Modulus? Remainder? #
What is the modulus operation? How is it different from remainder operation?
Euclidean Remainder #
Mathematically, the remainder operation generally refers to the Euclidean division and the subsequent remainder of that operation. In Euclidean division, the remainder will always be zero or positive. Let’s have a simplified example. Consider a dividend a of 27 and a divisor n of 4. 27 then divided by 4 would then give a quotient q of 6 and a remainder r of 3.
This looks straight forward and simple. However things get a bit tricky when one of either divisor or dividend or even both have negative integers (especially with respect to how they are implemented in different programming languages).
Let us extend it to negative numbers, without using too much technical or in this case mathematical jargons:
- positive dividend & negative divisor :
take 27 divided by -4 given euclidean division, this should give q = -6 and r = 3
- negative dividend & positive divisor :
take -27 divided by 4 given euclidean division, this should give q = -7 and r = 1
- negative dividend & negative divisor :
take -27 divided by -4 given euclidean division, this should give q = 7 and r = 1
Modulo operation #
In mathematics, the result of the modulo operation is an equivalence class, and any member of the class may be chosen as representative; however, the usual representative is the least positive residue, the smallest non-negative integer that belongs to that class (i.e., the remainder of the Euclidean division). However, other conventions are possible. Computers and calculators have various ways of storing and representing numbers; thus their definition of the modulo operation depends on the programming language or the underlying hardware.
Modulo is a mathematical jargon that was introduced into mathematics in the book Disquisitiones Arithmeticae by Carl Friedrich Gauss in 1801. Given the integers a, b and n, the expression “a ≡ b (mod n)”, pronounced “a is congruent to b modulo n”, means that a − b is an integer multiple of n, or equivalently, a and b both share the same remainder when divided by n. It is the Latin ablative of modulus, which itself means “a small measure.” The term has gained many meanings over the years—some exact and some imprecise.
As it turns out, modulo operation in fact represents an equivalence class, depending on different definitions implemented in computing. The different definitions that could be used are:
- division with truncation (which drops the decimal part generally)
- division with floor function (which floors down)
- division with ceiling function (which floors up generally)
- euclidean division (which we looked at just now)
All this said, let us now have a look at how these operations are performed in programming languages. Every programming language has its own implementation of the modulus operator, which has us, the users to know how the operation is implemented in the languages we use.
Python for instance implements division with floor function while Ocaml implements division with truncation. I’ll be linking resources at the bottom of this article as to why these specific implementations are chosen. In brief, a lot of it has to do with rounding, floating point arithmetic discrepancy in computing I might write an article explaining these topics later.
While it can be confusing, it is our duty as responsible programmers to understand these differences and know why they are implemented this way, so that we don’t get surprised when we see them. With that said, have a great day.