Lecture on Complex Numbers

Introduction

• Split complex numbers into three parts:
  • Conjugate and modulus
  • Problems based on those topics
  • Geometry problems based on locus and argument
• Focus on algebra of complex numbers
• Covered topics: conjugate, modulus, geometry, and algebra

Definition of Complex Numbers

• A complex number is of the form a + bi, where a and b are real numbers
• i is defined as the square root of -1
  • i^2 = -1
  • i^3 = -i
  • i^4 = 1
• Real part (a) and imaginary part (b) discussed
• Real numbers are a subset of complex numbers

Algebra of Complex Numbers

Operations

• Addition: Add corresponding real and imaginary parts
• Subtraction: Subtract corresponding real and imaginary parts
• Multiplication: Use distributive property and simplify using i^2 = -1
• Division: Multiply by the conjugate of the denominator

Conjugate and Modulus

• Conjugate: If z = x + iy, then z̅ = x - iy
  • Conjugate is a reflection about the x-axis
• Modulus: Distance from origin to the point (x, y) in the complex plane
  • |z| = √(x^2 + y^2)

Properties

• Equality: Two complex numbers are equal if their real and imaginary parts are equal
• Purely Real: z is real if the imaginary part is zero
• Purely Imaginary: z is purely imaginary if the real part is zero
• Triangle Inequality: |z1 + z2| ≤ |z1| + |z2|

Problems and Solutions

• Discussed various problems involving complex numbers, their properties, and operations
• Examples include calculating powers of i, finding conjugates, moduli, and solving equations involving complex numbers

Geometry of Complex Numbers

• Representation in 2D plane
• Reflection and rotation interpretations
• Locus and argument problems

Polar Form of Complex Numbers

• Conversion from Cartesian to Polar form
  • z = x + iy = r(cosθ + i sinθ) where r = |z| and θ is the argument
• Principal argument: Unique value of θ in (-π, π]
• Properties of polar form
  • Multiplication and division of complex numbers in polar form

Roots of Unity

• Cube roots of unity: Solutions to z^3 = 1
  • Divide the unit circle into three equal parts
  • Sum of cube roots of unity is zero
  • Product of cube roots of unity is one
• Fourth roots of unity: Solutions to z^4 = 1
  • Divide the unit circle into four equal parts
  • Forms a square in the complex plane

Problems on Roots of Unity

• Various problems involving cube roots and fourth roots of unity
• Finding values of expressions involving roots of unity

Advanced Problems and Applications

• Problems involving division, multiplication, and exponentiation in the context of roots of unity
• Use of binomial expansion to solve complex number equations
• Representation and simplification techniques for complex numbers

Conclusion

• Review of key concepts: complex number operations, polar form, roots of unity
• Emphasis on problem-solving techniques and properties of complex numbers
• Next topics include locus and geometry problems related to complex numbers