This document defines and describes properties of real numbers. It begins by defining real numbers as values that represent quantities along a number line and form a set with two basic operations: addition and multiplication. It then lists and explains key properties of equality for real numbers, including closure, commutativity, associativity, distributivity, identity, and inverse properties. It concludes by depicting the real number system as a hierarchical structure with natural numbers at the bottom and real numbers at the top, encompassing rational and irrational numbers.

1. Real Numbers
DEFINITION. PROPERTIES. SYSTEM.

2. Real Numbers
A value that represents a quantity along a number
line.
It is a set of undefined elements a, b, c, etc. or
symbollically
R = { a, b, c, … }
It has two basic operation: addition and
multiplication.

3. Examples

4. Properties of Equality
•Closure Property •Distributive Property
•Commutative Property
•Identity Property
•Additive Identity
•Multiplicative Identity
•Associative Property
•Inverse Property
•Additive Inverse
•Multiplicative Inverse

5. Properties of Equality
Closure Property
For any real number a, b € R,
a+b € R and ab € R.
Example:
1+2=3
R = {1,2,3}; 1,2,3 € R.

6. Properties of Equality
Commutative Property
For any real number a, b € R,
a+b = b+a and ab = ba.
Example:
4 + 2 = 2 + 4
4 x 2 = 2 x 4

7. Properties of Equality
Associative Property
For any real number a, b, c € R,
(a + b) + c = a + (b + c)
(ab)c = a(bc).
Example:
(4 + 2) + 3 = 4 + (2 + 3)
(4 x 2) 3 = 4 (2 x 3)

8. Properties of Equality
Distributive Property
For any real number a, b, c € R,
a (b + c) = ab + ac.
Example:
4 (5 + 3) = 4 (5) + 4 (3)

9. Properties of Equality
Identity Property
Additive Identity
For any real number a € R,
a+0=a.
Example:
4+0=4
Zero is the identity element for
addition or additive identity.
Multiplicative Identity
For any real number a € R,
a . 1 =a.
Example:
4 . 1 =4
One is the identity element for
multiplication or multiplicative identity.

10. Properties of Equality
Inverse Property
Additive Inverse
For any real number a, there is a
real number called additive
inverse of a denoted by “-a” such
that a + (-a) = 1.
Example:
-(ab) + (ab) = 0
Multiplicative Inverse
For any real number a, there is a
real number called multiplicative
inverse of a denoted by “1/a” such
that a . 1/a = 1.
Example:
4 . ¼ = 1

11. The Real Number System
Real Numbers
Rational Numbers
Integers
Positive
Natural
Numbers
Fractions
Zero Negative
Irrational Numbers

12. Thank you!
Presented by:
Kenneth R. Wasawas
BSED - 3