How python calculate this modulo?
python如何以数学方式计算这个模数？
>>>-1%10
9
The Wikipedia article on the modulo operation provides the following constraint for a % q :
a = nq + r
Substituting a = -1 , q = 10 and r = 9 , we see that n must be equal -1.
Plugging in -1 for n:
-1 % 10 # Python evaluates this as 9
-1 = n * 10 + r
-1 = -1 * 10 + r
9 = r
Testing with another example (again plugging in -1 for n):
-7 % 17 # Python evaluates this as 10
-7 = n * 17 + r
-7 = -17 + r
10 = r
A third example with a positive numerator and negative denominator:
7 % -17 # Python evaluates this as -10
7 = n * (-17) + r
7 = -1 * (-17) + r
7 = 17 + r
-10 = r
It appears that when a and q have different signs, we start with n = -1 and decrement n by 1 until we've found the n closest to zero such that n*q < a . We can test this by trying this out with an a and q such that |a| > |q| :
-100 % 11 # Python evaluates as 10
-100 = n * 11 + r
... -1 # -11 > -100
... -2 # -22 > -100
... -3 ...
... -4 ...
... -5 ...
... -6 ...
... -7 ...
... -8 ...
... -9 # -99 > -100
-100 = -10 * 11 + r # -110 < -100
-100 = -110 + r
10 = r
So while this might not be the algorithm Python actually uses to calculate the modulo, we at least have a useful mental model for reasoning about how a given result was arrived at.
Its is caluclated like this :-
-10 / 10 = -1 ,
hence remainder 9 .
-10 is greatest multiple of 10 which is less than -1 .
It is similar to 9 % 5 , will be greatest number less than dividend should be considers.
5/5 = 1 , hence 4 .
I'm not sure if you're asking the algorithm that python uses, or why the answer comes out this way.
If the latter, imagine that for modulo n you subtract or add n until you get a number between 0 and n-1 inclusive
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