1. Lesson 3.1
Real Numbers pp

2. Objectives:
1. To identify the different sets of numbers in our number system.
2. To study the properties of real numbers.
3. To define an equivalence relation and identify an example of one.
4. To define and apply absolute value.

3. Real Number System
Rational
Irrational
Integers
Whole
Natural

4. Real Number System Natural (N): {1, 2, 3, 4, . . .}
Whole (W): {0, 1, 2, 3, . . .}
Integer (Z): whole numbers and their opposites

5. Real Number System
Rational (Q): any number that can be written as a fraction
Irrational (Ir): any real number that is not rational

6. Commutative Property of addition: a + b = b + a of multiplication:

7. Associative Property of addition: a +(b + c) = (a + b)+ c
of multiplication:
a •(b • c) = (a • b)• c

8. Distributive Property
over addition:
a(b + c) = ab + ac

9. Identity Property
of addition:
a + 0 = a
of multiplication:
a • 1 = a

10. Inverse Property of addition: a + (-a) = 0 of multiplication: 1 a a •

11. Equality Properties Addition: If a = b, then a + c = b + c
Multiplication: If a = b, then ac = bc

12. Equality Properties Reflexive: a = a Symmetric: If a = b, then b = a
Transitive: If a = b and b = c, then a = c

13. Definition
An equivalence relation is a relation that is reflexive, symmetric, and transitive.

14. Order of Operations
Perform inside parentheses first, multiplications next, and additions last

15. Substitution Property
If a = b, then a can replace b in any mathematical statement.

16. Trichotomy Property
For any two real numbers a and b, exactly one of the following is true:
a = b, a > b, or a < b.

17. Absolute Value î í ì
|a| =
a if a  0
-a if a < 0

18. EXAMPLE
Find |3| and |-3|.
Answer
|3| = 3, since 3  0.
|-3| = 3, since -3  0.

19. Homework pp

20. ►A. Exercises
3. Classify the following real numbers as natural, whole, integers, rational, or irrational. Be as specific as possible.
5, -3, 0, -6, , 3, 8 2 1 1 7 3 5 2
-5, , , , 7, 

21. 3.
Real Numbers Q Ir 5 2 3 Z -3 -6 -5 3 1 W  N 1 7 5 7 8

22. ►A. Exercises State the property described by each example.
5. 5 • 8 = 8 • 5
1. Associative
2. Commutative
3. Multiplication
4. Reflexive
5. Transitive

23. ►A. Exercises State the property described by each example.
9. a > 5; therefore a  5
1. Identity
2. Inverse
3. Symmetric
4. Transitive
5. Trichotomy

24. ►A. Exercises
Compute the following.
11. |-8|

25. ►A. Exercises
Compute the following.
13. |7| + |10|

26. ►A. Exercises
Compute the following.
15. |189 – 207|

27. ►B. Exercises
Tell whether the following statements are true or false.
17. Each natural number is a whole number.
1. True
2. False

28. ►B. Exercises
Tell whether the following statements are true or false.
19. The addition property of equality is used to solve an equation such as a – 5 = 10.
1. True
2. False

29. ►B. Exercises
Tell whether the following statements are true or false.
21. If b is a positive number, then -b is a negative number.
1. True
2. False

30. ►C. Exercises
24. Give the reason (appropriate property) for each step in the solution of
5x + 7 = y
y = 9(x – 5)

31. a. 5x + 7 = 9(x – 5) b. 5x + 7 = 9x – 45 c. 5x+7-5x+45= 9x-45-5x+45
d. 5x-5x+7+45= 9x-5x-45+45
e = 4x + 0
f. 52 = 4x
g. ¼(4 ∙ 13) = ¼(4x)
h. (¼ ∙ 4)13 = (¼ ∙ 4)x
i. 1 ∙ 13 = 1 ∙ x
j. 13 = x
k. x = 13
l. 5(13) + 7 = y
Transitive Prop.
Distributive Prop.
Addition Prop.
Commutative Prop.
Inverse Prop.
Identity Prop.
Multiplication Prop.
Associative Prop.
Inverse Prop.
Identity Prop.
Symmetric Prop.
Substitution Prop.

32. m. 72 = y
n. y = 72
Order of operations
Symmetric Prop.

33. ■ Cumulative Review
List the possible solutions for the intersection of
26. a line and a plane.

34. ■ Cumulative Review
List the possible solutions for the intersection of
27. two planes.

35. ■ Cumulative Review
List the possible solutions for the intersection of
28. a line and a convex polygon.

36. ■ Cumulative Review
List the possible solutions for the intersection of
29. a line and a circular region.

37. ■ Cumulative Review
30. Discuss possible relations among four points.