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Integers Addition and Subtraction of Integers

Sec. 4.1 The set of integers is the set I = {... -3, -2, -1, 0, 1, 2, 3, ... }. The numbers 1, 2, 3, ... are called the The number -1, -2, -3, ... are called the Zero is

"Opposites" Model for Integers

So far, our number line has looked like: Now with the integers, it looks like:

0 1 2 3 4 5

Chip Model for Integers

Requires two colors of chips:

Black chips represent positive #s (credits) Red chips represent negative #s (debits)

When a black chip and a red chip meet, they cancel each other out, making 0.

1

Chip Model for Integers

Find the value of each set:

Addition with the Chip Model

Solve the following using chip model: 5 + ( ¯3) = ? ( ¯2) + ( ¯3) = ?

5 There are many ways to represent each number.

Charged-Field & Pattern Models

Charged-Field Model Just like the chip model but use + and

+++ _ _ _ _

Number-Line Model

Positives move to the right (forward). Negatives move backward. Ex: 3 + 2 = ?; 3 + (¯2) = ?; (¯1) + (¯2)= ?

Pattern Model

3 3 3 3 + + + + 3 2 1 0 = = = = 6 5 4 3 ¯5

¯4

¯3

¯2

¯1

0

1

2

3

4

5

2

Absolute Value

Absolute value is often thought of as the _________ from the number to __. Ex: | 3 | = ___ and | ¯ 4 | = ___ Definition: |x| = if x 0 |x| = if x < 0

Properties of Addition for Integers

Let a, b, and c Closure Commutative Associative Identity be integers a + b = c , c is an integer a+b=b+a a + (b + c) = (a + b) + c a+0=a=0+a

Properties of Additive Inverse

For any integers a and b : 1. ¯ ( ¯a) = a 2. ¯a + ¯b = ¯(a + b)

Subtraction for Integers using Chip Model

3 5 = ___

3 (¯5) =

3

Subtraction of Integers

Charged-Field Just like Chip Model 2 ( ¯1) = ? Patterns Model 32=1 31=2 30=3 3 ( ¯1) = 3 ( ¯2) = 3 ( ¯3) =

Number-Line Model

Ex: 3 2 = ?; 3 (¯2) = ?; (¯1) (¯2)= ?

¯5

¯4

¯3

¯2

¯1

0

1

2

3

4

5

Subtraction and a property

For integers a and b, a b is the unique integer n such that ____________

Subtraction as Opposite of Addition (Chip Model)

3 5 = ___

Subtraction is the inverse of addition:

3 5 = 3 + (-5)

4

Order of Operations

What is the value of: 6 + 4 ÷ 2 + 5 × 3 1 = ? Is it 11? 22?

Multiplication and Division of Integers

Sec. 4.2

Order or Operations:

PEMDAS (Please excuse my dear aunt Sally)

Multiplication of Integers Using the Set (Chip) Model

Recall that 3 × 4 means 3 groups of 4 Solve the following problems: 3 × 2 = ___ 3 × ( ¯ 2) = ___

Multiplication of Integers Using the Set (Chip) Model

What about ¯3 × 2? What does it mean? With 3 × 2, we_____________________ ¯ 3 × 2 = ___

5

Multiplication of Integers Using the Set (Chip) Model

Solve the following problems: ¯3 × (¯2) = ___ 3 × 0 = __

Multiplication of Integers

Charged-Field Model

Just like chip model

Number-Line Model Ex: 3 × (¯5)= __

¯3 × (¯5)= __

¯25 ¯20 ¯15 ¯10 ¯ 5

0

5

10

15

20

25

Which Properties Hold for Integer Multiplication

Closure Commutative Associative Identity Distributive over Addition Zero Multiplication Property

More Properties of Integer Multiplication

For any integers a and b, ( ¯ 1)a = ¯ a ( ¯ a)( ¯ b) = ab & (¯ a)b = (¯ab) Proof of ( ¯ a)( ¯ b) = ab

1. 2. 3. 4.

(¯a)(¯b) = [(¯1)a]× [(¯1)b] = [¯1×¯1](ab) = (1)(ab) = ab -

6

More Properties of Integer Multiplication

For any integers a and b, Proof of (¯ a)b = ¯(ab) Step Reason

1. 2. 3.

Distributive Property of Multiplication over Subtraction

For any integers a, b, & c : a (b c) = ab ac.

1.

2. 3. 4. 5. 6. 7. 8.

(¯a)(b) = [(¯1)a]×b = (¯1) (ab) = ¯(ab)

a (b c) = a (b + ¯c) = ab +a( ¯c) = ab +a(¯1c) = ab +(a ×¯1)c = ab +(¯1×a)c = ab +(¯1)ac = ab +¯(ac) = ab ac

A Couple of Examples

Find the value of ¯3(7 ¯2)= Using the Distributive property over :

¯3(7 ¯2)=

Difference-of-Squares

For any integers a and b: (a + b)(a b) = a 2 b 2.

Find the value (mentally) of 36 × 24 = __

1. (a+b)(ab) = (a+b)a (a+b)b 2. = (aa + ba) (ab+bb) 3. = a2 + ba (ab + b2) 4. = a2 + ab (ab + b2) 5. = a2 + ab ab b2 6. = a2 + 0 b2 7. = a2 b2

Proof:

7

Division of Integers

Whole numbers: a ÷ b = c iff _______ Integers: a ÷ b = c if & only if _______ 15÷(¯3) =__ ¯15÷3 =__ ¯15÷(¯3) =__

Ordering Integers (> & <)

Number line: Ex: ¯3 < ¯1 because

¯5

¯4 ¯3 ¯2 ¯1

0

1

2

3

4

5

Definition of Less than: a is less than b if & only if ___________ _________________________________ Ex: ¯3 < ¯1 because

Subtraction of Integers

Chip Model

Addition with the Chip Model

Solve the following using chip model: 5 + ( ¯3) = ? ( ¯2) + ( ¯3) = ?

2

8

Number-Line Model

Positives move to the right (forward). Negatives move backward.

¯5

¯4

¯3

¯2

¯1

0

1

2

3

4

5

9

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