Question

I am curious about these languages (Java, C …) which ignore mathematical definition of modulus operation.

What is the point of returning negative values in a module operation (that, by definition, should allways return a positive number)?

Answer 1

I doubt that the remainder operator was deliberately designed to have those semantics, which I agree aren’t very useful. (Would you ever write a calendar program that shows the weekdays Sunday, Anti-Saturday, Anti-Friday, …, Anti-Monday for dates before the epoch?)

Rather, negative remainders are a side effect of the way integer division is defined.

A rem B := A - (A div B) * B

If A div B is defined as trunc(A/B), you get C’s % operator. If A div B is defined as floor(A/B), you get Python’s % operator. Other definitions are possible.

So, the real question is:

Why do C++, Java, C#, etc. use truncating integer division?

Because that’s the way that C does it.

Why does C use truncating division?

Originally, C didn’t specify how / should handle negative numbers. It left it up to the hardware.

In practice, every significant C implementation used truncating division, so in 1999 these semantics were formally made a part of the C standard.

Why does hardware use truncating division?

Because it’s easier (=cheaper) to implement in terms of unsigned division. You just calculate abs(A) div abs(B) and flip the sign if (A < 0) xor (B < 0).

Floored division has the additional step of subtracting 1 from the quotient if the remainder is nonzero.