Download Chapter 8 Integers

Chapter 8
Integers
8.1 Addition and Subtraction
Negative numbers
• Negative numbers are helpful in:
– Describing temperature below zero
– Elevation below sea level
– Losses in the stock market
– Overdrawn checking accounts .
Definition
• Integers
The set of integers is the set
I={…,-3,-2,-1,0,1,2,3,…}
The numbers 1,2,3,…are called positive integers and
the numbers -1,-2,-3,… are called negative
integers. Zero is neither a positive nor a negative
integer .
Representing Integers
•
Chip Model
– One black chip
represents a credit of 1
– One red chip
represents a debit of 1
– Then one black chip
and one red chip will
cancel each other out
(or make zero) .
Representing Integers
• Integer number line
– Integers are equally spaced and arranged
symmetrically .
The opposite on an Integer
•
•
•
The opposite of the integer
a, written –a or (-a) is
defined:
Set Model – the opposite of
a is represented by the
same number of chips as a
(but of the opposite color)
Measurement Model – The
opposite of a is the integer
that is its mirror image about
0.
Addition and Its Properties
•
•
Definition
Let a and b be any integers.
1. Adding zero: a + 0 = 0 + a = a
2. Adding two positives: If a and b are positive,
they are added as whole numbers.
3. Adding two negatives; If a and b are positive
(hence –a and –b are negative), then
(-a) + (-b) = -(a + b), where a + b is the wholenumber sum of a and b .
Addition and Its Properties- cont
4. Adding a positive and a negative:
a. If a and b are positive and a>=b,
then a + (-b)= a – b, where a – b is the
whole-number difference of a and b .
b. If a and b are positive and a<b,
then a = (-b) = -(b – a), where b – a is the
whole-number difference of a and b .
Addition
•
•
Set Model – addition means
to put together or form the
union of two disjoint sets
Adding two positives .
Addition
•
•
Set Model – addition means
to put together or form the
union of two disjoint sets
Adding two negatives .
Addition
•
•
Set Model – addition means
to put together or form the
union of two disjoint sets
Adding a positive and a
negative: .
Addition
• Measurement Model – Addition means to put
directed arrows end to end starting at zero.
• Positive integers are represented by arrows pointing
to the right and negative integers by arrows pointing
to the left.
• Adding two positives .
Addition
• Measurement Model – Addition means to put
directed arrows end to end starting at zero.
• Positive integers are represented by arrows pointing
to the right and negative integers by arrows pointing
to the left.
• Adding two negatives: .
Addition
• Measurement Model – Addition means to put
directed arrows end to end starting at zero.
• Positive integers are represented by arrows pointing
to the right and negative integers by arrows pointing
to the left.
• Adding a positive and a negative: .
Properties of Integer Addition
• Let a, b, and c be any integers.
• Closure Property for Integer Addition
a + b is an integer
• Commutative Property for Integer Addition
a+b=b+a
• Associative Property for Integer Addition
(a + b) + c = a + (b + c) .
Properties of Integer Addition
• Let a, b, and c be any integers.
• Identity Property for Integer Addition
0 is the unique integer such that a + 0 = a = 0 + a
for all a
• Additive Inverse Property for Integer Addition
For each integer a there is a unique integer, written
–a, such that a + (-a) = 0
The integer –a is called the additive inverse of a .
Additive Cancellation for
Integers
• Let a, b, and c be any integers.
•
If a + c = b + c then a = b .
Theorem
• Let a be any integer
Then –(-a) = a .
Subtraction
•
Subtraction of integers can be
viewed in several ways.
•
Take-Away
•
6-2
•
-4 – (-1)
Subtraction
• Take-Away
• -2 – (-3)
Subtraction
• Adding the Opposite
inserting an equal number of red and black chips
before performing the operation
• 2–5
OR
Definition
• Subtraction of Integers: Adding the Opposite
Let a and b be any integers. Then
a – b = a + (-b)
• Adding the opposite is perhaps the most efficient
method for subtracting integers – replacing a
subtraction problem with an equivalent addition
problem .
Alternative Definition
• Subtraction of Integers: Missing-Addend
Approach
• Let a, b, and c be any integers.
Then a – b = c if and only if a = b + c .
Summary of Subtraction
Methods
•
Three equivalent ways to view subtraction of
integers
1. Take-away
2. Adding the opposite
3. Missing addend .
Summary
•
Find 4 – (-2) using all three
methods
•
Take-Away
Chapter 8
Integers
8.2 Multiplication, Division, and Order
Multiplication and Its Properties
• Integer multiplication can be viewed as extending
whole-number multiplication thus:
• 3 x 4 = 4 + 4 +4 = 12
• If you were selling tickets and you accepted three
bad checks worth $4 each then:
• 3 x (-4) = (-4) = (-4) = (-4) = -12
• Number line .
Multiplication and Its Properties
• Modeling integer multiplication with chips
1. 4 x -3 – combine 4 groups of red chips
2. Take-away -4 x 3 add an equal number of black
and red chips and then take away the black .
Multiplication and Its Properties
• Modeling integer multiplication with chips
2. Take-away -4 x 3 add an equal number of black
and red chips and then take away the black .
Multiplication of Integers
•
Let a and b be any integers.
1. Multiplying by 0: a x 0 = 0 = 0 x a
2. Multiplying two positives: If a and b are
positive, they are multiplied as whole
numbers .
Multiplication of Integers
•
Let a and b be any integers.
3. Multiplying a positive and a negative: If a is
positive and b is positive
(thus –b is negative), then
a(-1) = -(ab)
4. Multiplying two negatives: if a and b are
positive, then
(-a)(-b) = ab
when ab is the whole-number product of a
and b. That is, the product of two negatives
is positive .
Properties of integer
Multiplication
• Let a, b, and c be any integers.
• Closure Property for Integer Multiplication
– ab is an integer.
• Commutative Property for Integer Multiplication
– ab = ba
• Associative Property for Integer Multiplication
– (ab)c = a(bc)
• Identity property for integer Multiplication
– 1 is the unique integer such that a x 1 = a = 1 x a
for all a .
Properties of integer
Multiplication
• Distributivity of Multiplication over Addition of
Integers
• Let a, b, c be any integers. Then
– a(b + c) = ab + ac .
Theorem
• Let a be any integer. Then
a(-1) = -a
• “the product of negative one and any integer is
the opposite (or additive inverse) of that integer”
• On the integer number line, multiplication by -1
is equivalent geometrically to reflecting an
integer about the origin .
Theorem
• Let a and b be any integers. Then
(-a)b = -(ab)
• Let a and b be any integers. Then
(-a)(-b) = ab for all integers a,b .
Multiplicative Cancellation
Property
• Let a, b, c be any integers with
If ac = bc then a = b .
c≠0.
Zero Divisors Property
• Let a and b be integers. Then ab = 0 if and only if
a = 0 or b = 0 or a and b both equal zero .
Division
• Division of integers can be viewed as an extension
of whole-number division using the missing-factor
approach.
• Division of Integers
Let a and b be any integers, where b ≠ 0 .
Then a ÷ b = c if and only if a = b × c for a unique
integer c .
Following generalizations about
the division of integers:
•
Assume that b divides a; that is,
that b is a factor of a
1. Dividing by 1: a ÷1 = a
2. Dividing two positives (negatives): If a and b are
both positive (or both negative) then a ÷ b is
positive .
Following generalizations about
the division of integers:
•
Assume that b divides a; that is,
that b is a factor of a
1. Dividing a positive and a negative: If one of a or b
is positives and the other is negative,
then a ÷1 = a is negative
1. Dividing zero by a nonzero integer: a ÷ b = 0
where b ≠ 0 , since 0 = b × 0
AS with whole numbers, division by zero is
undefined for integers .
Ordering Integers
• The concepts of less than and greater than in the
integers are defined to be extensions of ordering in
the whole numbers
• Number-Line Approach the integer a is less than
the integer b, written a<b, if a is to the left of b on the
integer number line .
Ordering Integers
• Addition Approach The integer a is less than the
integer b, written a<b, if and only if there is a
positive integer p such that a + p = b.
• Thus -5<-3, since -5 +2 = -3
• And -7<2, since -7 + 9 = 2
• The integer a is greater than the integer b, written
a>b, if and only if b<a .
Properties of Ordering Integers
• Let a, b, and c be any integers, p a positive integer,
and n a negative integer.
•
Transitive Property for Less than
If a < b and b < c then a < c
Property of Less than and Addition
If a < b, then a + c < b + c .
Properties of Ordering Integers
• Property of Less Than and Multiplication by a
Positive
If a < b, then ap < bp
• Example -2 < 3 and 4 > 0 then (-2) x 4 < 3 x 4
Properties of Ordering Integers
Property of Less Than and Multiplication by a
Negative
If a < b, then an > bn
• -2 < 3 and -4 < 0 then (-2)(-4) > 3(-4)
Properties of Ordering Integers
Property of Less Than and Multiplication by a
Negative
If a < b, then an > bn
remember multiplying an integer a by -1 is
geometrically the same as reflecting a across the
origin on the integer number line. Applying this
idea
a < b then (-1)a > (-1)b