Integer division & modulo operation with negative operands in Python

Question

Questions arise when I type in these expressions to Python 3.3.0

-10 // 3  # -4
-10 % 3   #  2
10 // -3  # -4
10 % -3   # -2
-10 // -3 #  3

Answer 1

• Magic formula: a = (a // b) * b + (a % b)
• a: -10
• b: 3
• a // b: -4

• a % b: 2

  Substitute in magic formula: -10 = -4 * 3 + 2 = -12 + 2 = -10

• a: 10

• b: -3
• a // b: -4

• a % b: -2

  In magic formula: 10 = -4 * -3 - 2 = 12 - 2 = 10

So the magic formula seems to be correct.

If you define a // b as floor(a / b) (which it is), a % b should be a - floor(a / b) * b. Let's see:

• a: -10
• b: 3
• a % b = a - floor(a / b) * b = -10 - floor(-3.33) * 3 = -10 + 4 * 3 = 2

 

The fact that a // b is always floored is pretty easy to remember (please read Cthulhu's first link, it's an explanation by the creator of Python). For negative a in a % b.. try to imagine a table of numbers that starts at 0 and has b columns:

b = 3:

0  1  2
3  4  5
6  7  8
9 10 11
...

If a is the number in a cell, a % b would be the column number:

a         a % b
_______________
0  1  2   0 1 2
3  4  5   0 1 2
6  7  8   0 1 2
9 10 11   0 1 2

Now extend the table back in the negatives:

   a          a % b
 __________________
-12 -11 -10   0 1 2
 -9  -8  -7   0 1 2
 -6  -5  -4   0 1 2
 -3  -2  -1   0 1 2
  0   1   2   0 1 2
  3   4   5   0 1 2
  6   7   8   0 1 2
  9  10  11   0 1 2

-10 % 3 would be 2. Negative a in a % b would come up in these sorts of context. a % b with negative b doesn't come up much.