Overview

This topic explores the structure of the number system, focusing on complex numbers, their operations, and classification. You'll work with imaginary units, understand addition and multiplication of complex numbers, and review how numbers are categorized from natural numbers to complex forms.

Key Concepts and Structures

• Number Classifications:
  • Natural Numbers: 1, 2, 3, ...
  • Whole Numbers: 0, 1, 2, 3, ...
  • Integers: ..., -2, -1, 0, 1, 2, ...
  • Rational Numbers: Can be written as a ratio of integers
  • Irrational Numbers: Non-repeating, non-terminating decimals (e.g., √2, π)
  • Real Numbers: All rational and irrational numbers
  • Complex Numbers: Includes real and imaginary parts
• Imaginary Unit: i = √-1; i^2 = -1
• Complex Number Form: a + bi, where a is the real part and b is the imaginary part
• Adding/Subtracting: Combine like terms: (3 + 2i) + (1 - 5i) = 4 - 3i
• Multiplying: Use distributive property and i^2 = -1. Example: (2 + i)(3 - i) = 6 - 2i + 3i - i^2 = 6 + i + 1 = 7 + i
• Conjugates: Used to rationalize denominators. \overline{a + bi} = a - bi
• Modulus: |a + bi| = √(a² + b²)
• Quadratic Solutions: When the discriminant is negative, the roots are complex (e.g., x = 2 ± 3i)

Step-by-Step Example

Problem: Multiply (1 + 2i)(2 - 3i)

Step 1: Use distributive property:
1×2 + 1×(-3i) + 2i×2 + 2i×(-3i)

Step 2: Simplify:
2 - 3i + 4i - 6i^2

Step 3: Use i^2 = -1:
2 + i + 6 = 8 + i

Final Answer: 8 + i

Quick Tip