The modulo operation on negative numbers in Python

In python, modulo operator works like this.

>>> mod = n - math.floor(n/base) * base

so the result is (for your case):

mod = -5 - floor(-1.25) * 4
mod = -5 - (-2*4)
mod = 3

whereas other languages such as C, JAVA, JavaScript use truncation instead of floor.

>>> mod = n - int(n/base) * base

which results in:

mod = -5 - int(-1.25) * 4
mod = -5 - (-1*4)
mod = -1

If you need more information about rounding in python, read this.

Other answers, especially the selected one have clearly answered this question quite well. But I would like to present a graphical approach that might be easier to understand as well, along with python code to perform normal mathematical modulo in python.

Python Modulo for Dummies

Modulo function is a directional function that describes how much we have to move further or behind after the mathematical jumps that we take during division over our X-axis of infinite numbers. So let's say you were doing 7%3

So in forward direction, your answer would be +1, but in backward direction-

your answer would be -2. Both of which are correct mathematically.

Similarly, you would have 2 moduli for negative numbers as well. For eg: -7%3, can result both in -1 or +2 as shown -

Forward direction

Backward direction

In mathematics, we choose inward jumps, i.e. forward direction for a positive number and backward direction for negative numbers.

But in Python, we have a forward direction for all positive modulo operations. Hence, your confusion -

>>> -5 % 4 
3

>>> 5 % 4
1

Here is the python code for inward jump type modulo in python:

def newMod(a,b):
    res = a%b
    return res if not res else res-b if a<0 else res

which would give -

>>> newMod(-5,4)
-1

>>> newMod(5,4)
1

Many people would oppose the inward jump method, but my personal opinion is, that this one is better!!

It's also worth to mention that also the division in python is different from C: Consider

>>> x = -10
>>> y = 37

in C you expect the result

0

what is x/y in python?

>>> print x/y
-1

and % is modulo - not the remainder! While x%y in C yields

-10

python yields.

>>> print x%y
27

You can get both as in C

The division:

>>> from math import trunc
>>> d = trunc(float(x)/y)
>>> print d
0

And the remainder (using the division from above):

>>> r = x - d*y
>>> print r
-10

This calculation is maybe not the fastest but it's working for any sign combinations of x and y to achieve the same results as in C plus it avoids conditional statements.

You can use:

result = numpy.fmod(x,y)

it will keep the sign , see numpy fmod() documentation.

@Deekshant has explained it well using visualisation. Another way to understand %(modulo) is ask a simple question.

What is nearest smaller number to dividend that can be divisible by divisor on X-axis ?

Let's have a look at few examples.

5 % 3 

5 is Dividend, 3 is divisor. If you ask above question 3 is nearest smallest number that is divisible by divisor. ans would be 5 - 3 = 2. For positive Dividend, nearest smallest number would be always right side of dividend.

-5 % 3

Nearest smallest number that is divisible by 3 is -6 so ans would be -5 - (-6) = 1

-5 %4

Nearest smallest number that is divisible by 4 is -8 so ans would be -5 - (-8) = 3

Python answers every modulo expression with this method. Hope you can understand next how expression would be going to execute.