In this video, I explain what real numbers are, their different types, and the properties of real numbers.
What Are Real Numbers?
Real numbers include all the numbers that can be represented on a number line. These include:
- Positive and negative integers (e.g., -3, -1, 0, 1, 5)
- Fractions (e.g., 4/9, -8/11)
- Decimals (e.g., 0.43, 0.12)
- Irrational numbers (e.g., √2, π, e)
Types of Real Numbers
Real numbers can be categorized into several subsets:
- Natural Numbers (Counting Numbers): These are the numbers you use to count things, starting from 1. Examples: {1,2,3,4,…}.
- Whole Numbers: These are natural numbers including zero. Examples: {0, 1,2,3,4,…}.
- Integers: These include whole numbers and their negative counterparts. Examples: {…-3,-2, -1, 0, 1, 2, 3,…}.
- Rational Numbers: These include numbers that can be expressed as p/q, where q ≠ 0; include integers, terminating decimals, and repeating decimals. Examples: {1/2, -0.4, 8.9}.
- Irrational Numbers: Numbers that cannot be expressed as a simple fraction; have non-repeating, non-terminating decimal expansions. Examples: √2, π, e.
Properties of Real Numbers
Real numbers follow the following fundamental properties:
- Closure Property: Adding or multiplying two real numbers gives a real number.
- Commutative Property: For any two real numbers a and b,
- a + b = b + a
- a × b = b × a
- Associative Property: For any real numbers a, b, and c,
- (a + b) + c = a + (b + c)
- (a × b) × c = a × (b × c)
- Distributive Property: For any real numbers a, b, and c,
- a(b + c) = ab + ac
- Identity Property: For any real number a,
- Additive identity: a + 0 = a
- Multiplicative identity: a × 1 = a
- Inverse Property: For any real number a,
- Additive inverse: a + (-a) = 0.
- Multiplicative inverse: a × (1/a) = 1 (for a ≠ 0).