🎓 Deeper Dive — Adults

A complex number z = a + bi where i² = −1. Real part Re(z) = a, imaginary part Im(z) = b.

Modulus |z| = √(a² + b²). Conjugate z̄ = a − bi.

Multiplication: (a+bi)(c+di) = ac + adi + bci + bdi² = (ac−bd) + (ad+bc)i.

Euler's identity: e^(iπ) + 1 = 0 connects e, i, π, 1, 0 in one equation. Polar form: z = r·e^(iθ) — multiplication becomes adding angles, multiplying magnitudes.