Complex Numbers and Their Operations

Definition and Representation

• Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1 (i² = -1).
• 'a' is the real part and 'b' is the imaginary part of the complex number.

Geometric Representation

• Complex numbers can be represented geometrically on a coordinate plane called the complex plane.
• The horizontal axis represents the real part (a), and the vertical axis represents the imaginary part (b).
• Each complex number corresponds to a unique point in the complex plane.
• The magnitude (or modulus) of a complex number a + bi, denoted as |a + bi|, represents the distance from the origin (0,0) to the point (a,b) in the complex plane. It's calculated as √(a² + b²).

Operations on Complex Numbers

• Addition: To add two complex numbers (a + bi) and (c + di), add the real parts and the imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i
• Subtraction: To subtract two complex numbers (a + bi) and (c + di), subtract the real parts and the imaginary parts separately: (a + bi) - (c + di) = (a - c) + (b - d)i
• Multiplication: To multiply two complex numbers (a + bi) and (c + di), use the distributive property and the fact that i² = -1: (a + bi)(c + di) = ac + adi + bci + bdi² = (ac - bd) + (ad + bc)i
• Division: To divide two complex numbers (a + bi) and (c + di), multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate of (c + di) is (c - di).
  • [(a + bi)] / [(c + di)] = [(a + bi)(c - di)] / [(c + di)(c - di)] = [(ac + bd) + (bc - ad)i] / [c² + d²] = [(ac + bd)/(c² + d²)] + [(bc - ad)/(c² + d²)]i

Complex Conjugate

• The complex conjugate of a complex number a + bi is a - bi.
• When multiplying a complex number by its complex conjugate, the result is always a real number. (a + bi)(a - bi) = a² + b².

Polar Form

• Complex numbers can also be represented in polar form.
• A complex number z = a + bi can be expressed as z = r(cos θ + i sin θ), where:
  • r = |z| (the modulus or magnitude of z)
  • θ is an argument of z (an angle in the complex plane)
• The conversion between rectangular (a+bi) and polar form (r(cos θ + i sin θ)) is done using trigonometric functions:
  • a = r cos θ
  • b = r sin θ
  • θ = arctan(b/a)
• Polar form is useful for multiplying and dividing complex numbers. Using this representation:
  • z₁z₂ = r₁r₂(cos(θ₁ + θ₂) + i sin(θ₁ + θ₂))
  • z₁/z₂ = r₁/r₂(cos(θ₁ - θ₂) + i sin(θ₁ - θ₂))

De Moivre's Theorem

• De Moivre's Theorem states that for any complex number z = r(cosθ + i sinθ) and any positive integer n:
  • [r(cosθ + i sinθ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ))

Roots of Complex Numbers

• Finding the nth roots of a complex number involves using De Moivre's theorem.
• The nth roots of a complex number z = r(cosθ + i sinθ) are given by the formula:
  • z^(1/n) = r^(1/n)[cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)] , for k = 0, 1, 2, ..., n-1
• This formula gives n distinct roots.

This quiz focuses on the definition, representation, and operations involving complex numbers. You'll explore how to express complex numbers, their geometric representation on the complex plane, and the rules for operations such as addition and magnitude calculation.