Lecture Notes: Complex Numbers

Overview

• Section 3.5: Discussion on complex numbers
• Target Audience: Math Majors, Electrical Engineering, Computer Science, and Mechanical Engineering students
• Recommendation: Read textbook section 3.5, solve problems without looking at solutions, and review regularly

Complex Numbers Basics

• Definition: Complex numbers have a real part a and an imaginary part b associated with i (√(-1))
  • Example: a + bi
• Plotting: Plotted on a plane where:
  • Real part on the x-axis (Real Number Axis)
  • Imaginary part on the y-axis (Imaginary Axis)

Imaginary Unit i

• Definition: i = √(-1)
• Property: i² = -1

Examples and Exercises

1. Identifying Real and Imaginary Parts

   • From Complex Number 3 + 8i
     • Real part: 3
     • Imaginary part: 8
   • From Complex Number (-2/7) + (-4/7)i
     • Real part: -2/7
     • Imaginary part: -4/7

2. Arithmetic with Complex Numbers

   • Addition: ( -5 + 5i ) + ( 7 - i ) = 2 + 4i
     • Real part: 2
     • Imaginary part: 4
   • Subtraction: ( -1 + i ) - ( 6 - 9i ) = -7 + 10i
     • Real part: -7
     • Imaginary part: 10
   • Multiplication: ( 6 + i ) * ( 1 - 4i ) = 10 + 2i
     • Real part: 10
     • Imaginary part: 2

Complex Conjugate

• Definition: Change the sign between the real and imaginary parts
  • Example: Complex Conjugate of (7 + 2i) is (7 - 2i)
• Usage: Used to rationalize denominators in fractions involving complex numbers
• Example:
  • Rationalize (3 - 4i) / (1 - 3i)
    • Multiply by complex conjugate: (3 - 4i)(1 + 3i) / (1 - 3i)(1 + 3i)
    • Simplify: (15 + 5i) / 10 = (3/2) + (1/2)i

Roots of Complex Numbers

• Example: Solve the quadratic equation x² + 8x + 17 = 0 using the quadratic formula
  • Solution: x = 4 ± i

Key Property

• Negative Radicals: √a * √b ≠ √(ab) if a and b are negative and under separate radicals