Master complex arithmetic, polar form, and the geometry of the complex plane
Add one new number, i = √(-1), and the number line opens into a plane. On that plane, adding is sliding and multiplying is turning.
Choose addition or multiplication and move the sliders for z₁ and z₂. Addition slides one arrow onto the other; multiplication rotates and scales.
📐 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
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.
Move the sliders and watch a + bi and r∠θ change together. Euler's formula links them: 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)
Move z and watch its conjugate z̄ mirror it across the real axis. The product always lands on the real number 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!
Change n and watch the solutions of z^n = 1 spread into a regular polygon on the unit circle.
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)
Spot a mistake or have an idea?