Chapter 1: Complex Numbers and Complex Plane

Complex Numbers and Complex Plane

📖 Reading Time: 20-25 minutes 📊 Difficulty: Beginner 💻 Code Examples: 0 📝 Exercises: 0

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Fundamentals Mathematics Dojo > Complex Functions and Special Functions > Chapter 1

1.1 Basic Operations of Complex Numbers

Complex numbers are expressed in the form \(z = x + iy\) and have a real part \(x\) and an imaginary part \(y\). In Python, they can be handled with complex type or NumPy.

📐 Definition: Complex Numbers
Definition of complex numbers: \[z = x + iy, \quad i = \sqrt{-1}\] Basic operations:

💻 Code Example 1: Basic Operations of Complex Numbers

Python Implementation: Basic Operations of Complex Numbers

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

"""
Example: 💻 Code Example 1: Basic Operations of Complex Numbers

Purpose: Demonstrate data visualization techniques
Target: Intermediate
Execution time: 2-5 seconds
Dependencies: None
"""

import numpy as np
import matplotlib.pyplot as plt

# Define complex numbers
z1 = 3 + 4j
z2 = 1 - 2j

print(f"z1 = {z1}")
print(f"z2 = {z2}")
print(f"z1 + z2 = {z1 + z2}")
print(f"z1 * z2 = {z1 * z2}")
print(f"z1 / z2 = {z1 / z2}")

# Complex conjugate and absolute value
print(f"\nConjugate: z1.conjugate() = {z1.conjugate()}")
print(f"Absolute value: |z1| = {np.abs(z1)}")
print(f"Argument: arg(z1) = {np.angle(z1)} rad = {np.degrees(np.angle(z1)):.2f}°")

# Visualization on complex plane
fig, ax = plt.subplots(figsize=(8, 8))
ax.axhline(0, color='gray', linewidth=0.5)
ax.axvline(0, color='gray', linewidth=0.5)
ax.grid(True, alpha=0.3)

# Draw complex numbers as vectors
def plot_complex(z, label, color):
    ax.arrow(0, 0, z.real, z.imag, head_width=0.3, head_length=0.2,
             fc=color, ec=color, linewidth=2, label=label)
    ax.plot(z.real, z.imag, 'o', color=color, markersize=8)
    ax.text(z.real + 0.3, z.imag + 0.3, label, fontsize=12, color=color)

plot_complex(z1, 'z1', 'blue')
plot_complex(z2, 'z2', 'red')
plot_complex(z1 + z2, 'z1+z2', 'green')

ax.set_xlabel('Real part (Re)', fontsize=12)
ax.set_ylabel('Imaginary part (Im)', fontsize=12)
ax.set_title('Vector representation on complex plane', fontsize=14)
ax.legend()
ax.axis('equal')
ax.set_xlim(-1, 5)
ax.set_ylim(-3, 5)
plt.tight_layout()
plt.show()

1.2 Polar Form and Euler’s Formula

Complex numbers can also be expressed in polar form \(z = r e^{i\theta}\), which is based on Euler’s formula \(e^{i\theta} = \cos\theta + i\sin\theta\).

📐 Theorem: Euler’s Formula
Polar form representation: \[z = r e^{i\theta} = r(\cos\theta + i\sin\theta)\] where \(r = |z|\) (absolute value), \(\theta = \arg(z)\) (argument)
Special case: \[e^{i\pi} + 1 = 0 \quad \text{(Euler’s identity)}\]

Summary

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