Complex Number Implementation

The ComplexNumber class represents a complex number with real and imaginary parts, offering a range of operations for complex number arithmetic.

Features

• Representation of complex numbers in Cartesian format.
• Arithmetic operations: addition, subtraction, multiplication, division, and exponentiation.
• Utility methods: square root, conjugation, modulus (absolute value), argument (phase angle), and conversions between polar and cartesian forms.

Installation

To use the ComplexNumber class, include the complex_number.py file in your project and import it as needed.

from complex_number import ComplexNumber

Usage

Initialization

# Initialize a complex number with real and imaginary parts
z = ComplexNumber(3, 4)

Operations

Addition

Adds two complex numbers, or a complex number and a real number.

z1 = ComplexNumber(1, 2)
z2 = ComplexNumber(3, 4)
result = z1 + z2

Mathematically: $(a + bi) + (c + di) = (a+c) + (b+d)i$

Subtraction

Subtracts two complex numbers, or a complex number and a real number.

result = z1 - z2

Mathematically: $(a + bi) - (c + di) = (a-c) + (b-d)i$

Multiplication

Multiplies two complex numbers, or a complex number and a real number.

result = z1 * z2

Mathematically: $(a + bi) \cdot (c + di) = (ac-bd) + (ad+bc)i$

Division

Divides two complex numbers, or a complex number by a real number.

result = z1 / z2

Mathematically: $\frac{a + bi}{c + di} = \frac{ac+bd}{c^2+d^2} + \frac{bc-ad}{c^2+d^2}i$, assuming $c^2 + d^2 \neq 0$

Exponentiation

Raises a complex number to the power of a real number. Complex exponents are not implemented.

result = z1 ** 2

Mathematically: $r(\cos(\theta) + i\sin(\theta)))^n = r^n(\cos(n\theta) + i\sin(n\theta))$, where $r$ is the modulus and $\theta$ is the argument of the complex number.

Utility Methods

Square Root

Calculates the square root of a complex number.

result = z1.sqrt()

Conjugate

Returns the conjugate of a complex number.

result = z1.conjugate()

Mathematically: If $z = a + bi$, then $\bar{z} = a - bi$

Modulus

Calculates the modulus (absolute value) of a complex number.

result = z1.modulus()

Mathematically: $|a + bi| = \sqrt{a^2 + b^2}$

Argument

Returns the argument (phase angle) of a complex number in radians.

result = z1.argument()

Mathematically: $\theta = \tan^{-1}(\frac{b}{a})$

Polar Conversion

Converts a complex number to its polar form (magnitude and phase).

result = z1.to_polar()

Class Methods

From Polar

Creates a complex number instance from polar coordinates.

z = ComplexNumber.from_polar(magnitude, angle)

Tests

Comprehensive tests covering arithmetic operations, utility methods, and special cases are included in the tests directory. Ensure to run these tests to validate functionality.