The complex plane: every number as a point or a rotation
i² = -1: why negative squares make sense geometrically
Multiplying by i is a 90-degree counterclockwise rotation. Multiplying by i twice (i.e. by i²) is a 180-degree rotation, which turns 1 into -1. So i² = -1 is not an algebraic trick; it is a rotation.
Complex multiplication: rotate and scale simultaneously
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Fundamental Theorem of Algebra: every polynomial splits completely
Table showing polynomials over reals versus complex numbers, demonstrating every degree-n polynomial has exactly n complex roots
POLYNOM
REELLE NULLSTELLEN
KOMPLEX
x - 3 = 0
1 (x=3)
1
x² - 4 = 0
2 (±2)
2
x² + 1 = 0
0 reelle Nullstellen
2 (±i)
x³ - 1 = 0
1 reelle Nullstelle
3
x⁴ + 4 = 0
0 reelle Nullstellen
4
Jedes Polynom vom Grad n hat genau n komplexe Nullstellen, Vielfachheiten mitgezählt