Why is $4 \mod{-5} = -1$?

Question

I understand that modulo follows the formula "Quotient × Divisor + Remainder = Dividend," but what I do not understand is how $4 \mod{-5}$ fits as $-1 \times -5 + (-1) = 4$ in the equation. Why isn't it $0 \times (-5) + 4 = 4$, giving us a remainder of $4$ like it does when we do $4 \mod{5}$ in $0 \times 5 + 4 = 4$. What changes when we negate the divisor? How does this make sense?

If the number was greater than five, such as $6$, then negating the $-5$ with a $-1$ like the formula does would make sense, because then you could treat the whole thing as if everything was positive. However I don't understand why $4 \mod (-5)$ is treated the same, why not make the Quotient "$-0$" instead?

Tell us where you are seeing this behavior. Depends on how you define the mod operator - there are a lot of differences in coding about it. It is non-standard. — Thomas Andrews, Sep 08 '22 at 20:44
For some basic information about writing mathematics at this site see, e.g., here, here, here and here. — Another User, Sep 08 '22 at 20:54
See https://torstencurdt.com/tech/posts/modulo-of-negative-numbers/ — Brian Moehring, Sep 08 '22 at 20:55
@DavidK oh ok, I’ll have a look at it and delete the comment. — Seeker, Sep 08 '22 at 21:18
See also here one the relation between mod as an operation vs. congruence relation - which goes to the heart of the matter. — Bill Dubuque, Sep 09 '22 at 01:11

score 1 · Answer 1 · answered Sep 08 '22 at 21:15

Computer languages with integer division generally support a pair of operators:

a DIV b (spelled // in Python 2.2+) is the quotient a/b converted to an integer.
a MOD b (often spelled %) is the remainder of the division.

For the sake of consistency, MOD must be defined as:

$$a \operatorname{mod} b := a - (a \operatorname{div} b) \cdot b$$

When $a$ and $b$ are both positive, everybody generally agrees that $a \operatorname{div} b = \lfloor \frac{a}{b} \rfloor$. The confusion comes when negative numbers are involved.

In C, 4 / -5 is truncated towards zero. So 4 % -5 $= 4 - (0)(-5) = 4$.
In Python, 4 // -5 is floored to $-1$. So 4 % -5 $= 4 - (-1)(-5) = -1$.

Please strive not to post more (dupe) answers to dupes of FAQs, cf. recent site policy announcement here. — Bill Dubuque, Sep 08 '22 at 23:21
@BillDubuque At what point is "recent" no longer the proper descriptor? ;-) — Brian Moehring, Sep 09 '22 at 00:21

Why is $4 \mod{-5} = -1$?

1 Answers1