This document is a lecture on real numbers presented by Ms. Cherry Rose R. Estabillo. It introduces the different subsets of real numbers from natural numbers to irrational numbers. Diagrams are shown to illustrate the relationships between these number sets. Examples are provided to demonstrate classifying numbers within these sets. Properties of real numbers like closure, commutativity, associativity and distributivity are discussed. The order of operations and examples applying it are also covered.

1. ALGEB-X: REAL NUMBER SYSTEM
MATH 101
REAL NUMBERS
Lecture by: Ms. Cherry Rose R. Estabillo
MATH101 C. ESTABILLO

2. Real Numbers
Rational Numbers
Irrational Numbers
Non- Integers
Integers
Negative
Integers
Whole Numbers
Zero
Natural Numbers
MATH101 C. ESTABILLO

3. Real Numbers
Rational Numbers
Irrational Numbers
Non- Integers
Integers
Negative
Integers
Whole Numbers
Zero
Natural Numbers
MATH101 C. ESTABILLO

4. Schematic Diagram
Natural
Numbers
(N)
N = {1, 2, 3, 4, …}
MATH101 C. ESTABILLO

5. Schematic Diagram
Whole
Numbers (W)
Natural
Numbers
(N)
W = {0, 1, 2, 3, 4, …}
MATH101 C. ESTABILLO

6. Schematic Diagram
Integers (Z)
Whole
Numbers (W)
Natural
Numbers
(N)
Z = {… -2, -1, 0, 1, 2, …}
MATH101 C. ESTABILLO

7. Schematic Diagram
Rational Numbers (Q)
Integers (Z)
Whole
Numbers (W)
Natural
Numbers
(N)
A rational number can be expressed
as a quotient of two integers, a/b
where b is not equal to 0. The set of
rational numbers includes terminating
and repeating decimals.
Q = { … -3, … 0, … 3, … }
-3.75
5/7
-2
4.21
MATH101 C. ESTABILLO

8. Schematic Diagram
- 2
Irrational Numbers
3 3
(Q’) Q’ = { … -3, … 0, … 3, … }
-1.1234567890023… p
An irrational number is a non-repeating,
non-terminating decimal.
MATH101 C. ESTABILLO

9. Schematic Diagram
Rational Numbers (Q)
Integers (Z)
Whole
Numbers (W)
Natural
Numbers
(N)
Irrational Numbers
Real Numbers
MATH101 C. ESTABILLO

10. Example
Given:
30, 3.14, 8 p i
, 3,,2.71253...,0, 1
2
Name the set of non-integer, rational numbers.
Name the set of irrational numbers.
Name the set of integers.
Name the set of whole numbers.
Name the set of natural numbers.
þ ý ü
î í ì
- - - ,25,9.253253...,2 ,- 81
2
,
11
MATH101 C. ESTABILLO

11. Properties of Real Numbers
• Closure Property
• Commutative Property
• Associative Property
• Distributive Property of Multiplication over
Addition
• Identity Property
• Inverse Property
• Zero Property
MATH101 C. ESTABILLO

12. Name the axiom that justifies the ff. statement
(30 ×5) × 20 = 20 × (30 ×5)
(4 + x) ×(2 + x) = 4×(2 + x) + x ×(2 + x)
x + (5 +13) = ( x + 5) +13
0 + (50 + 40) = 50 + 40
7 + 3ÎÂ
MATH101 C. ESTABILLO

13. Application
• Is the action of undressing and taking a bath
commutative?
• Is the action of tying your left shoe and tying
your right shoe commutative?
MATH101 C. ESTABILLO

14. ORDER OF OPERATIONS
P E M D A S
MATH101 C. ESTABILLO

15. EXAMPLES
1.) 56 ÷ 23 x 7 –1 + 3 x 8
2.)
( )
¸ × + ×
40 2 3 4 2
- × ¸ + -
( 3) 8 4 (1 3)
MATH101 C. ESTABILLO