The Python math Module: Everything You Need to Know

Pi

Pi (π) is the ratio of a circle’s circumference (c) to its diameter (d):

π = c/d

This ratio is always the same for any circle.

Pi is an irrational number, which means it can’t be expressed as a simple fraction. Therefore, pi has an infinite number of decimal places, but it can be approximated as 22/7, or 3.141.

You can access pi as follows:

>>> math.pi
3.141592653589793

As you can see, the pi value is given to fifteen decimal places in Python. The number of digits provided depends on the underlying C compiler. Python prints the first fifteen digits by default, and math.pi always returns a float value.

So what are some of the ways that pi can be useful to you? You can calculate the circumference of a circle using 2πr, where r is the radius of the circle:

>>> r = 3
>>> circumference = 2 * math.pi * r
>>> f"Circumference of a Circle = 2 * {math.pi:.4} * {r} = {circumference:.4}"
'Circumference of a Circle = 2 * 3.142 * 3 = 18.85'

You can use math.pi to calculate the circumference of a circle. You can also calculate the area of a circle using the formula πr² as follows:

>>> r = 5
>>> area = math.pi * r * r
>>> f"Area of a Circle = {math.pi:.4} * {r} * {r} = {area:.4}"
'Area of a Circle = 3.142 * 5 * 5 = 78.54'

You can use math.pi to calculate the area and the circumference of a circle. When you are doing mathematical calculations with Python and you come across a formula that uses π, it’s a best practice to use the pi value given by the math module instead of hardcoding the value.

Tau

Tau (τ) is the ratio of a circle’s circumference to its radius. This constant is equal to 2π, or roughly 6.28. Like pi, tau is an irrational number because it’s just pi times two.

Many mathematical expressions use 2π, and using tau instead can help simplify your equations. For example, instead of calculating the circumference of a circle with 2πr, we can substitute tau and use the simpler equation τr.

The use of tau as the circle constant, however, is still under debate. You have the freedom to use either 2π or τ as necessary.

You can use tau as below:

>>> math.tau
6.283185307179586

Like math.pi, math.tau returns fifteen digits and is a float value. You can use tau to calculate the circumference of a circle with τr, where r is the radius, as follows:

>>> r = 3
>>> circumference = math.tau * r
>>> f"Circumference of a Circle = {math.tau:.4} * {r} = {circumference:.4}"
'Circumference of a Circle = 6.283 * 3 = 18.85'

You can use math.tau in place of 2 * math.pi to tidy up equations that include the expression 2π.

Euler’s Number

Euler’s number (e) is a constant that is the base of the natural logarithm, a mathematical function that is commonly used to calculate rates of growth or decay. As with pi and tau, Euler’s number is an irrational number with infinite decimal places. The value of e is often approximated as 2.718.

Euler’s number is an important constant because it has many practical uses, such as calculating population growth over time or determining rates of radioactive decay. You can access Euler’s number from the math module as follows:

>>> math.e
2.718281828459045

As with math.pi and math.tau, the value of math.e is given to fifteen decimal places and is returned as a float value.

Infinity

Infinity can’t be defined by a number. Rather, it’s a mathematical concept representing something that is never-ending or boundless. Infinity can go in either direction, positive or negative.

You can use infinity in algorithms when you want to compare a given value to an absolute maximum or minimum value. The values of positive and negative infinity in Python are as follows:

>>> f"Positive Infinity = {math.inf}"
'Positive Infinity = inf'
>>> f"Negative Infinity = {-math.inf}"
'Negative Infinity = -inf'

Infinity is not a numerical value. Instead, it’s defined as math.inf. Python introduced this constant in version 3.5 as an equivalent to float("inf"):

>>> float("inf") == math.inf
True

Both float("inf") and math.inf represent the concept of infinity, making math.inf greater than any numerical value:

>>> x = 1e308
>>> math.inf > x
True

In the above code, math.inf is greater than the value of x, 10³⁰⁸ (the maximum size of a floating-point number), which is a double precision number.

Similarly, -math.inf is smaller than any value:

>>> y = -1e308
>>> y > -math.inf
True

Negative infinity is smaller than the value of y, which is -10³⁰⁸. No number can be greater than infinity or smaller than negative infinity. That’s why mathematical operations with math.inf don’t change the value of infinity:

>>> math.inf + 1e308
inf
>>> math.inf / 1e308
inf

As you can see, neither addition nor division changes the value of math.inf.

Not a Number (NaN)

Not a number, or NaN, isn’t really a mathematical concept. It originated in the computer science field as a reference to values that are not numeric. A NaN value can be due to invalid inputs, or it can indicate that a variable that should be numerical has been corrupted by text characters or symbols.

It’s always a best practice to check if a value is NaN. If it is, then it could lead to invalid values in your program. Python introduced the NaN constant in version 3.5.

You can observe the value of math.nan below:

>>> math.nan
nan

NaN is not a numerical value. You can see that the value of math.nan is nan, the same value as float("nan").

Find Factorials With Python factorial()

You may have seen mathematical expressions like 7! or 4! before. The exclamation marks don’t mean that the numbers are excited. Rather, “!” is the factorial symbol. Factorials are used in finding permutations or combinations. You can determine the factorial of a number by multiplying all whole numbers from the chosen number down to 1.

The following table shows the factorial values for 4, 6, and 7:

You can see from the table that 4!, or four factorial, gives the value 24 by multiplying the range of whole numbers from 4 to 1. Similarly, 6! and 7! give the values 720 and 5040, respectively.

You can implement a factorial function in Python using one of several tools:

1. for loops
2. Recursive functions
3. math.factorial()

First you are going to look at a factorial implementation using a for loop. This is a relatively straightforward approach:

def fact_loop(num):
    if num < 0:
        return 0
    if num == 0:
        return 1

    factorial = 1
    for i in range(1, num + 1):
        factorial = factorial * i
    return factorial

You can also use a recursive function to find the factorial. This is more complicated but also more elegant than using a for loop. You can implement the recursive function as follows:

def fact_recursion(num):
    if num < 0:
        return 0
    if num == 0:
        return 1

    return num * fact_recursion(num - 1)

The following example illustrates how you can use the for loop and recursive functions:

>>> fact_loop(7)
5040

>>> fact_recursion(7)
5040

Even though their implementations are different, their return values are the same.

However, implementing functions of your own just to get the factorial of a number is time consuming and inefficient. A better method is to use math.factorial(). Here’s how you can find the factorial of a number using math.factorial():

>>> math.factorial(7)
5040

This approach returns the desired output with a minimal amount of code.

factorial() accepts only positive integer values. If you try to input a negative value, then you will get a ValueError:

>>> math.factorial(-5)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: factorial() not defined for negative values

Inputting a negative value will result in a ValueError reading factorial() not defined for negative values.

factorial() doesn’t accept decimal numbers, either. It will give you a ValueError:

>>> math.factorial(4.3)
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
ValueError: factorial() only accepts integral values

Inputting a decimal value results in a ValueError reading factorial() only accepts integral values.

You can compare the execution times for each of the factorial methods using timeit():

>>> import timeit
>>> timeit.timeit("fact_loop(10)", globals=globals())
1.063997201999996

>>> timeit.timeit("fact_recursion(10)", globals=globals())
1.815312818999928

>>> timeit.timeit("math.factorial(10)", setup="import math")
0.10671788000001925

The sample above illustrates the results of timeit() for each of the three factorial methods.

timeit() executes one million loops each time it is run. The following table compares the execution times of the three factorial methods:

As you can see from the execution times, factorial() is faster than the other methods. That’s because of its underlying C implementation. The recursion-based method is the slowest out of the three. Although you might get different timings depending on your CPU, the order of the functions should be the same.

Not only is factorial() faster than the other methods, but it’s also more stable. When you implement your own function, you have to explicitly code for disaster cases such as handling negative or decimal numbers. One mistake in the implementation could lead to bugs. But when using factorial(), you don’t have to worry about disaster cases because the function handles them all. Therefore, it’s a best practice to use factorial() whenever possible.

Find the Ceiling Value With ceil()

math.ceil() will return the smallest integer value that is greater than or equal to the given number. If the number is a positive or negative decimal, then the function will return the next integer value greater than the given value.

For example, an input of 5.43 will return the value 6, and an input of -12.43 will return the value -12. math.ceil() can take positive or negative real numbers as input values and will always return an integer value.

When you input an integer value to ceil(), it will return the same number:

>>> math