Post on 03-Jan-2016
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)