Python Floor Division

Summary: in this tutorial, you’ll learn about Python floor division operator (//) or mod.

Introduction to Python floor division #

Suppose you have a division of two integers:

101 / 4Copy

In this division, 101 is called a numerator (N) and 4 is called a denominator (D).

The integer division 101 / 4 returns 25 with the remainder 1. In other words:

101 / 4 = 25 with remainder 1Copy

Or put it in another way:

101 = 4 * 25 + 1Copy

Python uses two operators // and % that returns the result of the division:

101 // 4 = 25
101 % 4 = 1Copy

The // is called the floor division operator or div. And the % is called the modulo operator or mod.

This tutorial focuses on the floor division operator. You’ll learn about the modulo operator in the following tutorial.

Both floor division and modulo operators satisfy the following equation:

101 = 4 * (101 // 4) + (101 % 4)
101 = 4 * 25         + 1Copy

Generally, if N is the numerator and D is the denominator, then the floor division and modulo operators always satisfy the following equation:

N = D * ( N // D) + (N % D)Copy

The floor division in Python #

To understand the floor division, you first need to understand the floor of a real number.

The floor of a real number is the largest integer less than or equal to the number. In other words:

floor(r) = n, n is an integer and n <= rCopy

For example, the floor of 3.4 is 3 because 3 is the largest integer less than or equal to 3.4. The floor of 3.9 is also 3. And the floor of 3 is 3 obviously:

floor(3.4) = 4
floor(3.9) = 3
floor(3) = 3Copy

For the positive numbers, it would be easy to understand the definition. However, you should pay attention when it comes to negative numbers.

For example, the floor of -3.4 returns -4, not -3 based on the floor definition. Similarly, the floor of -3.9 also returns -4 .

floor(-3.4) = -4
floor(-3.9) = -4
floor(-3) = -3Copy

The floor division can be defined as:

n // d = floor(n/d)Copy

Notice that the floor division of a number is not always the same as truncation. The floor division is the same as truncation only when the numbers are positive.

Python floor division operator examples #

The following example uses the floor division operators with positive and negative integers:

a = 10
b = 3

print(a//b)  # 3

a = -10
b = -3

print(a//b)  # 3

a = 10
b = -3
print(a//b)  # -4

a = -10
b = 3
print(a//b)  # -4Copy

Try it

Output:

3
3
-4
-4Copy

The following table illustrates the floor division of two integers a and b:

Python math.floor() function #

The floor() function of the math module returns the floor division of two integers. For example:

from math import floor

a = 10
b = 3

print(a//b)  # 3
print(floor(a/b))  # 3Copy

Try it

Output:

3
3Copy

The floor() function returns the same result as the floor division operator (//). It’s also true for the negative numbers:

from math import floor

a = 10
b = -3

print(a//b)  # -4
print(floor(a/b))  # -4Copy

Try it

Output:

-4
-4Copy

Summary #

• Python uses // as the floor division operator and % as the modulo operator.
• If the numerator is N and the denominator D, then this equation N = D * ( N // D) + (N % D) is always satisfied.
• Use floor division operator // or the floor() function of the math module to get the floor division of two integers.