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Adding and Subtracting Real
Numbers
Section 1-5
Goals
Goal
• To find sums and
differences of real numbers.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• Absolute value
• Opposite
• Additive inverses
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.
Example: Adding &
Subtracting on a Number Line
Add or subtract using a number line.
–4 + (–7)
Start at 0. Move left to –4.
To add –7, move left 7 units.
+ (–7)
11 10 9
–4 + (–7) = –11
8 7
–4
6
5
4 3
2 1 0
Example: Adding &
Subtracting 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
3 – (–6) = 9
0
1 2
3 4
5 6
7
8 9
Your Turn:
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
–3 + 7 = 4
0
1 2
3 4
5 6
7
8 9
Your Turn:
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
–3 – 7 = –10
8 7
6
5
4 3
2 1
0
Your Turn:
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 – (–6.5) = 1.5
5
4 3
2 1
0 1 2
Definition
• Absolute Value – The distance between a
number and zero on the number line.
– Absolute value is always nonnegative since
distance is always nonnegative.
– The symbol used for absolute value is | |.
• Example:
– The |-2| is 2 and the |2| is 2.
Absolute Value on the
Number Line
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
- 6 - 5 - 4 - 3 - 2 -1
|–5| = 5
5 units
0
1
2
|5| = 5
3
4
5
6
Rules For Adding
Example: 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.
Your Turn:
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)
(13.5 + 22.3 = 35.8)
–35.8
Same signs: add the absolute values.
Both numbers are negative, so the
sum is negative.
Your Turn:
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.
Definition
• Additive Inverse – The negative of a
designated quantity.
– The additive inverse is created by multiplying
the quantity by -1.
• Example:
– The additive inverse of 4 is -1 ∙ 4 = -4.
Opposites
• 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.
Additive Inverse Property
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.
• Example:
– The expressions 3 – 5 and 3 + (-5) are
equivalent.
Subtracting Real Numbers
A number and its opposite are additive inverses.
To subtract signed numbers, you can use additive
inverses.
Subtracting 6 is the same
as adding the inverse of 6.
Additive inverses
11 – 6 = 5
11 + (–6) = 5
Subtracting a number is the same as adding the
opposite of the number.
Rules
For
Subtracting
Subtracting Real Numbers
Example: Subtracting Real
Numbers
Subtract.
–6.7 – 4.1
–6.7 – 4.1 = –6.7 + (–4.1)
(6.7 + 4.1 = 10.8)
–10.8
To subtract 4.1, add –4.1.
Same signs: add absolute values.
Both numbers are negative, so the sum
is negative.
Example: Subtracting Real
Numbers
Subtract.
5 – (–4)
5 − (–4) = 5 + 4
To subtract –4, add 4.
Same signs: add absolute values.
(5 + 4 = 9)
9
Both numbers are positive, so the sum
is positive.
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.
Your Turn:
Subtract.
13 – 21
13 – 21 = 13 + (–21)
To subtract 21, add –21.
Different signs: subtract absolute values.
(21 – 13 = 8)
–8
Use the sign of the number with the greater
absolute value.
Your Turn:
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.
Example: 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 top of
iceberg
minus
elevation at bottom
of iceberg
75
–
–247
75 – (–247)
75 – (–247) = 75 + 247
= 322
To subtract –247, add 247.
Same signs: add the absolute
values.
The height of the iceberg is 322 feet.
Your Turn:
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
–
–12,468
550
550 – (–12,468)
550 – (–12,468) = 550 + 12,468
= 13,018
To subtract –12,468,
add 12,468.
Same signs: add the
absolute values.
Distance from the top of the iceberg to the Titanic is 13,018 feet.
Joke Time
• What’s brown and sticky?
• A stick.
• What happened when the wheel was invented?
• It caused a revolution.
• Why was the calendar depressed?
• Because it’s days were numbered.
Assignment
• 1.5 Exercises Pg. 41 – 43: #10 – 76 even