Perform Modulo with Negative Values in Python

In Python, modulo operator % always ensures result has same sign as the divisor. The modulo operator (%) gives remainder after division. With negative numbers, different programming languages handle result differently, which often causes confusion.

This rule explains why -5 % 4 gives 3 instead of -1. Understanding this behavior is key when working with negative numbers in modulo operations.

Example:

print(-5 % 4)   
print(5 % -4)  

3
-3

Let’s carefully break this down.

Mathematics Behind Negative Modulo

Suppose we want to calculate -5 % 4. First, divide -5 by 4:

-5 / 4 = -1.25

If we take the floor of this division:

math.floor(-1.25) = -2

Now multiply back:

(-2 * 4) = -8

Then calculate the remainder:

-5 - (-8) = 3

So,

-5 % 4 = 3

Notice that instead of returning -1 (which some other languages would), Python gives 3 because it ensures the remainder has the same sign as the divisor (4, which is positive).

Distributive Property of Modulo

Python internally uses the distributive law of modulo:

(a + b) \bmod n \;=\; \big( (a \bmod n) + (b \bmod n) \big) \bmod n

For example:

-5 % 4 = (-2 * 4 + 3) % 4
= ( -8 + 3 ) % 4
= -5 % 4
= 3

This ensures that multiples of the divisor (like -2*4) reduce cleanly, and the remainder is adjusted to match the divisor’s sign.

More Examples

Let’s see some more calculations manually before checking them in Python.

-3 % 7 = 4
-5 % 2 = 1
-12 % 4 = 0

Python follows the same logic consistently.

Example 1: Basic Negative Modulo 

Here we compare Python’s % result with the distributive property manually.

res1 = -5 % 4
res2 = ((-2 * 4) + 3) % 4

print(res1)
print(res2)

3
3

Both methods give the same result.

Example 2: More Negative Modulo Calculations

Here we calculate modulo for two different negative values to see how Python handles them.

res1 = -3 % 7
res2 = -12 % 4

print(res1)
print(res2)

4
0

Here, -12 % 4 = 0 because -12 is already a perfect multiple of 4.

Using math.fmod() for Alternative Behavior

Python also provides another way to perform modulo using the math.fmod() function.

Note: math.fmod() always returns a float value, even if both inputs are integers.

Let’s see an example:

import math
x = -10
y = 3

result = math.fmod(x, y)
print(result)  

-1.0

Here, the remainder is -1 (negative) because the numerator -10 is negative.

% vs math.fmod() in Python

So depending on whether you want the result to follow the divisor’s sign or the numerator’s sign, you can choose between % and math.fmod().

Real-World Use Cases

1. Clock Arithmetic (Modulo with Positive Result)

A common use case is handling time values, where wrapping around to positive values is important.

hours = -5
print(hours % 12)  

7

2. Signal Processing (fmod with Negative Result)

In scientific applications like signal processing, preserving the sign of the input is often necessary.

import math
phase = -10
print(math.fmod(phase, 360)) 

-10.0