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Adding and Subtracting Real Numbers
Preview
Warm Up
California Standards
Lesson Presentation
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Adding and Subtracting Real Numbers
Warm Up
Simplify.
1. |–3|
3
2. –|4|
–4
Write an improper fraction to represent each
mixed number.
6
2
14
55
3. 4 3
4. 7 7
3
7
Write a mixed number to represent each
improper fraction.
5.
12
5
2
2
5
6.
24
9
2
2
3
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Adding and Subtracting Real Numbers
California
Standards
2.0 Students understand and use such
operations as taking the opposite, finding the
reciprocal, taking a root, and raising to a fractional
power. They understand and use the rules of
exponents.
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Adding and Subtracting Real Numbers
Vocabulary
real numbers
absolute value
opposites
additive inverse
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Adding and Subtracting Real Numbers
The set of all numbers that can be represented
on a number line are called real numbers. You
can use a number line to model addition and
subtraction of real numbers.
Addition
To model addition of a positive number, move
right. To model addition of a negative number,
move left.
Subtraction
To model subtraction of a positive number, move
left. To model subtraction of a negative number,
move right.
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Adding and Subtracting Real Numbers
Additional Example 1A: Adding and Subtracting
Numbers on a Number Line
Add or subtract using a number line.
–4 + (–7)
+ (–7)
Start at 0. Move left to –4.
To add –7, move left 7 units.
11 10 9 8 7 6 5 4 3
–4 + (–7) = –11
–4
2 1 0
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Adding and Subtracting Real Numbers
Additional Example 1B: Adding and Subtracting
Numbers on a Number Line
Add or subtract using a number line.
3 – (–6)
Start at 0. Move right to 3.
To subtract –6, move right 6 units.
–(–6)
+3
-3 -2 -1 0 1 2 3 4 5 6 7 8 9
3 – (–6) = 9
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Adding and Subtracting Real Numbers
Check It Out! Example 1a
Add or subtract using a number line.
–3 + 7
Start at 0. Move left to –3.
To add 7, move right 7 units.
+7
–3
-3 -2 -1 0 1 2 3 4 5 6 7 8 9
–3 + 7 = 4
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Adding and Subtracting Real Numbers
Check It Out! Example 1b
Add or subtract using a number line.
–3 – 7
Start at 0. Move left to –3.
To subtract 7, move left 7 units.
–7
–3
11 10 9 8 7 6 5 4 3 2 1 0
–3 – 7 = –10
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Adding and Subtracting Real Numbers
Check It Out! Example 1c
Add or subtract using a number line.
Start at 0. Move left to –5.
–5 – (–6.5)
To subtract –6.5, move right
6.5 units.
– (–6.5)
–5
8 7 6 5 4 3 2 1 0 1 2
–5 – (–6.5) = 1.5
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Adding and Subtracting Real Numbers
The absolute value of a number is the
distance from zero on a number line. The
absolute value of 5 is written as |5|.
5
units
5 units
-6 -5 - 4 -3 -2 -1 0 1 2 3 4 5 6
|–5| = 5
|5| = 5
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Adding and Subtracting Real Numbers
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Adding and Subtracting Real Numbers
Additional Example 2: Adding Real Numbers
Add.
A.
Different signs: subtract the
absolute values.
Use the sign of the number with
the greater absolute value.
B. –6 + (–2)
(6 + 2 = 8)
–8
Same signs: add the absolute values.
Both numbers are negative, so the
sum is negative.
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Adding and Subtracting Real Numbers
Check It Out! Example 2
Add.
a. –5 + (–7)
(5 + 7 = 12)
–12
Same signs: add the absolute
values.
Both numbers are negative, so
the sum is negative.
b. –13.5 + (–22.3)
Same signs: add the absolute
(13.5 + 22.3 = 35.8)
values.
Both numbers are negative, so
–35.8
the sum is negative.
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Adding and Subtracting Real Numbers
Check It Out! Example 2c
Add.
c. 52 + (–68)
(68 – 52 = 16)
–16
Different signs: subtract the
absolute values.
Use the sign of the number with
the greater absolute value.
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Adding and Subtracting Real Numbers
Two numbers are opposites if their sum
is 0. A number and its opposite are
additive inverses and are the same
distance from zero. They have the same
absolute value.
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Adding and Subtracting Real Numbers
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Adding and Subtracting Real Numbers
To subtract signed numbers, you can use additive
inverses. Subtracting a number is the same as
adding the opposite of the number.
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Adding and Subtracting Real Numbers
Subtracting Real Numbers
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Adding and Subtracting Real Numbers
Additional Example 3A: Subtracting Real Numbers
Subtract.
–6.7 – 4.1
–6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1.
(6.7 + 4.1 = 10.8)
–10.8
Same signs: add absolute
values.
Both numbers are negative, so
the sum is negative.
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Adding and Subtracting Real Numbers
Additional Example 3B: Subtracting Real Numbers
Subtract.
5 – (–4)
5 − (–4) = 5 + 4
(5 + 4 = 9)
9
To subtract –4, add 4.
Same signs: add absolute values.
Both numbers are positive, so
the sum is positive.
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Adding and Subtracting Real Numbers
Additional Example 3C: Subtracting Real Numbers
Subtract.
To subtract
Rewrite
of 10.
,, add
with a denominator
Same signs: add absolute
values .
–5.3
.
Both numbers are negative,
so the sum is negative.
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Adding and Subtracting Real Numbers
Helpful Hint
On many scientific and graphing calculators,
there is one button to express the opposite of a
number and a different button to express
subtraction.
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Adding and Subtracting Real Numbers
Check It Out! Example 3a
Subtract.
13 – 21
13 – 21 = 13 + (–21)
To subtract 21, add –21.
(21 – 13 = 8)
Different signs: subtract
absolute values.
–8
Use the sign of the number
with the greater absolute
value.
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Adding and Subtracting Real Numbers
Check It Out! Example 3b
Subtract.
To subtract –3 1 , add 3 1 .
2
2
Same signs: add absolute
values.
4
Both numbers are positive,
so the sum is positive.
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Adding and Subtracting Real Numbers
Check It Out! Example 3c
Subtract.
–14 – (–12)
–14 – (–12) = –14 + 12
(14 – 12 = 2)
–2
To subtract –12, add 12.
Different signs: subtract
absolute values.
Use the sign of the number
with the greater absolute
value.
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Adding and Subtracting Real Numbers
Additional Example 4: Oceanography Application
An iceberg extends 75 feet above the sea. The
bottom of the iceberg is at an elevation of
–247 feet. What is the height of the iceberg?
Find the difference in the elevations of the top of the iceberg and
the bottom of the iceberg.
elevation at bottom
elevation at
minus
of iceberg
top of iceberg
–
75
–247
75 – (–247)
75 – (–247) = 75 + 247
= 322
To subtract –247, add 247.
Same signs: add the
absolute values.
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Adding and Subtracting Real Numbers
Additional Example 4 Continued
An iceberg extends 75 feet above the sea. The
bottom of the iceberg is at an elevation of
–247 feet. What is the height of the iceberg?
The height of the iceberg is 322 feet.
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Adding and Subtracting Real Numbers
Check It Out! Example 4
What if…? The tallest known iceberg in the North
Atlantic rose 550 feet above the ocean's surface.
How many feet would it be from the top of the
tallest iceberg to the wreckage of the Titanic,
which is at an elevation of –12,468 feet?
elevation at
top of iceberg
minus
elevation of the
Titanic
550
–
–12,468
550 – (–12,468)
550 – (–12,468) = 550 + 12,468
= 13,018
To subtract –12,468,
add 12,468.
Same signs: add the
absolute values.
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Adding and Subtracting Real Numbers
Check It Out! Example 4 Continued
What if…? The tallest known iceberg in the North
Atlantic rose 550 feet above the ocean's surface.
How many feet would it be from the top of the
tallest iceberg to the wreckage of the Titanic,
which is at an elevation of –12,468 feet?
Distance from the top of the iceberg to the Titanic
is 13,018 feet.
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Adding and Subtracting Real Numbers
Lesson Quiz
Add or subtract using a number line.
1. –2 + 9
7
Add or subtract.
3. –23 + 42
19
2. –5 – (–3) –2
4. 4.5 – (–3.7) 8.2
5.
6. The temperature at 6:00 A.M. was –23°F.
At 3:00 P.M., it was 18°F. Find the difference
in the temperatures. 41°F