Complex Numbers

Complex numbers expand the real number system by introducing the imaginary unit $i$, where $i^2 = -1$.

Arithmetic in the Complex Plane

A complex number $z$ is written as $x + iy$, where $x$ is the real part and $y$ is the imaginary part. Addition and multiplication follow standard algebraic rules, with $i^2 = -1$.

# Python supports complex numbers natively using 'j'
z1 = 3 + 4j
z2 = 1 - 2j

# Addition
print(f"z1 + z2 = {z1 + z2}")

# Multiplication (3+4j)*(1-2j) = 3 - 6j + 4j - 8j^2 = 3 - 2j + 8 = 11 - 2j
print(f"z1 * z2 = {z1 * z2}")

# Magnitude (Absolute Value)
print(f"|z1| = {abs(z1)}")

Polar Form and Euler’s Formula

Complex numbers can be represented in polar coordinates $(r, \theta)$. Euler’s Formula connects the exponential function to trigonometry: $e^{i\theta} = \cos \theta + i \sin \theta$ Thus, $z = r e^{i\theta}$.

import cmath
import math

z = 1 + 1j
r, theta = cmath.polar(z)
print(f"r: {r:.3f}, theta: {theta:.3f} rad")

# Euler's Identity: e^(i*pi) + 1 = 0
identity = cmath.exp(1j * math.pi) + 1
print(f"e^(i*pi) + 1 = {identity.real:.1f} + {identity.imag:.1f}j")

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that every non-constant polynomial of degree $n$ with complex coefficients has exactly $n$ complex roots (counting multiplicity).