Master complex arithmetic, polar form, and the geometry of the complex plane
Complex numbers extend the real number system by introducing i = √(-1), the imaginary unit. Every complex number can be written as z = a + bi where a is the real part and b is the imaginary part.
Unlike real numbers that live on a line, complex numbers live on a plane - the complex plane. This gives them rich geometric structure that makes complex analysis incredibly powerful and beautiful.
See how complex numbers add and multiply geometrically. Addition is vector addition, while multiplication rotates and scales. Watch the magic happen!
📐 Addition as Vector Addition:
Complex addition works like vector addition. To find z₁ + z₂, place the tail of z₂ at the head of z₁. The result is the vector from the origin to this final point.
Formula: (a + bi) + (c + di) = (a + c) + (b + d)i
Every complex number can be expressed in two forms: Cartesian (a + bi) or Polar (r∠θ). Discover Euler's beautiful formula: e^(iθ) = cos(θ) + i·sin(θ)
z = r · eiθ = r · (cos θ + i sin θ)
= 2.50 · (cos(36.9°) + i sin(36.9°))
= 2.50 · (0.80 + i·0.60)
= 2.00 + 1.50i
🎯 Two Ways to Describe the Same Number:
Cartesian: z = a + bi (horizontal + vertical components)
Polar: z = r∠θ (distance from origin + angle from positive real axis)
✨ Euler's Beautiful Formula:
eiθ = cos(θ) + i·sin(θ) connects exponentials, trigonometry, and complex numbers in one elegant equation. This makes multiplication and powers much easier in polar form!
Conversion: a = r·cos(θ), b = r·sin(θ), r = √(a² + b²), θ = arctan(b/a)
The conjugate z̄ reflects z across the real axis. The modulus |z| measures distance from the origin. Together they satisfy the beautiful identity: z · z̄ = |z|²
2.00 - 1.50i
Reflects across real axis: (a + bi)̄ = a - bi
2.500
Distance from origin: √(a² + b²)
6.250
= |z|² = 6.250
z · z̄ = (2.00 + 1.50i) · (2.00 + -1.50i)
= 6.250
|z|² = (2.500)² = 6.250
✓ Identity holds!
🔁 Complex Conjugate (z̄):
The conjugate reflects z across the real axis. If z = a + bi, then z̄ = a - bi. Notice that both z and z̄ lie on the same circle (same distance from origin).
📏 Modulus (|z|):
The modulus is the distance from the origin to z. It's always non-negative and equals 0 only when z = 0. The green circle shows all points with the same modulus as z.
⚡ The Identity z · z̄ = |z|²:
This beautiful identity shows that multiplying a number by its conjugate always gives a real number (the square of its modulus). This is incredibly useful for simplifying complex fractions!
The nth roots of unity are the solutions to z^n = 1. They form a beautiful regular polygon on the unit circle and connect to group theory - they form a cyclic group!
ω = e2πi/6
= 0.500 + 0.866i
All roots are powers of ω: 1, ω, ω², ..., ω5
Angle: 360°/6 = 60.0°
= 2π/6 ≈ 1.047 rad
Forms a regular 6-gon on the unit circle
(0.500 + 0.866i)6 = 1.000
✓ Confirmed!
🔷 Beautiful Symmetry:
The nth roots of unity are evenly spaced around the unit circle, forming a perfect regular polygon. They're given by: zk = e2πik/n for k = 0, 1, 2, ..., n-1.
🔄 Connection to Group Theory:
These roots form a cyclic group under multiplication! The generator ω has order n (ωn = 1), and every root is a power of ω. If you studied group theory with the Rubik's Cube module, you've seen cyclic groups before!
Formula: zk = cos(2πk/n) + i·sin(2πk/n)
You've mastered the fundamentals of complex numbers! You now understand:
Next: We'll explore complex functions and discover the stunning world of domain coloring!