Python - Mathematics

Mathematics in Python

Introduction

Python is not only a powerful general-purpose programming language but also a robust tool for performing mathematical operations. From basic arithmetic to advanced calculus, Python provides a variety of tools to handle mathematical computations efficiently. Whether through built-in operations, modules like math and cmath, or libraries like NumPy and SymPy, Python is a valuable asset in scientific, engineering, data science, and financial domains.

1. Basic Arithmetic Operations

Operators

Python supports the following arithmetic operators:

• +: Addition
• -: Subtraction
• *: Multiplication
• /: Division
• //: Floor Division
• %: Modulus
• **: Exponentiation

Examples

x = 10
y = 3

print(x + y)   # 13
print(x - y)   # 7
print(x * y)   # 30
print(x / y)   # 3.333...
print(x // y)  # 3
print(x % y)   # 1
print(x ** y)  # 1000

2. Mathematical Functions Using the math Module

Introduction to math Module

The math module provides access to mathematical functions like trigonometry, logarithms, and constants. It works on float and integer numbers (not complex numbers).

Importing the Module

import math

Constants

• math.pi: 3.14159...
• math.e: 2.71828...
• math.tau: 6.28318...

Common Functions

math.sqrt(25)        # 5.0
math.ceil(4.2)       # 5
math.floor(4.8)      # 4
math.factorial(5)    # 120
math.fabs(-7)        # 7.0
math.pow(2, 3)       # 8.0
math.log(100, 10)    # 2.0

Trigonometric Functions

math.sin(math.pi/2)  # 1.0
math.cos(0)          # 1.0
math.tan(math.pi/4)  # 1.0
math.degrees(math.pi) # 180.0
math.radians(180)     # 3.14159...

3. Complex Numbers with cmath

Introduction to Complex Math

Python has native support for complex numbers. The cmath module provides functions for complex number mathematics.

Creating Complex Numbers

z = complex(3, 4)
print(z)       # (3+4j)
print(z.real)  # 3.0
print(z.imag)  # 4.0

cmath Functions

import cmath

z = complex(3, 4)

print(cmath.polar(z))        # (5.0, 0.927...)
print(cmath.phase(z))        # 0.927...
print(cmath.rect(5, 0.927))  # (3.0+4.0j)

Exponential and Logarithmic Functions

cmath.exp(1j * cmath.pi)      # (-1+1.224e-16j)
cmath.log(1 + 1j)             # (0.3465+0.7853j)

4. Rounding and Number Manipulation

Rounding Functions

round(4.567, 2)    # 4.57
round(4.567)       # 5

Absolute and Sign Functions

abs(-15)           # 15
math.copysign(3, -1)  # -3.0

Divmod Function

divmod(10, 3)      # (3, 1)

5. Working with Random Numbers

The random Module

import random

random.seed(10)  # Set seed for reproducibility
print(random.randint(1, 10))    # Random integer between 1 and 10
print(random.uniform(1, 10))    # Random float
print(random.choice([1, 2, 3])) # Random choice from list
print(random.sample(range(100), 5)) # List of 5 unique random numbers

Shuffling and Sampling

items = [1, 2, 3, 4, 5]
random.shuffle(items)
print(items)

6. Advanced Mathematics with NumPy

Introduction

NumPy is the foundational package for numerical computing in Python. It offers support for large, multi-dimensional arrays and matrices, along with a collection of mathematical functions.

Installation

pip install numpy

Basic Operations

import numpy as np

a = np.array([1, 2, 3])
b = np.array([4, 5, 6])

print(a + b)        # [5 7 9]
print(a * b)        # [ 4 10 18]
print(np.dot(a, b)) # 32 (dot product)

Statistical Functions

data = np.array([1, 2, 3, 4, 5])

print(np.mean(data))    # 3.0
print(np.median(data))  # 3.0
print(np.std(data))     # 1.4142...

Trigonometric and Exponential Functions

angles = np.array([0, np.pi/2, np.pi])
print(np.sin(angles))   # [0. 1. 0.]
print(np.exp([1, 2]))   # [2.718 7.389]

7. Symbolic Mathematics with SymPy

Introduction

SymPy is a Python library for symbolic mathematics. It can handle algebraic expressions, calculus, equations, and more.

Installation

pip install sympy

Algebraic Operations

from sympy import symbols, expand, factor

x, y = symbols('x y')
expr = (x + y)**2

print(expand(expr))     # x**2 + 2*x*y + y**2
print(factor(x**2 + 2*x*y + y**2))  # (x + y)**2

Solving Equations

from sympy import Eq, solve

eq = Eq(x**2 - 4, 0)
solutions = solve(eq, x)
print(solutions)        # [-2, 2]

Calculus

from sympy import diff, integrate, sin

print(diff(sin(x), x))           # cos(x)
print(integrate(sin(x), x))      # -cos(x)

Limits and Series

from sympy import limit, oo

print(limit(1/x, x, oo))         # 0

8. Fractions and Decimals

Using the fractions Module

from fractions import Fraction

f1 = Fraction(3, 4)
f2 = Fraction(1, 2)

print(f1 + f2)   # 5/4
print(f1 * f2)   # 3/8

Using the decimal Module

from decimal import Decimal, getcontext

getcontext().prec = 4
d1 = Decimal('1.123')
d2 = Decimal('2.456')
print(d1 + d2)   # 3.579

9. Geometry and Coordinate Math

Distance Formula

import math

def distance(x1, y1, x2, y2):
    return math.sqrt((x2 - x1)**2 + (y2 - y1)**2)

print(distance(0, 0, 3, 4))  # 5.0

Midpoint Formula

def midpoint(x1, y1, x2, y2):
    return ((x1 + x2) / 2, (y1 + y2) / 2)

print(midpoint(0, 0, 4, 4))  # (2.0, 2.0)

10. Probability and Combinatorics

Factorials and Permutations

import math

# Combinations C(n, k)
def combinations(n, k):
    return math.factorial(n) // (math.factorial(k) * math.factorial(n - k))

# Permutations P(n, k)
def permutations(n, k):
    return math.factorial(n) // math.factorial(n - k)

print(combinations(5, 2))  # 10
print(permutations(5, 2))  # 20

Python offers a rich and diverse set of tools for mathematical computations. Whether you're performing basic arithmetic, exploring symbolic algebra, or applying statistical analysis, Python’s built-in features and third-party libraries make it a powerful language for mathematical tasks. From math and cmath for general operations, random for probability, to NumPy and SymPy for advanced numeric and symbolic math, Python scales from beginner to expert-level mathematical applications.