Download Dividing Signed Numbers

Math 300
Basic College Mathematics
Chapter 10
Signed Numbers
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
Page 1 of 16
10.1 Signed numbers
 Integers consist of whole numbers and their
opposites. (see diagram below)
Neither positive nor negative (zero)
Negative Integers
–4
–3
–2
–1
Positive Integers
0
1
2
3
4
The natural numbers to the right of zero are called
positive integers.
The opposites of the natural numbers to the left of zero
on the number line are called negative integers.
Zero is neither positive nor negative.
Therefore, the integers to the left of zero are opposites
of the integers to the right of zero and zero serves as
its own opposite.
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
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10.1 Signed numbers (cont)
Integers and the Real World
 Integers correspond to many real-world problems
and situations.
 Some common uses for negative integers are as
follows:
Time: Before an event
Temperature: Degrees below zero
Money: Amount lost, spent, owed, or withdrawn
Elevation: Depth below sea level
Travel: Motion in the backward (reverse) direction
Otherwise, the integer is probably represented as a
positive number.
Example: Represent each quantity by a signed
number.
1. A scuba diver is swimming 35 feet below the surface
of the water in the Gulf of Mexico.
2. The Minnesota Viking football team lost 15 yards on
a play.
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
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10.1 Signed numbers (cont)
Graphing Signed Numbers
 To graph signed numbers on a number line
means to make a dot on the number line and
label the dot with its corresponding number.
Example: Graph the signed numbers in the list on a
number line.
1. 4, 0, 2,  3
–4
–3
–2
1
4
–1
0
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
1
2
3
4
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10.1 Signed numbers (cont)
Order
 Real numbers are named in order on the number
line. Things you will need to know to compare
them:
a. positive numbers > negative numbers.
b. negative numbers < positive numbers.
c. As you travel right on the number line, the
numbers always get larger.
(i.e.: -6 is greater than -9, because -6 is farther
to the right)
-9
-6
Using < and > symbols:
-3
0
-6 > -9 and -9 < -6
Example: Use either < or > or for
to write a true
sentence. Picture where each number lies on a
number line.
1. 0
–7
2. 6
–6
3. –3
4. –10
–4
–14
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
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10.1 Signed numbers (cont)
Absolute value
 The absolute value of a number is its distance
from zero on a number line. The symbol x is
used to represent the absolute value of a number
x.
 The absolute value of a number is always positive.
 If there is a negative sign on the outside of the
absolute value symbol, then the answer would be
negative (though the absolute value part is positive).
Example: Find the value.
1.
7
2.
0
3.
8
4.
 10
3

5.
8
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
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10.1 Signed numbers (cont)
The opposite of a number
 When two numbers are the same distance from
zero on a number line but are on opposite sides of
zero, they are called opposites.
 The phrase “the opposite of” is written in symbol
as “ – ”.
If a is a number, then – (a) = – a.
If a is a number, then – (– a) = a.
Example: Find the opposite of each number.
Question
Equivalent to:
Opposite of:
1. 8
2. 
3.
4
9
 43
4.   14
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
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10.2 Adding Signed Numbers
ADDING 2 NEGATIVE INTEGERS
 Add the numbers and make the answer negative.
Example #1: 7 + (10) =  17
{Think: 7 + 10 = 17,
now write the answer
with a negative sign as  17.}
ADDING THE NUMBER ZERO TO ANY NUMBER
 When adding any number (positive or negative) to
zero, your answer will be that number.
Example #2: 8 + 0 = 8
Example #3:
4+0=4
ADDING A POSITIVE AND A NEGATIVE INTEGER
 Subtract the numbers and take the sign of the
larger number.
Example #4: 11 + (5) = 6
{Think: 11 – 5 = 6
and since 11 is the larger number,
the answer will be positive.}
1 + (6) = 5
{Think: 6 – 1 = 5
and since 6 is the larger number,
the answer will be negative.}
Example #5:
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
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10.2 Adding Signed Numbers (cont)
ADDING OPPOSITES
 Any number added to its opposite is equal to
zero.
Example #6:
20 + (20) = 0
Example #7: 12 + 12 = 0
ADDING SEVERAL INTEGERS BOTH POSTIVE &
NEGATIVE
 First, add all the positive numbers together.
 Then, add all the negative numbers together.
 Finally, add the results.
Example #8:
(15) + (37) + 25 + 42 + (59) + (14)
Step 1: Add all of the positive numbers.
25 + 42 = 67
Step 2: Add all of the negative numbers.
15 + (37) + (59) + (14) = 125
Step 3: Add the results.
67 + (125) = 58
So, (15) + (37) + 25 + 42 + (59) + (14) = 58
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
Page 9 of 16
10.3 Subtracting Signed Numbers
The first thing you will need to know in order to
subtract integers is finding the opposite or
additive inverse of a number.
 To find the opposite, or additive inverse, you
would change only the sign of that number.
Examples: Find the opposite of each number
a. 34
b.  8.3
c. 0
Answers:
a. The opposite of 34 is  34
b. The opposite of  8.3 is 8.3
c. The opposite of 0 is 0
Note:
The opposite or additive inverse of a
positive number is negative.
The opposite or additive inverse of a
negative number is positive.
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
Page 10 of 16
10.3 Subtracting Signed Numbers (cont)
Now to subtract signed numbers, you must add
its opposite or additive inverse of the number
that is being subtracted.
 To subtract signed numbers, change the
minus sign to a plus sign and change the
number behind the minus sign to its
opposite.
Examples: Subtract each of the following.
a. 2  6
**In this problem, the 6 is positive.**
b.  4   9
**In this problem, the 9 is negative.**
Answers:
a. 2   6  4
**After taking the opposite of 6 from the original
problem, it becomes negative.**
b.  4  9  5
**After taking the opposite of  9 from the
original problem, it becomes positive.**
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
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10.4 Multiplying and Dividing Signed Numbers
Multiplying Signed Numbers
Positive  Positive = Positive
Example: 5  5  25
Positive  Negative = Negative
Example: 3   4  12
Negative  Positive = Negative
Example:  2  5  10
Negative  Negative = Positive
Example:  2 4  8
Multiplying by Zero
Any number (positive or negative) multiplied by
zero is equal to zero.
Example: 0  9  0
15  0  0
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
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10.4 Multiplying and Dividing Signed Numbers
(cont)
Dividing Signed Numbers
Positive
 Positive = Positive
Example:
Positive
14  2  7
 Negative = Negative
Example: 6   2  3
Negative
 Positive = Negative
Example:  24  8  3
Negative
 Negative = Positive
 15
Example:  3  5
Division by Zero
Zero divided by any number (positive or negative) is
equal to zero.
(Hint: When the numerator is zero, the answer is zero!)
Examples: 0  21  0
0
0
2
Any number divided by zero is undefined.
(Hint: When the denominator is zero, the answer is undefined!)
Examples: 3  0  undefined
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
6
 undefined
0
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10.5 Order of Operations
Parentheses
Exponents / Roots
Multiplication / Division
(Left to Right)
Addition / Subtraction
(Left to Right)
Example: Simplify.
1.
2.
9   27 
3
8  4  3
 11  7
3.
4.
79
12  42  23
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
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Math 300 Chapter 10
Glossary
Negative numbers - numbers less than 0
Positive numbers - numbers greater than 0
Signed numbers - the collection of numbers that included
positive numbers, negative numbers and zero
Integers - consists of zero, the natural numbers, and the
opposites of the natural numbers.
Absolute Value - the distance the number is from zero
Opposite - if two numbers are the same distance from zero
on the number line then they are opposites.
Properties
Adding two numbers with the same sign
1. add their absolute value
2. use the common sign as the sign of the sum
Adding two numbers with different signs
1. find the larger absolute value minus the smaller absolute
value
2. use the sign of the larger absolute value as the sign of the
sum
Subtracting two numbers: If a and b are numbers, then a – b
= a + ( -b)
The product of two numbers having the same sign is positive
(+)(+) = + (-)(-) = +
The product of two numbers having different signs is negative
(+)(-) = - (-)(+) = -
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
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Math 300 Chapter 10
Glossary
Hints
The absolute value of a number is always positive.
If you think of < and > as arrowheads, they always point to the
smaller value.
When subtracting a negative number, think of it as adding a
positive, 5 – ( -2) = 5 + 2
When multiplying and dividing, if you have an even number of
negatives, the result is positive. If you have an odd number of
negatives, the result is negative.
Be aware of the difference: (-5)
52 = -(5)(5) = -25
Math 300 M-G 4e Chapter 10; Rev: Mar 2011
= (-5)(-5) = 25. However, -
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