Chapter 8
Integers
Negative numbers
• Negative numbers are helpful in: – Describing temperature below zero – Elevation below sea level
– Losses in the stock market
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
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
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
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
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.
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.
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.
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
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
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
Subtraction
• Take-Away • -2 – (-3)
Subtraction
• Adding the Opposite
inserting an equal number of red and black chips before performing the operation
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.
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
Chapter 8
Integers
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
Multiplicative Cancellation
Property
• Let a, b, c be any integers with . If ac = bc then a = b .
0
≠
Zero Divisors Property
• Let a and b be integers. Then ab = 0 if and only if
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 .
Then if and only if for a unique integer c .
c
b
a
÷
=
b ≠ 0c
b
a
=
×
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:
2. Dividing two positives (negatives): If a and b are
both positive (or both negative) then is positive .
b
a
÷
a
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 is negative
1. Dividing zero by a nonzero integer:
where , since
AS with whole numbers, division by zero is undefined for integers .
0
≠
b
0
=
÷ b
a
a
a
÷1
=
0
0
= b
×
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
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
Properties of Ordering Integers
• Property of Less Than and Multiplication by a
Positive
If a < b, then ap < bp
Properties of Ordering Integers
Property of Less Than and Multiplication by a Negative
If a < b, then an > bn
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
