Chapter 5

Integers

Review

• a is a factor of b if . . .

• m is a multiple of n if . . .

• p is a divisor of q if . . .

Review

• A number is divisible by 2 if . . .• A number is divisible by 3 if . . . • A number is divisible by 4 if . . . • A number is divisible by 5 if . . . • A number is divisible by 7 if . . .• A number is divisible by 8 if . . .• A number is divisible by 9 if . . .• A number is divisible by 11 if . . .

• How do we “come up with” other divisibility rules?

• What is the difference in listing all the factors of a number and writing the prime factorization of the number?

• What is a prime number?

• What is a composite number?

• How do you know if a number is prime?

• What numbers do you check to find out?

• How do you know when you are finished checking?

• What does the GCF mean? What have you found when you have it?

• What does the LCM mean? What have you found when you have it?

Homework QuestionsChapter 4

Lab Questions

• Counting Numbers = {1, 2, 3, . . . }• Whole Numbers = {0, 1, 2, 3, . . . }

Counting Numbers 1. Closed with respect to addition 2. Closed with respect to multiplication3. Not closed with respect to subtraction4. Not closed with respect to division

Whole Numbers

CountingNumbers

• Counting Numbers = {1, 2, 3, . . . }• Whole Numbers = {0, 1, 2, 3, . . . }

Whole Numbers 1. Closed with respect to addition 2. Closed with respect to multiplication3. Not closed with respect to subtraction4. Not closed with respect to division

Whole Numbers

CountingNumbers

{ . . . -2, -1, 0, 1, 2, . . .}

• False Numbers

• Numbers of Integrity

• Integers

• Counting Numbers = {1, 2, 3, . . . }• Whole Numbers = {0, 1, 2, 3, . . . }• Integers = { . . . , -2, -1, 0, 1, 2 . . . }

Integers

• Counting Numbers = {1, 2, 3, . . . }• Whole Numbers = {0, 1, 2, 3, . . . }• Integers = { . . . , -2, -1, 0, 1, 2 . . . }

Integers 1. Closed with respect to addition 2. Closed with respect to multiplication3. Closed with respect to subtraction4. Not closed with respect to division

Integers

• Ancient Asian Notation

• Ancient Asian Notation

• Ancient Asian Notation: +3

• Indian Notation

-5 = 5

Chip Model

• Counters colored black on one side, red on the other.

• Drop 10 of them.

Chip Model

• Counters colored black on one side, red on the other.

• Drop 10 of them. Result is -2

Hot Air Balloon4

2

-2

0

Hot Air Balloon

4

2

-2

0

Hot Air Balloon4

2

-2

0

I Walk the Line

• Face a positive direction and stand at 0

• Addition:– Walk forward for a positive integer, backward

for a negative integer

• Subtraction:– Walk forward for a positive integer, backward

for a negative integer– To subtract, do the inverse so turn around

+3 + -2 =

+3 + -2 = +1

-3 + -2 =

+3 + -2 = +1

-3 + -2 = -5

+4 - +6 =

+3 + -2 = +1

-3 + -2 = -5

+4 - +6 = -2

-5 - +2 =

+3 + -2 = +1

-3 + -2 = -5

+4 - +6 = -2

-5 - +2 = -7

+2 - -3 =

+3 + -2 = +1

-3 + -2 = -5

+4 - +6 = -2

-5 - +2 = -7

+2 - -3 = +5

-4 - -7 =

+3 + -2 = +1

-3 + -2 = -5

+4 - +6 = -2

-5 - +2 = -7

+2 - -3 = +5

-4 - -7 = +3

Absolute Value of an IntegerPage 290

The absolute value of an integer is the distance that integer is from 0 on the number line.

|-11| = |13| =

|0| = |-9| =

Absolute Value of an IntegerPage 290

The absolute value of an integer is the distance that integer is from 0 on the number line.

|-11| = 11 |13| = 13

|0| = 0 |-9| = 9

|x| = x if x ≥ 0

|x| = -x if x < 0

| 5 + (-7)| =“The absolute value of 5 + -7.”| 5 + (-7)| = | -2 | = 2

| 5 | + | -7 | =“The absolute value of 5 plus the absolute

value of -7.”| 5 | + | -7 | = 5 + 7 = 12

Mail-Time Model

• At mail time you are delivered a check for $20. What happens to your net worth.

• At mail time you are delivered a bill for $35. What happens to you net worth?

• At mail time you receive a check for $10 and a bill for $10. What happens to your net worth?

Example 5.10Page 297

Example 5.19Page 306

• Adding Integers

• Subtracting Integers

Multiplication by repeated addition

(3)(-4)

= (-4) + (-4) + (-4)

= -12

Multiplication by patterns: (4)(-3)

(4)(3) = 12

Multiplication by patterns: (4)(-3)

(4)(3) = 12

(4)(2) = 8

(4)(1) = 4

(4)(0) = 0

Multiplication by patterns: (4)(-3)

(4)(3) = 12

(4)(2) = 8

(4)(1) = 4

(4)(0) = 0

(4)(-1) =

(4)(-2) =

(4)(-3) =

Multiplication by patterns:

(4)(3) = 12

(4)(2) = 8

(4)(1) = 4

(4)(0) = 0

(4)(-1) = -4

(4)(-2) = -8

(4)(-3) = -12

Multiplication by patterns: (-3)(-2)

(3)(-2) = -6

Multiplication by patterns: (-3)(-2)

(3)(-2) = -6

(2)(-2) = -4

(1)(-2) = -2

(0)(-2) = 0

Multiplication by patterns: (-3)(-2)

(3)(-2) = -6

(2)(-2) = -4

(1)(-2) = -2

(0)(-2) = 0

(-1)(-2) =

(-2)(-2) =

(-3)(-2) =

Multiplication by patterns:

(3)(-2) = -6

(2)(-2) = -4

(1)(-2) = -2

(0)(-2) = 0

(-1)(-2) = +2

(-2)(-2) = +4

(-3)(-2) = +6

Multiplication of Integers:

• (+)(+) = +

• (+)(-) = -

• (-)(+) = -

• (-)(-) = +

Division: Family of Facts

• (3)(-4) = -12

• (-4)(3) = -12

• (-12) ÷ 3 = -4

• (-12) ÷ (-4) = 3

Multiplication of Integers:• (+)(+) = +• (+)(-) = -• (-)(+) = -• (-)(-) = +• Division of Integers:• (+) ÷ (+) = +• (-) ÷ (-) = +• (-) ÷ (+) = -• (+) ÷ (-) = -

More Mail-TimeExample 5.23, Page 317

PropertiesPage 296

• Closure• Commutative • Associative• Identity Element• Existence of Negative – For every integer

n, there exists –n called “the additive inverse of n” or “the opposite of n” such that n + -n = 0 (the identity element for addition)