chapter 8 integers. 8.1 addition and subtraction definition: the set of integers is the set the...
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Chapter 8
Integers
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8.1 Addition and Subtraction
Definition: The set of integers is the set
The numbers 1, 2, 3, … are called positive integers and the numbers -1, -2, -3, … are called negative integers. Zero is neither a positive or a negative integer.
.,,,,,,,, 3210123 I
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Set Model
In a set model, two different colored chips can be used, one color for positive numbers and another color for negative numbers.
+5 --3
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Using Chips
One black chip represents a credit of one and one red chip represents a debit of 1. One chip of each color cancel each other out making 0.
0
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Using ChipsEach integer has infinitely many representations
using chips.
All three examples represent +3.
+3 +3 +3
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Number Line Representation
The integers are equally spaced and arranged symmetrically about 0.
Due to this symmetry, we have the concept of “opposite.”
-5 -4 -3 -2 5-1 0 1 2 3 4
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Opposite
MeasurementModel:
+3
-3
Opposites
Set Model:
-2 -1 0 1 2
Opposites
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Addition of Integers
Definition:Let a and b be any integers.1.
2. If a and b are positive, they are added as whole numbers.
3. If a and b are positive (thus –a and –b are negative), then where a+b is the whole number sum of a and b.
aaa 00
baba
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Addition of Integers Continued
4. Adding a positive and a negativea. If a and b are positive and then
where a – b is the whole number difference of a and b.
5. b. If a and b are positive and then
where b – a is the whole number difference of a and b.
ba
abba
baba
ba
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Addition using the Set Model
Example: 43
-7
Example:
-4-3
43
+1
+4
-3
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Properties
1. Closure Property for Integer Addition.
2. Commutative Property for Integer Addition
3. Associative Property for Integer Addition
4. Identity Property for Integer Addition
5. Additive Inverse Property for Integer Addition
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Additive Cancellation for Integers
Theorem: Let a, b, and c be any integers.
If then ,cbca .ba
Proof: Let Then .cbca ccbcca
ccbcca 00 ba
ba
Addition
Associativity
Additive Inverse
Additive Identity
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Theorem:
Let a be any integer. Then .aa
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Subtraction
Pattern:
624
The first column remains 4.
The second column decreases by 1 each time.
224 314 404
514
The column after the = increases by 1 each time.
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Subtraction
Take-Away:
35
Take Away 3
Leaves 2
235
43
Take Away 4
Leaves –1
143
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Subtraction
Adding the Opposite Let a and b be any integers. Then
Missing-Addend ApproachLet a, b, and c be any integers. Then
if and only if
.baba
cba .cba
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8.2 Multiplication, Division, and Order
Positive Times a Negative
The first column remains 3.
The second column decreases by 1 each time.
The column after the = decreases by by 3 each time.
1243 933 623 313 003
313
623
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Negative Times a Positive
The first column remains –3 .
The second column decreases by 1 each time.
The column after the = increases by by 3 each time.
1243 933 623 313 003
313
623
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Positive Times a Negative
Chip Model
1234
Combine 4 groups of 3 red chips
1234
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Take away 4 groups of 3 black chips.
Negative Times a Positive
Chip Model
1234
0 0 --12
Insert 12 chips of each color. Take away 4 groups of 3 blacks.
Leaves 12 reds.
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Multiplication of Integers
Definition:Let a and b be any integers.1.
2. If a and b are positive, they are multiplied as whole numbers.
3. If a and b are positive (thus–b is negative), then where is the whole
number product of a and b.
000 aa
abba ab
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Multiplication of Integers Continued
4. Multiplying two negativesa. If a and b are positive then
where is the whole number product of a and b. abba ab
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Properties1. Closure Property for Integer
Multiplication
2. Commutative Property for Integer Multiplication
3. Associative Property for Integer Multiplication
4. Identity Property for Integer Multiplication
5. Distributive Property of Multiplication over Addition
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Some Theorems
Theorem: Let a be any integer. Then
Theorem: Let a and b be any integers. Then1.
2.
aa 1
abba
abba
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Two More Properties
6. Multiplicative Cancellation PropertyLet a, b, c be any integers with If then
7. Zero Divisors PropertyLet a and b be integers. Then if and only if or a and b both equal zero.
0c,bcac .ba
0ab
0 or 0 ba
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Division
Definition: Let a and b be any integers, where Then if and only if
for a unique integer c.
.0b cba
cba
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Negative Exponentsaaaa 3
aaa 2
aa 1
10 a
aa
11
2
2 1a
a
3
3 1a
a
aaa
a
a
a
Definition:
Negative Integer Exponent
Let a be any nonzero number and n be a positive integer.
Then
n
n
aa
1
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Scientific Notation
A number is said to be in scientific notation when expressed in the form where and n is any integer.
The number a is called the mantissa and the exponent n is the characteristic.
na 10101 a
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Ordering Integers
Less Than: Number Line Approach
-5 -4 -3 -2 5-1 0 1 2 3 4
The integer a is less than the integer b, written if a is to the left of b on the integer number line.
Since –2 is to the left of 4 on the number line, --2 is less than 4.
,ba
42
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Ordering Integers
Less Than: Addition Approach
The integer a is less than the integer b, written if and only if there is a positive integer p such that
,ba .bpa
Since –2 +6=4, .42
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Properties of Ordering Integers
Let a, b and c be any integers, p a positive integer and n a negative integer.
1. Transitive Property for Less Than
2. Property of Less than and Addition
3. Property of Less Than and Multiplication by a Positive.
4. Property of Less Than and Multiplication
. then , and If cacbba
. then , If cbcaba
. then , If bpapba
. then , If bnanba