In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit numbers that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps.
As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps. The abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit.
Contents
History
Related algorithms have existed since the 12th century.
Al-Samawal al-Maghribi (1125–1174) performed calculations with decimal numbers that essentially require long division, leading to infinite decimal results, but without formalizing the algorithm.
Caldrini (1491) is the earliest printed example of long division, known as the Danda method in medieval Italy, and it became more practical with the introduction of decimal notation for fractions by Pitiscus (1608).
The specific algorithm in modern use was introduced by Henry Briggs c. 1600.
Education
Inexpensive calculators and computers have become the most common tools for performing division in educational and professional contexts worldwide, reducing reliance on traditional paper-and-pencil techniques. Internally, these devices implement various division algorithms, many of which rely on iterative approximations and multiplication to improve computational efficiency.
Educational approaches to teaching division vary across countries and regions, reflecting differing curricular priorities. In North America, long division has been de-emphasized or, in some cases, removed from portions of the curriculum as part of reform mathematics, which emphasizes conceptual understanding and the use of technology.
In contrast, many education systems in Europe and Asia continue to emphasize mastery of standard algorithms, including long division, as a foundational arithmetic skill. For example, curricula in countries such as Japan and Germany typically introduce and reinforce long division during primary education, often alongside mental arithmetic strategies and problem-solving techniques.
International assessments such as the Trends in International Mathematics and Science Study (TIMSS) highlight these differences, showing variation in how procedural fluency and conceptual understanding are balanced across educational systems.
These differing approaches reflect broader educational philosophies regarding the balance between procedural fluency, conceptual understanding, and the role of technology in mathematics education.
Method
In English-speaking countries, long division does not use the division slash ⟨∕⟩ or division sign ⟨÷⟩ symbols but instead constructs a tableau. The divisor is separated from the dividend by a right parenthesis ⟨)⟩ or vertical bar ⟨|⟩; the dividend is separated from the quotient by a vinculum (i.e., an overbar). The combination of these two symbols is sometimes known as a long division symbol, division bracket, or even a bus stop. It developed in the 18th century from an earlier single-line notation separating the dividend from the quotient by a left parenthesis. This symbol is available in Unicode as U+27CC ⟌ LONG DIVISION.
The process is begun by dividing the left-most digit of the dividend by the divisor. The quotient (rounded down to an integer) becomes the first digit of the result, and the remainder is calculated (this step is notated as a subtraction). This remainder carries forward when the process is repeated on the following digit of the dividend (notated as 'bringing down' the next digit to the remainder). When all digits have been processed and no remainder is left, the process is complete.
An example is shown below, representing the division of 500 by 4 (with a result of 125).
125 (Explanations)
4)500
4 ( 4 × 1 = 4)
10 ( 5 - 4 = 1)
8 ( 4 × 2 = 8)
20 (10 - 8 = 2)
20 ( 4 × 5 = 20)
0 (20 - 20 = 0)
A more detailed breakdown of the steps goes as follows:
Find the shortest sequence of digits starting from the left end of the dividend, 500, that the divisor 4 goes into at least once. In this case, this is simply the first digit, 5. The largest number that the divisor 4 can be multiplied by without exceeding 5 is 1, so the digit 1 is put above the 5 to start constructing the quotient.
Basic procedure for long division of n ÷ m
Find the location of all decimal points in the dividend n and divisor m.
If necessary, simplify the long division problem by moving the decimals of the divisor and dividend by the same number of decimal places, to the right (or to the left), so that the decimal of the divisor is to the right of the last digit.
When doing long division, keep the numbers lined up straight from top to bottom under the tableau.
After each step, be sure the remainder for that step is less than the divisor. If it is not, there are three possible problems: the multiplication is wrong, the subtraction is wrong, or a greater quotient is needed.
In the end, the remainder, r, is added to the growing quotient as a fraction, r⁄m.
Invariant property and correctness
The basic presentation of the steps of the process (above) focuses on what steps are to be performed,
rather than the properties of those steps that ensure the result will be correct
(specifically, that q × m + r = n, where q is the final quotient and r the final remainder).
A slight variation of presentation requires more writing,
and requires that we change, rather than just update, digits of the quotient,
but can shed more light on why these steps actually produce the right answer
by allowing evaluation of q × m + r at intermediate points in the process.
This illustrates the key property used in the derivation of the algorithm
(below).
Specifically, we amend the above basic procedure so that
we fill the space after the digits of the quotient under construction with 0's, to at least the 1's place,
and include those 0's in the numbers we write below the division bracket.
This lets us maintain an invariant relation at every step:
q × m + r = n, where q is the partially-constructed quotient (above the division bracket)
Example with multi-digit divisor
A divisor of any number of digits can be used. In this example, 1260257 is to be divided by 37. First the problem is set up as follows:
37)1260257
Digits of the number 1260257 are taken until a number greater than or equal to 37 occurs. So 1 and 12 are less than 37, but 126 is greater. Next, the greatest multiple of 37 less than or equal to 126 is computed. So 3 × 37 = 111 < 126, but 4 × 37 > 126. The multiple 111 is written underneath the 126 and the 3 is written on the top where the solution will appear:
3
37)1260257
111
Note carefully which place-value column these digits are written into. The 3 in the quotient goes in the same column (ten-thousands place) as the 6 in the dividend 1260257, which is the same column as the last digit of 111.
The 111 is then subtracted from the line above, ignoring all digits to the right:
3
37)1260257
111
15
Now the digit from the next smaller place value of the dividend is copied down and appended to the result 15:
3
37)1260257
111
150
The process repeats: the greatest multiple of 37 less than or equal to 150 is subtracted. This is 148 = 4 × 37, so a 4 is added to the top as the next quotient digit. Then the result of the subtraction is extended by another digit taken from the dividend:
Mixed mode long division
For non-decimal currencies (such as the British £sd system before 1971) and measures (such as avoirdupois) mixed mode division must be used. Consider dividing 50 miles 600 yards into 37 pieces:
mi - yd - ft - in
1 - 634 1 9 r. 15"
37) 50 - 600 - 0 - 0
37 22880 66 348
13 23480 66 348
1760 222 37 333
22880 128 29 15
170 ===
148
22
66
==
Each of the four columns is worked in turn. Starting with the miles: 50/37 = 1 remainder 13. No further division is
possible, so perform a long multiplication by 1,760 to convert miles to yards, the result is 22,880 yards. Carry this to the top of the yards column and add it to the 600 yards in the dividend giving 23,480. Long division of 23,480 / 37 now proceeds as normal yielding 634 with remainder 22. The remainder is multiplied by 3 to get feet and carried up to the feet column. Long division of the feet gives 1 remainder 29 which is then multiplied by twelve to get 348 inches. Long division continues with the final remainder of 15 inches being shown on the result line.
Interpretation of decimal results
When the quotient is not an integer and the division process is extended beyond the decimal point, one of two things can happen:
The process can terminate, which means that a remainder of 0 is reached; or
A remainder could be reached that is identical to a previous remainder that occurred after the decimal points were written. In the latter case, continuing the process would be pointless, because from that point onward the same sequence of digits would appear in the quotient over and over. So a bar is drawn over the repeating sequence to indicate that it repeats forever (i.e., every rational number is either a terminating or repeating decimal).
Notation in non-English-speaking countries
China, Japan, Korea use the same notation as English-speaking nations including India. Elsewhere, the same general principles are used, but the figures are often arranged differently.
Latin America
In Latin America (except Argentina, Bolivia, Mexico, Colombia, Paraguay, Venezuela, Uruguay and Brazil), the calculation is almost exactly the same, but is written down differently as shown below with the same two examples used above. Usually the quotient is written under a bar drawn under the divisor. A long vertical line is sometimes drawn to the right of the calculations.
500 ÷ 4 = 125 (Explanations)
4 ( 4 × 1 = 4)
10 ( 5 - 4 = 1)
8 ( 4 × 2 = 8)
20 (10 - 8 = 2)
20 ( 4 × 5 = 20)
0 (20 - 20 = 0)
and
127 ÷ 4 = 31.75
124
30 (bring down 0; decimal to quotient)
28 (7 × 4 = 28)
20 (an additional zero is added)
20 (5 × 4 = 20)
0
In Mexico, the English-speaking world notation is used, except that only the result of the subtraction is annotated and the calculation is done mentally, as shown below:
125 (Explanations)
4)500
10 ( 5 - 4 = 1)
20 (10 - 8 = 2)
Eurasia
In Spain, Italy, France, Portugal, Lithuania, Romania, Turkey, Greece, Belgium, Belarus, Ukraine, and Russia, the divisor is to the right of the dividend, and separated by a vertical bar. The division also occurs in the column, but the quotient (result) is written below the divider, and separated by the horizontal line. The same method is used in Iran, Vietnam, and Mongolia.
127|4
−12 |31,75
7
−4
30
−28
20
-20
0
In Cyprus, as well as in France, a long vertical bar separates the dividend and subsequent subtractions from the quotient and divisor, as in the example below of 6359 divided by 17, which is 374 with a remainder of 1.
6359|17
−51 |374
125 |
−119 |
69|
−68|
1|
Decimal numbers are not divided directly, the dividend and divisor are multiplied by a power of ten so that the division involves two whole numbers. Therefore, if one were dividing 12,7 by 0,4 (commas being used instead of decimal points), the dividend and divisor would first be changed to 127 and 4, and then the division would proceed as above.
In Austria, Germany and Switzerland, the notational form of a normal equation is used. <dividend> : <divisor> = <quotient>, with the colon ":" denoting a binary infix symbol for the division operator (analogous to "/" or "÷"). In these regions the decimal separator is written as a comma. (cf. first section of Latin American countries above, where it's done virtually the same way):
Algorithm for arbitrary base
Every natural number
n
{\displaystyle n}
can be uniquely represented in an arbitrary number base
b
>
1
{\displaystyle b>1}
as a sequence of digits
n
=
α
0
α
1
α
2
.
.
.
α
k
−
1
{\displaystyle n=\alpha _{0}\alpha _{1}\alpha _{2}...\alpha _{k-1}}
where
0
≤
α
i
<
b
{\displaystyle 0\leq \alpha _{i}<b}
Examples
In base 10, using the example above with
n
=
1260257
{\displaystyle n=1260257}
and
m
=
37
{\displaystyle m=37}
, the initial values
q
−
1
=
0
{\displaystyle q_{-1}=0}
and
r
−
1
=
1
{\displaystyle r_{-1}=1}
.
Thus,
q
=
34061
{\displaystyle q=34061}
and
r
=
0
{\displaystyle r=0}
Rational quotients
If the quotient is not constrained to be an integer, then the algorithm does not terminate for
i
>
k
−
l
{\displaystyle i>k-l}
. Instead, if
i
>
k
−
l
{\displaystyle i>k-l}
then
α
i
=
0
{\displaystyle \alpha _{i}=0}
by definition. If the remainder
r
i
{\displaystyle r_{i}}
is equal to zero at any iteration, then the quotient is a
b
{\displaystyle b}
-adic fraction, and is represented as a finite decimal expansion in base
b
{\displaystyle b}
Performance
On each iteration, the most time-consuming task is to select
β
i
{\displaystyle \beta _{i}}
. We know that there are
b
{\displaystyle b}
possible values, so we can find
β
i
{\displaystyle \beta _{i}}
using
O
(
log
(
b
)
)
{\displaystyle O(\log(b))}
comparisons. Each comparison will require evaluating
d
i
−
m
β
i
{\displaystyle d_{i}-m\beta _{i}}
. Let
k
{\displaystyle k}
Generalizations
Rational numbers
Long division of integers can easily be extended to include non-integer dividends, as long as they are rational. This is because every rational number has a repeating decimal expansion. The procedure can also be extended to include divisors which have a finite or terminating decimal expansion (i.e. decimal fractions). In this case the procedure involves multiplying the divisor and dividend by the appropriate power of ten so that the new divisor is an integer – taking advantage of the fact that a ÷ b = (ca) ÷ (cb) – and then proceeding as above.
Polynomials
A generalised version of this method called polynomial long division is also used for dividing polynomials (sometimes using a shorthand version called synthetic division).