Complex Numbers
Complex Numbers
Definition of complex numbers, Cartesian and polar forms, Argand diagram, modulus and argument, arithmetic operations, De Moivre's theorem, roots of complex numbers.
Complex Numbers
Definition: A complex number z = a + bi where:
• a = Re(z) = real part
• b = Im(z) = imaginary part
• i = −1, so i² = −1
• a = Re(z) = real part
• b = Im(z) = imaginary part
• i = −1, so i² = −1
Arithmetic Operations
Let z₁ = a+bi and z₂ = c+di:
Addition: z₁+z₂ = (a+c) + (b+d)i
Subtraction: z₁−z₂ = (a−c) + (b−d)i
Multiplication: z₁z₂ = (ac−bd) + (ad+bc)i
Division: z₁z₂ = (ac+bd) + (bc−ad)ic²+d²
Complex Conjugate
z̄ = a − bi (conjugate: flip sign of imaginary part)
z·z̄ = a² + b² = |z|² (always real and non-negative)
z·z̄ = a² + b² = |z|² (always real and non-negative)
Modulus and Argument
|z| = a²+b² (modulus = distance from origin on Argand diagram)
arg(z) = θ = arctan(ba) (argument = angle from positive real axis)
arg(z) = θ = arctan(ba) (argument = angle from positive real axis)
Polar Form
z = r(cos θ + i sin θ) = re^(iθ) where r=|z|, θ=arg(z)
Euler's formula: e^(iθ) = cos θ + i sin θ
Euler's formula: e^(iθ) = cos θ + i sin θ
Example: z = 1+i
|z| = 1+1 = 2, θ = arctan(11) = 45° = π/4
Polar: z = 2 (cos 45° + i sin 45°) = 2 · e^(iπ/4)
|z| = 1+1 = 2, θ = arctan(11) = 45° = π/4
Polar: z = 2 (cos 45° + i sin 45°) = 2 · e^(iπ/4)
De Moivre's Theorem
[r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ)
Equivalently: (re^(iθ))ⁿ = rⁿe^(inθ)
Equivalently: (re^(iθ))ⁿ = rⁿe^(inθ)
Example: (1+i)⁴ using De Moivre:
r = 2, θ = π/4 → (2)⁴(cos π + i sin π) = 4(−1+0i) = −4
r = 2, θ = π/4 → (2)⁴(cos π + i sin π) = 4(−1+0i) = −4
nth Roots of a Complex Number
The n roots of zⁿ = re^(iθ) are:
zₖ = r^(1/n) · e^(i(θ+2πk)/n) for k = 0, 1, 2, …, n−1
Roots are equally spaced on a circle of radius r^(1/n)
zₖ = r^(1/n) · e^(i(θ+2πk)/n) for k = 0, 1, 2, …, n−1
Roots are equally spaced on a circle of radius r^(1/n)
Cube roots of 1 (z³=1): r=1, θ=0
z₀ = 1, z₁ = e^(2πi3) = cos(120°)+i·sin(120°), z₂ = e^(4πi3) = cos(240°)+i·sin(240°)
z₀ = 1, z₁ = e^(2πi3) = cos(120°)+i·sin(120°), z₂ = e^(4πi3) = cos(240°)+i·sin(240°)