Euclidean division

History

Although "Euclidean division" is named after Euclid, it seems that he did not know the existence and uniqueness theorem, and that the only computation method that he knew was the division by repeated subtraction.

Before the invention of Hindu–Arabic numeral system, which was introduced in Europe during the 13th century by Fibonacci, division was extremely difficult, and only the best mathematicians were able to do it . Presently, most division algorithms, including long division, are based on this numeral system or its variants, such as binary numerals. A notable exception is Newton–Raphson division, which is independent from any numeral system.

The term "Euclidean division" was introduced during the 20th century as a shorthand for "division of Euclidean rings". It has been rapidly adopted by mathematicians for distinguishing this division from the other kinds of division of numbers.

Intuitive example

Suppose that a pie has 9 slices and they are to be divided evenly among 4 people. Using Euclidean division, 9 divided by 4 is 2 with remainder 1. In other words, each person receives 2 slices of pie, and there is 1 slice left over.

This can be confirmed using multiplication, the inverse of division: if each of the 4 people received 2 slices, then 4 × 2 = 8 slices were given out in total. Adding the 1 slice remaining, the result is 9 slices. In summary: 9 = 4 × 2 + 1.

In general, if the number of slices is denoted ${\displaystyle a}$ and the number of people is denoted ${\displaystyle b}$, then one can divide the pie evenly among the people such that each person receives ${\displaystyle q}$ slices (the quotient), with some number of slices ${\displaystyle r<b}$ being the leftover (the remainder). In which case, the equation ${\displaystyle a=bq+r}$ holds.

If 9 slices were divided among 3 people instead of 4, then each would receive 3 and no slice would be left over, which means that the remainder would be zero, leading to the conclusion that 3 evenly divides 9, or that 3 divides 9.

Euclidean division can also be extended to negative dividend (or negative divisor) using the same formula; for example −9 = 4 × (−3) + 3, which means that −9 divided by 4 is −3 with remainder 3.

Examples

• If a = 7 and b = 3, then q = 2 and r = 1, since 7 = 3 × 2 + 1.
• If a = 7 and b = −3, then q = −2 and r = 1, since 7 = −3 × (−2) + 1.
• If a = −7 and b = 3, then q = −3 and r = 2, since −7 = 3 × (−3) + 2.
• If a = −7 and b = −3, then q = 3 and r = 2, since −7 = −3 × 3 + 2.

Proof

The following proof of the division theorem relies on the fact that a decreasing sequence of non-negative integers stops eventually. It is separated into two parts: one for existence and another for uniqueness of ${\displaystyle q}$ and ${\displaystyle r}$. Other proofs use the well-ordering principle (i.e., the assertion that every non-empty set of non-negative integers has a smallest element) to make the reasoning simpler, but have the disadvantage of not providing directly an algorithm for solving the division (see § Effectiveness for more).

Existence

For proving the existence of Euclidean division, one can suppose ${\displaystyle b>0,}$ since, if ${\displaystyle b<0,}$ the equality ${\displaystyle a=bq+r}$ can be rewritten ${\displaystyle a=(-b)(-q)+r.}$ So, if the latter equality is a Euclidean division with ${\displaystyle -b>0,}$ the former is also a Euclidean division.

Given ${\displaystyle b>0}$ and ${\displaystyle a,}$ there are integers ${\displaystyle q_1}$ and ${\displaystyle r_1\ge 0}$ such that ${\displaystyle a=b{q}_1+r_1;}$ for example, ${\displaystyle q_1=0}$ and ${\displaystyle r_1=a}$ if ${\displaystyle a\ge 0,}$ and otherwise ${\displaystyle q_1=a}$ and ${\displaystyle r_1=a-ab.}$

Let ${\displaystyle q}$ and ${\displaystyle r}$ be such a pair of numbers for which ${\displaystyle r}$ is nonnegative and minimal. If ${\displaystyle r<b,}$ we have Euclidean division. Thus, we have to prove that, if ${\displaystyle r\ge b,}$ then ${\displaystyle r}$ is not minimal. Indeed, if ${\displaystyle r\ge b,}$ one has ${\displaystyle a=b(q+1)+(r-b),}$ with ${\displaystyle 0\le r-b<r,}$ and ${\displaystyle r}$ is not minimal

This proves the existence in all cases. This provides also an algorithm for computing the quotient and the remainder, by starting from ${\displaystyle q=0}$ (if ${\displaystyle a\ge 0}$) and adding ${\displaystyle 1}$ to it until ${\displaystyle a-bq<b.}$ However, this algorithm is not efficient, since its number of steps is of the order of ${\displaystyle a/b}$

Uniqueness

The pair of integers ${\displaystyle r}$ and ${\displaystyle q}$ such that ${\displaystyle a=bq+r}$ is unique, in the sense that there can be no other pair of integers that satisfy the same condition in the Euclidean division theorem. In other words, if we have another division of ${\displaystyle a}$ by ${\displaystyle b}$, say ${\displaystyle a=b{q}^{\prime }+r^{\prime }}$ with ${\displaystyle 0\le r^{\prime }<|b|}$, then we must have that

${\displaystyle q^{\prime }=q\text{ and }{r}^{\prime }=r}$.

To prove this statement, we first start with the assumptions that

${\displaystyle 0\le r<|b|}$

${\displaystyle 0\le r^{\prime }<|b|}$

${\displaystyle a=bq+r}$

${\displaystyle a=b{q}^{\prime }+r^{\prime }}$

Subtracting the two equations yields

${\displaystyle b(q-q^{\prime })=r^{\prime }-r}$.

So ${\displaystyle b}$ is a divisor of ${\displaystyle r^{\prime }-r}$. As

${\displaystyle |r^{\prime }-r|<|b|}$

by the above inequalities, one gets

${\displaystyle r^{\prime }-r=0}$,

and

${\displaystyle b(q-q^{\prime })=0}$.

Since ${\displaystyle b\ne 0}$, we get that ${\displaystyle r=r^{\prime }}$ and ${\displaystyle q=q^{\prime }}$, which proves the uniqueness part of the Euclidean division theorem.

Effectiveness

In general, an existence proof does not provide an algorithm for computing the existing quotient and remainder, but the above proof does immediately provide an algorithm (see Division by repeated subtraction), even though it is not a very efficient one as it requires as many steps as the size of the quotient. This is related to the fact that it uses only additions, subtractions and comparisons of integers, without involving multiplication, nor any particular representation of the integers such as decimal notation.

In terms of decimal notation, long division provides a much more efficient algorithm for solving Euclidean divisions. Its generalization to binary and hexadecimal notation provides further flexibility and possibility for computer implementation. However, for large inputs, algorithms that reduce division to multiplication, such as Newton–Raphson division, are usually preferred, because they only need a time which is proportional to the time of the multiplication needed to verify the result—independently of the multiplication algorithm which is used (for more, see Fast division methods).

Variants

The Euclidean division admits a number of variants, some of which are listed below.

Other intervals for the remainder

In Euclidean division with d as divisor, the remainder is supposed to belong to the interval [0, d) of length |d|. Any other interval of the same length may be used. More precisely, given integers ${\displaystyle m}$, ${\displaystyle a}$, ${\displaystyle d}$ with ${\displaystyle m>0}$, there exist unique integers ${\displaystyle q}$ and ${\displaystyle r}$ with ${\displaystyle d\le r<m+d}$ such that ${\displaystyle a=mq+r}$.

In particular, if ${\displaystyle d=-\lfloor \frac{m}{2}\rfloor }$ then ${\displaystyle -\lfloor \frac{m}{2}\rfloor \le r<m-\lfloor \frac{m}{2}\rfloor }$ . This division is called the centered division, and its remainder ${\displaystyle r}$ is called the centered remainder or the least absolute remainder.

This is used for approximating real numbers: Euclidean division defines truncation, and centered division defines rounding.

Montgomery division

Given integers ${\displaystyle a}$, ${\displaystyle m}$ and ${\displaystyle R,}$ with ${\displaystyle m>0}$ and ${\displaystyle gcd(R,m)=1,}$ let ${\displaystyle R^{-1}}$ be the modular multiplicative inverse of ${\displaystyle R}$ (i.e., ${\displaystyle 0<R^{-1}<m}$ with ${\displaystyle R^{-1}R-1}$ being a multiple of ${\displaystyle m}$), then there exist unique integers ${\displaystyle q}$ and ${\displaystyle r}$ with ${\displaystyle 0\le r<m}$ such that ${\displaystyle a=mq+R^{-1}\cdot r}$. This result generalizes Hensel's odd division (1900).

The value ${\displaystyle r}$ is the N-residue defined in Montgomery reduction.

In Euclidean domains

Euclidean domains (also known as Euclidean rings) are defined as integral domains which support the following generalization of Euclidean division:

Given an element ${\displaystyle a}$ and a non-zero element ${\displaystyle b}$ in a Euclidean domain ${\displaystyle R}$ equipped with a Euclidean function ${\displaystyle d}$ (also known as a Euclidean valuation or degree function), there exist ${\displaystyle q}$ and ${\displaystyle r}$ in ${\displaystyle R}$ such that ${\displaystyle a=bq+r}$ and either ${\displaystyle r=0}$ or ${\displaystyle d(r)<d(b)}$.

Uniqueness of ${\displaystyle q}$ and ${\displaystyle r}$ is not required. It occurs only in exceptional cases, typically for univariate polynomials, and for integers, if the further condition ${\displaystyle r\ge 0}$ is added.

Examples of Euclidean domains include fields, polynomial rings in one variable over a field, and the Gaussian integers. The Euclidean division of polynomials has been the object of specific developments.