Different kinds of integer division

Suppose we wish to define an integer division operation, that is, a division operation which accepts two integers and returns another integer. This blog post will address two issues:

1. how to decide whether a particular operation can sensibly be described as ‘integer division’, and
2. why there are a few different options, and what the differences between them are.

Before we start, some vocabulary. If we write $a / b$, then $a$ is called the dividend, and $b$ is called the divisor.

What is a sensible notion of ‘integer division’?

First, I want to argue that when we define a form of integer division, we really ought to consider the result to be a pair of integers; the first is called the quotient, and it represents the number of times the divisor ‘goes into’ the dividend, and the second is called the remainder, and it represents what is left over after taking away that multiple of the divisor.

We can formalise this: suppose we have some form of integer division, and we use it to divide $a$ by $b$, obtaining a quotient $q$ and a remainder $r$. The first thing we need to do in order to ensure that we have an operation which can sensibly be described as ‘division’ is to require that $a=qb+r$. This requirement sort of corresponds to saying that “the number of times $b$ goes into $a$ is $q$, and the remainder is $r$”. For example, $5$ goes into $13$ twice with remainder $3$, so if we choose $a=13$ and $b=5$, we get $q=2$ and $r=3$.

After adding this requirement, $q$ and $r$ become closely related. In particular, once we’ve chosen what we want one to be, there is only one choice for the other.

Note that if we fix $a$ and $b$, and we have a pair of integers $q$ and $r$ such that $a=qb+r$, then we can always find a different choice for $q$ and $r$ which also satisfy our requirement. For example, we can perform the substitutions $q \to q-1$ and $r \to r+b$, since $(q-1)b + (r+b) = qb - b + r + b = qb + r = a$. Similarly, we can substitute $q \to q+1$, and $r \to r-b$. Therefore, when we are defining an integer division operation, since we have many different options for $q$ and $r$, we must decide which one we are going to choose. In fact, at this point, we have an infinite number of choices for $q$ and $r$ given $a$ and $b$; this suggests that we need to narrow down our options a little more by adding more constraints.

There is another additional constraint we can add now to ensure that we have something that can sensibly be described as ‘integer division’: we can require that $q$ is close to the result of exact division (which usually will not be an integer). More specifically, we can require that the difference between $q$ and the exact result $a/b$ is less than $1$. We can express this in symbols: $|q - a/b| < 1$.

One consequence of this requirement is that if $b$ goes exactly into $a$, we must take $q$ to be the exact result of the division, and $r$ to be $0$. For example, if we have $a=-8$ and $b=2$, then the result of exact division is $-4$, so in this case we must take $q=-4$ and $r=0$.

There is one more constraint we can add, due to the fact that pretty much everyone agrees on what the result of integer division should be when $a$ and $b$ are both positive. In this case, we should take $r$ such that $0 \leq r < b$, and we should take $q$ to be the largest integer satisfying $qb \leq a$. Equivalently, we obtain $q$ by taking the exact result of dividing $a$ by $b$ and rounding down to the nearest integer. Going back to one of our previous examples, if $a=13$ and $b=5$, the largest $q$ we can take is $2$, because $5 \times 2=10$, and $10$ is the largest multiple of $5$ which is less than $13$.

So we end up with the following definition of a ‘sensible’ integer division operation. It is a function which takes a dividend $a$ and a nonzero divisor $b$ as inputs, and returns a quotient $q$ and a remainder $r$, subject to the following constraints:

1. $a = qb + r$,
2. $\lvert q - a/b \rvert < 1$,
3. If $a,b > 0$, then $q = \max \{ t \in \mathbb{Z} : bt \leq a \}$.

So what are the options?

As we have seen, there is only one option which satisfies all three of these constraints if both of $a$ and $b$ are nonnegative, or if $b$ goes into $a$ exactly. However, in the case where either (or both) of $a$ or $b$ are negative AND when $b$ does not go into $a$ exactly, there are a few different options to choose from.

One of the most common options taken by programming languages is called “truncating” division, because we obtain $q$ by taking the exact result of division and “truncating” (rounding towards zero). This can be implemented in JavaScript as follows:

// Truncating division
function tdiv(x,y) {
return Math.trunc(x / y);
}
function tmod(x,y) {
return x % y;
}


To give an example, suppose we have $a=-2$ and $b=3$, and we want to divide $a$ by $b$. The exact result of division is $-2/3$; rounding this towards zero, we obtain $q=0$. Then, we must choose $r=-2$ so that the first constraint is satisfied.

The other most common option is called “flooring” or “Knuthian” division. It works by taking the exact result of division and then rounding towards negative infinity. In JavaScript:

// Flooring/Knuthian division
function fdiv(x,y) {
return Math.floor(x / y);
}
function fmod(x,y) {
return ((x % y) + y) % y;
}


Notice that, if the exact result of division is positive, flooring division is identical to truncating division. However, if the exact result is negative, then the results of flooring and truncating division will be slightly different. For example, consider $a=-2$ and $b=3$ again. We saw that truncating division produces $q=0$ in this case, but flooring division rounds towards negative infinity and so we obtain $q=-1$. Now we just need to find what remainder flooring division gives us in this case. To do this, we can use the requirement from earlier; we need to choose $r$ so that the equation $-2 = (-1 \times 3) + r$ is satisfied, and of course the only such choice is $r=1$.

Now consider a similar example, where we wish to perform flooring division with $a=2$ and $b=-3$. In this case, we again obtain $q=-1$. Since $a=2$, we need to choose $r=-1$ to satisfy our requirement. Notice that flooring division gave us a positive $r$ before, but this time, it gave us a negative $r$.

In fact, it turns out that, with truncating division, the remainder $r$ always has the same sign as the dividend $a$, and that with flooring division, the remainder $r$ always has the same sign as the divisor $b$.

It is arguably a severe drawback of truncating division that the remainder takes the sign of the dividend; as any number theorist will tell you, when performing division with a dividend $a$ and a divisor $b$, it makes the most sense to consider only $|b|$ different possibilities for the remainder. For example, if we are dividing by $3$, there are precisely $3$ possibilities for how the remainder can turn out: either the dividend goes exactly and there is no remainder, or there is a remainder of $1$, or there is a remainder of $2$.

However, with truncating division, the fact that the remainder can be either positive or negative means that we actually have $2|b| - 1$ possibilities.

Flooring division improves on this situation in that if we fix a divisor $b$, there are $|b|$ possibilities for the remainder $r$, as we wanted. However, as we have seen, we don’t know whether $r$ will be positive or negative without knowing what $b$ is.

It is arguably more useful to have a form of division in which the remainder is always nonnegative. There is in fact a form of division which satisfies this, and it is called Euclidean division:

// Euclidean division
function ediv(x,y) {
return Math.sign(y) * Math.floor(x / Math.abs(y));
}
function emod(x,y) {
var yy = Math.abs(y);
return ((x % yy) + yy) % yy;
}


The rounding behaviour of Euclidean division is a little more complex, in order to accommodate our additional requirement that the remainder should always be nonnegative. With Euclidean division, the type of rounding depends on the sign of the divisor. If the divisor is positive, Euclidean division rounds towards negative infinity. If the divisor is negative, Euclidean division rounds towards positive infinity.

If this seems a bit overly-complicated, there is another way of understanding these different kinds of division, by considering the signs of the dividend and the divisor.

• If both dividend and divisor are positive, then all three definitions agree.
• If the dividend is positive and the divisor is negative, then the truncating and Euclidean definitions agree.
• If the dividend is negative and the divisor is positive, then the flooring and Euclidean definitions agree.
• If both dividend and divisor are negative, then the truncating and flooring definitions agree.

You can try it out using the table below.

 Dividend Divisor Exact result Result with truncating division Result with flooring division Result with Euclidean division

Comparison of some programming languages

Java’s / and % operators, when applied to integers, implement truncating division. The Math class provides flooring division via Math.floorDiv and Math.floorMod.

Ruby’s / and % operators both implement flooring division, as do Python’s // and % operators.

Haskell’s div and mod implement flooring division, whereas quot and rem implement truncating division.

JavaScript’s % operator implements remainder with respect to truncating division, but there isn’t a truncating integer division operator to go with it, probably because JavaScript doesn’t really have integers (yet). However, you can implement one yourself quite easily via Math.trunc(x/y).

The main reason I ended up researching and writing this is because we are considering changing PureScript’s behaviour in an upcoming release. Currently, the functions div and mod, when specialised to the builtin Int type, implement truncating division, but in a future release they may implement Euclidean division.

Bibliography

Much of the insight in this post comes from the paper

“The Euclidean definition of the functions div and mod”, Raymond T. Boute, ACM Transactions on Programming Languages and Systems, Vol 14, No. 2, April 1992, pages 127-144.

In particular, if you’re after a more detailed discussion of which form of integer division is the ‘best’, this paper is a good place to start.

Wikipedia is also a good resource; see https://en.wikipedia.org/wiki/Modulo_operation.

Guido van Rossum wrote in 2010 about why integer division in Python floors. His post also offers a possible explanation for why truncating division is common, despite it not really having many nice mathematical properties.