5 1
5 1
5
c. 2 ( 3 5)
d. 2 ( 2)
e. 4
5 6
6. Using the formula |a – b|, find the distance between
7. Simplify.
a. 15 and 8
a. 3 • 5
b. –15 and 8
b. ( 3)( 5)
c. 15 ( 3)
c. Beatrice on Floor –55 and Ted on Floor 22.
d. 15 ( 5)
e.
15
5
8. Write the expression as a multiplication, then evaluate.
9. Simplify.
a. 63
a. 7x
2x
b. ( 6)3
b. 7(x
2)
c. 63
c. 7(x
2)
d. 7(x
2) 7(x 2)
d. ( 6)4
© 2010 Cheryl Wilcox