Download b - iyang

Chapter
5
Integers
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.
5-1 Addition and Subtraction of
Integers
• The meaning of integers and their representation on a
number line.
• Models for addition and subtraction of integers.
• Properties of addition and subtraction of integers.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide
2
Representations of Integers
The set of integers is denoted by I:
The negative integers are opposites of
the positive integers.
–4
is the opposite of positive 4
3 is the opposite of –3
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide
3
Example
For each of the following, find the opposite of x.
a. x = 3
−x
= −3
b. x = −5
−x
=5
c. x = 0
−x
=0
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide
4
Integer Addition
Chip Model
Black chips represent positive integers and red
chips represent negative integers. Each pair of
black/red chips neutralize each other.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide
5
Integer Addition
Charged-Field Model
Similar to the chip model. Positive integers are
represented by +’s and negative integers by –’s.
Positive charges neutralize negative charges.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide
6
Integer Addition
Number-Line Model
Positive integers are represented by moving
forward (right) on the number line; negative
integers are represented by moving backward
(left).
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide
7
Example
The temperature was −4°C. In an
hour, it rose 10°C. What is the new
temperature?
−4
+ 10 = 6
The new temperature is 6°C.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide
8
Integer Addition
Pattern Model
Beginning with whole number facts, a table of
computations is created by following a pattern.
Basic
facts
4+3=7
4+2=6
4+1=5
4+0=4
4 + −1 = 3
4 + −2 = 2
4 + −3 = 1
4 + −4 = 0
4 + −5 = −1
4 + −6 = −2
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide
9
Absolute Value
The absolute value of a number a, written |a|, is
the distance on the number line from 0 to a.
|4| = 4 and |−4| = 4
Absolute value is always positive or zero.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 10
Definition
For any integer x,
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 11
Example
Evaluate each of the following expressions.
a. |20|
|20| = 20
b. |−5|
|−5| = 5
c. |0|
|0| = 0
d.
−|−3
|
e. |2 + −5|
−|−3|
= −3
|2 + −5| = |−3| = 3
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 12
Properties of Integer Addition
Integer addition has all the properties of wholenumber addition.
Given integers a, b, and c.
Closure property of addition of integers
a + b is a unique integer.
Commutative property of addition of integers
a + b = b + a.
Associative property of addition of integers
(a + b) + c = a + (b + c).
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 13
Properties of Integer Addition
Identity property of addition of integers
0 is the unique integer such that, for all integers
a, 0 + a = a = a + 0.
Additive Inverse Property of Integers
For every integer a, there exists a unique
integer −a, the additive inverse of a, such that
a + −a = 0 = −a + a.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 14
Properties of the Additive Inverse
By definition, the additive inverse, −a, is the
solution of the equation x + a = 0.
For any integers a and b, the equation x + a = b
has a unique solution, b + −a.
For any integers a and b
−(−a) = a and −a + −b = −(a + b).
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 15
Example
Find the additive inverse of each of the following.
a. −(3 + x)
3+x
b. a + −4
−(a
−3
−(−3
c.
+ −x
+ −4) = −a + 4
+ −x) = 3 + x
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 16
Integer Subtraction
Chip Model for Subtraction
To find 3 − −2, add 0 in the form 2 + −2 (two
black chips and two red chips) to the three
black chips, then take away the two red chips.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 17
Integer Subtraction
Charged-Field Model for Subtraction
To find −3 − −5, represent −3 so that at least five
negative charges are present. Then take away
the five negative charges.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 18
Integer Subtraction
Number-Line Model
While integer addition is modeled by maintaining
the same direction and moving forward or
backward depending on whether a positive or
negative integer is added, subtraction is modeled
by turning around.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 19
Integer Subtraction
Pattern Model for Subtraction
Using inductive reasoning and starting with
known subtraction facts, find the difference of
two integers by following a pattern.
3−2=1
3−3=0
3 − 4 =−1
3 − 5 =−2
3−2=1
3−1=2
3−0=3
3 − −1 =4
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 20
Integer Subtraction
Subtraction Using the Missing Addend Approach
Subtraction of integers, like subtraction of whole
numbers, can be defined in terms of addition.
We compute 3 – 7 as follows:
3 – 7 = n if and only if 3 = 7 + n.
Because 7 + –4 = 3, then n = –4.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 21
Definition
Subtraction
For integers a and b, a − b is the unique integer
n such that a = b + n.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 22
Example
Use the definition of subtraction to compute the
following:
a. 3 − 10
Let 3 − 10 = n. Then 10 + n = 3, so n = −7.
b. −2 − 10
Let −2 − 10 = n. Then 10 + n = −2, so n = −12.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 23
Integer Subtraction
Subtraction Using Adding the Opposite
Approach
Subtracting an integer is the same as adding its
opposite.
For all integers a and b
a − b = a + −b.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 24
Example
Using the fact that a − b = a + −b, compute each of
the following:
a. 2 − 8
2 − 8 = 2 + −8 = −6
b. 2 − −8
2 − −8 = 2 + −(−8) = 2 + 8 = 10
c.
−12
− −5
−12
− −5 = −12 + −(−5) = −12 + 5 = −7
d.
−12
−5
−12
− 5 = −12 + −5 = −17
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 25
Example
Rewrite each expression without parentheses.
a.
−(b
− c)
−(b
− c) = −(b + −c) = −b + −(−c) = −b + c
b. a − (b + c)
a − (b + c) = a + −(b + c) = a + (−b + −c)
= (a + −b) + −c = a + −b + −c
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 26
Example
Simplify each of the following expressions.
a. 2 − (5 − x)
2 − (5 − x) = 2 + −(5 + −x) = 2 + −5 + −(−x)
= 2 + −5 + x = −3 + x or x − 3
b. 5 − (x − 3)
5 − (x − 3) = 5 + −(x + −3) = 5 + −x + −(−3)
= 5 + −x + 3 = 8 + −x = 8 − x
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 27
Example (continued)
Simplify each of the following expressions.
c.
−(x
− y) − y
−(x
− y) − y = −(x + −y) + −y = [−x + −(−y)] + −y
= (−x + y) + −y = −x + (y + −y) = −x
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 28
Order of Operations
Recall that subtraction is neither commutative nor
associative.
An expression such as 3 − 15 − 8 is ambiguous
unless we know in which order to perform the
subtractions.
Mathematicians agree that 3 − 15 − 8 means
(3 − 15) − 8.
Subtractions are performed in order from left
to right.
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 29
Example
Compute each of the following.
a. 2 − 5 − 5
2 − 5 − 5 = −3 − 5 = −8
b. 3 − 7 + 3
3 − 7 + 3 = −4 + 3 = −1
c. 3 − (7 − 3)
3 − (7 − 3) = 3 − 4 = −1
Copyright © 2016, 2013, 2010 Pearson Education, Inc.
Slide 30