I know this has been explained many times in the past, but interestingly this bit me in the back a few days ago while coding, and, the same as homogeneous coordinates we will find plenty of different explanations depending on whom we ask to.
Mathematically speaking, we define a modulus or \(a \mod n\) to be the remainder \(r\) when \(a\) is divided by \(n\), in other words:
This was defined by Gauss in 1801, thought the Chinese and Greeks probably knew about it thousand years before. One important thing about this is that \(0 \le r \lt n\).
This last part is important, because it defines a modulus as a positive number, for example:
Why the second case is 4? You could say the result is -3 (because the sign), but because we already establish that \(0 \le r \lt n\) the number cannot be negative, so we say \(7-3 = 4\). We usually say -129 is congruent with 4 mod 7 or with the notation \(-129 \equiv 4 \pmod 7\) which means both \(-129\) and \(4\) have the same remainders when divided by 7.
Notice, while we use the word reminder, the modulus is not the remainder, \(-3\) is the remainder while \(4\) is the modulus.
And now computers…
Almost every programming language I had seen have the modulus operator, most of the time is expressed by the percentage symbol, so
35 % 12 is the same as \(35 \mod 12\). What about the operation against a negative number?
-129 % 7 // Returns -3, it should return 4
% is the remainder operator while
modulus is the modulus operator.
The Wikipedia page has a good list and sort of explanation why this happens, but I can see in the same way as homogeneous coordinates everybody says both are right conceptions, my math books disagree with them though.
We could assume every programming language uses the
% as the remainder instead of the modulus? oh my friend, you will get a nice surprise when using Python and R, both operators (
%% in R) are true modulus operators, so
-129 % 7 will return 4, and yes, they are named modulus as well.
I highly recommend the Wikipedia page, at least for the list of implementations in different programming languages (notice some languages like C/C++ the sign is implementation specific, so it could be different depending on the compiler). Rob Conery wrote a blog post some time ago about the same topic. If you want to go really down in the rabbit hole I recommend the section 1.6 Theory of congruences from the book Number Theory for Computing by Song Y. Yang, its explanation is massive and very mathematical but it is worth reading if you find a copy of the book in the library or something like that.