Chapter 8

Integers

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

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

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

Using ChipsEach integer has infinitely many representations

using chips.

All three examples represent +3.

+3 +3 +3

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

Opposite

MeasurementModel:

+3

-3

Opposites

Set Model:

-2 -1 0 1 2

Opposites

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

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

Addition using the Set Model

Example: 43

-7

Example:

-4-3

43

+1

+4

-3

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

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

Theorem:

Let a be any integer. Then .aa

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.

Subtraction

Take-Away:

35

Take Away 3

Leaves 2

235

43

Take Away 4

Leaves –1

143

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

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

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

Positive Times a Negative

Chip Model

1234

Combine 4 groups of 3 red chips

1234

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.

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

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

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

Some Theorems

Theorem: Let a be any integer. Then

Theorem: Let a and b be any integers. Then1.

2.

aa 1

abba

abba

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

Division

Definition: Let a and b be any integers, where Then if and only if

for a unique integer c.

.0b cba

cba

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

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

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

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

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