Complex Numbers and Quadratic Equations

• Introduction & Motivation

  • Need for Complex Numbers ($x^2 + 1 = 0$)
  • Imaginary Unit ($i = \sqrt{-1}$)
  • Powers of $i$ ($i^2 = -1, i^3 = -i, i^4 = 1$)

• Algebra of Complex Numbers

  • Definition: $z = a + ib$ ($a, b \in \mathbb{R}$)
  • Real part ($\text{Re}(z) = a$)
  • Imaginary part ($\text{Im}(z) = b$)
  • Equality of Complex Numbers
  • Basic Operations
    • Addition ($(a+ib) + (c+id) = (a+c) + i(b+d)$)
    • Subtraction ($(a+ib) - (c+id) = (a-c) + i(b-d)$)
    • Multiplication ($(a+ib)(c+id) = (ac-bd) + i(ad+bc)$)
    • Division (Multiply numerator & denominator by conjugate)
  • Algebraic Properties
    • Closure, Commutative, Associative
    • Distributive law
    • Existence of additive identity & inverse
    • Existence of multiplicative identity & inverse

• The Modulus and Conjugate

  • Conjugate of $z$: $\bar{z} = a - ib$
    • Key property: $z\bar{z} = |z|^2$
  • Modulus of $z$: $|z| = \sqrt{a^2 + b^2}$
  • Properties of Modulus and Conjugate
    • $|z_1 z_2| = |z_1||z_2|$
    • $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$
    • $\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2$

• Argand Plane and Polar Representation

  • Argand Plane (Complex Plane)
    • X-axis as Real Axis
    • Y-axis as Imaginary Axis
  • Polar Representation
    • $z = r(\cos\theta + i\sin\theta)$
    • Modulus $r = |z|$
    • Argument $\theta = \arg(z)$
  • Principal Argument ($-\pi < \theta \le \pi$)

• Quadratic Equations

  • Standard Form: $ax^2 + bx + c = 0$ ($a \neq 0$)
  • Discriminant: $D = b^2 - 4ac$
  • Solutions for $D < 0$ (Complex Roots)
    • Formula: $x = \frac{-b \pm \sqrt{D}}{2a}$
    • Roots are complex conjugates of each other
  • Fundamental Theorem of Algebra
    • A polynomial equation of degree $n$ has $n$ roots