Complex Numbers

Complex numbers extend the real number system by introducing the imaginary unit i, where $i^{2}$ = -1. Every complex number z = a + ib has a real part Re(z) = a and imaginary part Im(z) = b. The set of complex numbers C is algebraically closed, meaning every polynomial equation of degree n has exactly n roots in C (Fundamental Theorem of Algebra).

The conjugate of z = a + ib is z-bar = a - ib. Key properties: z * z-bar = |z|² = $a^{2}$ + $b^{2}$, z + z-bar = 2Re(z), z - z-bar = 2iIm(z). The conjugate distributes over all operations: (z1 + z2)-bar = z1-bar + z2-bar, (z1z2)-bar = z1-bar * z2-bar, and (z1/z2)-bar = z1-bar / z2-bar.

The modulus |z| = $\sqrt{a^{2} + b^{2}}$ represents the distance from the origin in the Argand plane.
The argument arg(z) = $\theta$ is the angle with the positive real axis, with principal value in (-$\pi$, $\pi$].
The polar form z = r(cos $\theta$ + i sin $\theta$) = r * e^(i*$\theta$) (Euler's formula) is essential for multiplication and division. Multiplying complex numbers in polar form: multiply moduli and add arguments.

De Moivre's Theorem: (cos $\theta$ + i sin $\theta$)^n = cos(n*$\theta$) + i sin(n*$\theta$). This is the backbone for computing powers and finding nth roots.
The nth roots of unity are e^(2*$\pi$ik/n) for k = 0, 1, ..., n-1.
They form a regular n-gon on the unit circle and satisfy: sum = 0 and product = (-1)^(n+1).

The Argand plane geometry is crucial: |z - z1| = r is a circle with center z1 and radius r. |z - z1| = |z - z2| is the perpendicular bisector of the segment joining z1 and z2. arg(z - z1) = $\theta$ is a ray from z1 making angle $\theta$ with the positive direction. The triangle inequality |z1 + z2| <= |z1| + |z2| and its reverse ||z1| - |z2|| <= |z1 + z2| are fundamental.

The key problem-solving concept is switching fluently between algebraic (a+ib), polar (re^(i$\theta$)), and geometric (Argand plane) representations, choosing whichever simplifies the given problem most effectively.

---