Complex Numbers and Quadratic Equations

Learning Objectives

• Understand the concept of complex numbers as an extension of the real number system, including the definition of the imaginary unit $i$ and representation in the form $a + ib$.
• Perform algebraic operations on complex numbers such as addition, subtraction, multiplication, and division, and comprehend properties like closure, commutativity, associativity, and the existence of identities and inverses.
• Calculate the modulus and conjugate of a complex number and interpret their geometric significance on the Argand plane.
• Represent complex numbers geometrically on the Argand plane and understand their polar representation.
• Compute powers of the imaginary unit $i$ and determine square roots of negative real numbers.
• Apply identities and solve equations involving complex numbers.
• Simplify complex expressions into the standard form $a + ib$.
• Solve quadratic equations with complex roots when the discriminant $D = b^2 - 4ac < 0$, using complex numbers and square roots of negative numbers.

Chapter Notes

Complex Numbers

• Definition: A complex number is of the form $a + ib$, where $a$ and $b$ are real numbers.
• Imaginary Unit: Denoted by $i$, where $i^2 = -1$.
• Real Part: For $z = a + ib$, the real part is $a$, denoted by $\text{Re}(z)$.
• Imaginary Part: For $z = a + ib$, the imaginary part is $b$, denoted by $\text{Im}(z)$.

Algebra of Complex Numbers

• Addition: $(a + ib) + (c + id) = (a + c) + i(b + d)$
  • Properties:
    • Closure: Sum is a complex number.
    • Commutative: $z_1 + z_2 = z_2 + z_1$
    • Associative: $(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$
    • Additive Identity: $z + 0 = z$
    • Additive Inverse: $z + (-z) = 0$
• Subtraction: $z_1 - z_2 = z_1 + (-z_2)$
• Multiplication: $(a + ib)(c + id) = (ac - bd) + i(ad + bc)$
  • Properties:
    • Closure: Product is a complex number.
    • Commutative: $z_1z_2 = z_2z_1$
    • Associative: $(z_1z_2)z_3 = z_1(z_2z_3)$
    • Multiplicative Identity: $z \cdot 1 = z$
    • Multiplicative Inverse: For $z \neq 0$, $z^{-1}$ exists.
• Division: $\frac{z_1}{z_2} = \frac{z_1 \cdot \overline{z_2}}{|z_2|^2}$, where $z_2 \neq 0$.

Modulus and Conjugate of Complex Numbers

• Modulus: $|z| = \sqrt{a^2 + b^2}$
• Conjugate: $\overline{z} = a - ib$
• Properties:
  • $z \cdot \overline{z} = |z|^2$
  • $|z_1z_2| = |z_1||z_2|$
  • $|z_1/z_2| = |z_1|/|z_2|$ for $z_2 \neq 0$

Argand Plane and Polar Representation

• Argand Plane: A geometric representation of complex numbers.
• Polar Form: $z = r(\cos \theta + i\sin \theta)$, where $r = |z|$ and $\theta$ is the argument of $z$.

Powers and Roots of Complex Numbers

• Powers of $i$:
  • $i^1 = i$
  • $i^2 = -1$
  • $i^3 = -i$
  • $i^4 = 1$
• Square Roots of Negative Numbers: $\sqrt{-a} = \sqrt{a}i$ for $a > 0$.

Identities and Equations Involving Complex Numbers

• Identities:
  • $(z_1 + z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2$
  • $(z_1 - z_2)^2 = z_1^2 - 2z_1z_2 + z_2^2$
• Equations: Solve using substitution and equating real and imaginary parts.

Standard Form and Simplification of Complex Expressions

• Standard Form: Reduce expressions to $a + ib$.
• Example: $(5 - 3i)^3 = 10 - 198i$.

Quadratic Equations with Complex Roots

• Discriminant: If $D = b^2 - 4ac < 0$, roots are complex.
• Roots: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $\sqrt{b^2 - 4ac}$ involves $i$.