Easy Python Program to Calculate Leibniz Formula for π

by
5/5 - (2 votes)

The Leibniz formula to calculate an approximation of the value of π is an infinite alternating series that adds and subtracts smaller and smaller terms of the form 1/x until it converges to the value of π/4. You can then multiply the result by 4 to obtain an approximation of π.

Here’s the simple formula that calculates π/4:

You can write a simple Python program that starts with pi = 0 and then iterates the following in a for loop, e.g., 10 million times:

  • In the first, third, fifth, … iteration, add 1/i whereas i is a series going over all odd values i=1, 3, 5, …
  • In the second, fourth, sixth, … iteration, subtract 1/i whereas i is a series going over all odd values i=1, 3, 5, …

Here’s how you can efficiently implement the Leibniz formula for Pi in Python code: 🐍

def calculate_pi():
    pi = 0
    n = 10**7
    add = True

    for i in range(1, n, 2):
        if add:
            pi += 1/i
        else:
            pi -= 1/i
        add = not add
    return pi*4
        

print(calculate_pi())
# 3.1415924535897797

Of course, you can get more precision by changing the value of n to a possibly much higher value than 10 million.

If you care about conciseness, here’s a shorter but equally efficient way to calculate the Leibniz formula in Python:

def calculate_pi(n = 10**7):
    pi = 0
    add = True
    for i in range(1, n, 2):
        pi = pi + (1/i if add else -1/i)
        add = not add
    return pi*4
        

print(calculate_pi())
# 3.1415924535897797

The main optimization in this code snippet compared to the previous one is that we used the ternary operator in Python. More here:

🌍 Recommended Tutorial: The Ternary Operator in Python

Thanks for reading this tutorial—if you want to keep learning, feel free to join our free email academy. We have cheat sheets too! 👾

Related Tutorials