Real Numbers in Mathematics

Real numbers are the set of numbers that include both rational and irrational numbers. They can be plotted on a number line, which makes them easy to understand and visualize.

Types of Real Numbers:

• Natural Numbers (N): Numbers we use for counting. Examples: 1, 2, 3, 4, …
• Whole Numbers (W): Natural numbers + 0. Examples: 0, 1, 2, 3, …
• Integers (Z): Whole numbers and their negatives. Examples: … -3, -2, -1, 0, 1, 2, 3 …
• Rational Numbers (Q): Numbers that can be written as a fraction of two integers (denominator ≠ 0).
  Examples: ½, -3/4, 0.25 (which is 1/4), 5 (which is 5/1)
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Their decimal expansions are non-terminating and non-repeating. Examples: √2, π (pi), e

Properties of Real Numbers:

• Commutative Property:
  • a + b = b + a
  • a × b = b × a
• Associative Property:
  • (a + b) + c = a + (b + c)
  • (a × b) × c = a × (b × c)
• Distributive Property:
  • a × (b + c) = a × b + a × c
• Identity Elements:
  • 0 is the additive identity
  • 1 is the multiplicative identity
• Closure Property:
  • The result of adding, subtracting, multiplying, or dividing (except by 0) two real numbers is always a real number.