Python & PI

Have you ever wondered about the enigmatic number “pi” in mathematics calculations and problems? The ratio of a circle’s circumference to its diameter is represented by the mathematical constant pi, which has an infinite number of decimal places. In this post, we’ll examine pi and the various Python functions that can be used to get its value.

Math Library

The built-in Python math package is the easiest way to calculate pi. The constant math.pi in the math library supplies the value of pi to 15 decimal places. Using the format function, we can specify the number of decimal places we wish to utilize to calculate pi to the nth decimal place. Here is an illustration showing how to use the math library to calculate pi to the fifth decimal place:

# Import math Library
import math

# Print the value of pi to five digits
print(format(math.pi, f".{5}f"))

NumPy.pi

Python’s NumPy library is well-liked and frequently used for numerical computation, including data analysis and scientific computing. Numpy.pi, which faithfully depicts the number pi, is one of the many valuable functions of the NumPy library.

The proportion of a circle’s circumference to its diameter is known as the NumPy.pi constant. It is a mathematical constant utilized in many mathematics, science, and engineering areas, including physics, geometry, and trigonometry.

The fact that numpy.pi offers a high-precision representation of pi is one benefit of utilizing it. Numpy.pi is defined as a NumPy array with a data type of float64, which offers more accuracy than a conventional floating-point number. This is in contrast to the built-in math.pi constant in Python, which represents pi in floating-point.Here’s an example of how you can use numpy.pi in your Python code:

# Import numpy Library
import numpy

# Print the value of pi to five digits
print(format(numpy.pi, f".{5}f"))

This code will print the value of numpy.pi to the console, which will be a high-precision representation of pi. The numpy.pi constant in the NumPy library is a valuable and convenient way to access an accurate representation of the mathematical constant pi for numerical computing and data analysis in Python.

If you need the value of Pi beyond the limitations of these two libraries, you will need to turn to algorithms written for calculating Pi. Here I will go over four other methods of determining Pi. We start with Chudnovsky’s Algorithm and go into Infinite Series, Arcsine/Inverse Sine Functions, and then the Monte Carlo Method.

Chudnovsky Algorithm

The Chudnovsky algorithm is a fast and accurate method for calculating the value of pi. It is based on the formula for pi as an infinite series, discovered by the brothers Gregorio and David Chudnovsky in the late 20th century.

The Chudnovsky formula for pi can be expressed as follows:

π = C * (sum from k=0 to infinity of ((-1)^k * (6k)! * (13591409 + 545140134k)) / ((3k)! * (k!)^3 * 640320^(3k + 3/2)))

Where C is a constant equal to 10005^(1/2) / 4270934400

The Chudnovsky algorithm uses the series’s fast convergence to calculate pi to high precision. To calculate pi to n digits of precision, the algorithm calculates the first m terms of the series, where m is chosen such that the terms beyond m contribute less than n decimal digits to the final value of pi.

Here’s an example implementation of the Chudnovsky algorithm in Python:

def calc_pi(n):

    C = 426880 * pow(10005, 0.5)
    k = 1
    M = 1
    L = 13591409
    X = 1
    pi = C / M

    while n > 0:
        M = (k**3 - 16 * k) / M
        L += 545140134
        X *= -262537412640768000
        pi += Decimal(C / M * (L / X))
        k += 1
        n -= 1

    return pi

This code calculates the value of pi to n decimal digits of precision. The calc_pi function uses the pow function from the built-in math library to calculate the power of 10005 to the square root. The Decimal type from the decimal library represents the intermediate values of pi with a high degree of precision.

Infinite Series Method

The Infinite Series Method is a method for calculating the pi value because pi can be expressed as an infinite series. There are several infinite series that can be used to calculate pi, but one of the most well-known is the following:

π = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - ...

This series expresses pi as the sum of an infinite sequence of terms, each alternating between positive and negative values. The magnitude of the terms decreases as the sequence continues, so the series converges to the value of pi.

To calculate pi to n decimal places, the Infinite Series Method calculates the first m terms of the series, where m is chosen such that the terms beyond m contribute less than n decimal places to the final value of pi.

Here’s an example implementation of the Infinite Series Method in Python:

def calc_pi(n):
    pi = 0
    for i in range(0, n):
        pi += 4/((2*i) + 1) * (-1)**i

    return pi

This code calculates the value of pi to n decimal places of precision. The calc_pi function calculates the first n terms of the series by looping n times and adding up the terms as it goes. The terms are calculated using the formula 4/((2*i) + 1) * (-1)**i, where i is the index of the term in the series and (-1)**i alternates between positive and negative values.

Arcsine Function/Inverse Sine Function

The Arcsine Function, also known as the Inverse Sine Function, is a method for calculating the pi value based on the relationship between pi and the sine function. The basic idea is to use the sine function to calculate pi by finding the value of pi that makes the sine function equal to 1.

Here’s an example implementation of the Arcsine Function/Inverse Sine Function in Python:

import math

def calc_pi(n):
    pi = 2 * math.asin(1)
    return round(pi, n)

This code calculates the value of pi to n decimal places of precision. The calc_pi function uses the math library to calculate the value of 2 * math.ASIN (1), equal to pi. The round function is then used to round the value of pi to n decimal places of precision.

The Arcsine Function/Inverse Sine Function is a simple and effective method for calculating the value of pi in Python. It takes advantage of the relationship between pi and the sine function and uses the math library to perform the calculation.

Monte Carlo Method

The Monte Carlo Method is a technique for finding the value of pi that simulates points in a two-dimensional plane using random values. The core notion is creating random points inside a square and counting how many of those points fall inside a circle drawn inside the square. By comparing the number of points that fall inside the circle to the total number of points produced, we may calculate the value of pi.

Here’s an example implementation of the Monte Carlo Method in Python:

import random

def calc_pi(n):
    inside = 0
    total = 0

    for i in range(n):
        x = random.uniform(-1, 1)
        y = random.uniform(-1, 1)
        if x**2 + y**2 <= 1:
            inside += 1
        total += 1
    pi = 4 * (inside / total)

    return round(pi, n)

This code calculates the value of pi to n decimal places of precision. The calc_pi function uses the random library to generate n random points in a two-dimensional plane. The x and y coordinates of each point are calculated using the random.uniform function, which generates random numbers between -1 and 1.

The code then checks if each point falls within the circle inscribed inside the square using the equation x**2 + y**2 <= 1. If the point falls within the circle, the inside counter is incremented. The total number of points generated is tracked using the total counter.

Finally, the value of pi is estimated by dividing the number of points within the circle by the total number of points generated and multiplying the result by 4. The round function is then used to round the value of pi to n decimal places of precision.

Conclusion

Which approach works the best? The math.pi, numpy, and pi libraries are your best bet if you need a rapid estimation of PI for up to twenty decimal places. Both libraries respond quickly and won’t make your application run more slowly. On the other hand, any additional functions would be a fine choice if you need to calculate the value of PI beyond the scope of those two libraries. These functions’ limitations state that more than 250,000 repetitions are required to provide a precise approximation of the value of Pi.

The built-in math library, the Chudnovsky algorithm, the infinite series approach, the arcsine function/inverse sine function, and the Monte Carlo method are just a few of the several ways that you can compute the value of pi using Python. The most effective approach for a given issue will rely on the conditions and limitations of that situation because each of these strategies has advantages and disadvantages of its own. Regardless of the approach employed, computing pi is an interesting and worthwhile endeavor that aids in our understanding of this enigmatic and crucial mathematical constant.