Integer division & modulo operation with negative operands in Python

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

It appears as though it takes the approximate floating point (-3.33)? and rounds down either way in integer division but in the modulo operation it does something totally different. It seems like it returns the remainder +/-1 and only switches the sign depending on where the negative operand is. I am utterly confused, even after looking over other answers on this site! I hope someone can clearly explain this too me! The book says hint: recall this magic formula a = (a//b)(b)+(a%b) but that doesn't seem to clear the water for me at all.

-Thanks in advance!

Edit: Those are just my personal assessments of what happens (above), I know, I'm completely off!

Solution 1:

The integer division there is just taking the floor of the number obtained at the end.

10/3  -> floor(3.33)  ->  3
-10/3 -> floor(-3.33) -> -4

(Why it floors)

The modulo operation on the other hand is following the mathematical definition.

Solution 2:

• 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.

Solution 3:

A simple rule: for a % b = c, if c is not zero, then should have the same sign as b.

And apply the magic formula:

10 % -3 = -2 => 10 // -3 = (10 - (-2)) / (-3) = -4

-10 % 3 = 2 => -10 // 3 = (-10 - 2) / 3 = -4

-10 % -3 = -1 => -10 // -3 = (-10 - (-1)) / (-3) = 3

Solution 4:

OK, so I did some digging and I think that the problem isn't Python, but rather the Modulo function. I'm basing this answer off of http://mathforum.org/library/drmath/view/52343.html

10 % 3 Uses the highest multiple of 3 that is LESS THAN 10. In this case, 9. 10 - 9 = 1

-10 % 3 does the same thing. It's still looking for a multiple of 3 that is LESS THAN -10. In this case, -12. (-10) - (-12) = 2