Long division

# Long division

Discussion

Encyclopedia

In arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

, long division is a standard procedure
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

suitable for dividing simple or complex multidigit numbers. It breaks down a division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

problem into a series of easier steps. As in all division problems, one number, called the dividend
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...

, is divided by another, called the divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

, producing a result called the quotient
Quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...

. 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
Short division
In arithmetic, short division is a procedure which breaks down a division problem into a series of easy steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. Short division is an abbreviated form of...

, which is almost always used instead of long division when the divisor has only one digit.

## Place in education

Today, inexpensive calculators and computers have become the most common way to solve division problems, decreasing the traditional educational imperative to know how to do so by paper and pencil techniques. (Internally, those devices use one of a variety of division algorithm
Division (digital)
Several algorithms exist to perform division in digital designs. These algorithms fall into two main categories: slow division and fast division. Slow division algorithms produce one digit of the final quotient per iteration. Examples of slow division include restoring, non-performing restoring,...

s). In the United States, long division has been especially targeted for de-emphasis, or even elimination from the school curriculum, by reform mathematics
Reform mathematics
Reform mathematics is an approach to mathematics education, particularly in North America. It is based on principles explained in 1989 by the National Council of Teachers of Mathematics . The NCTM document, Curriculum and Evaluation Standards for School Mathematics, attempted to set forth a vision...

, though traditionally introduced in the 4th or 5th grades. Some curricula such as Everyday Mathematics
Everyday Mathematics
Everyday Mathematics is a pre-K and elementary school mathematics curriculum developed by the University of Chicago School Mathematics Project....

teach non-standard methods, or in the case of TERC
TERC
TERC may refer to:*Telomerase RNA component, a human gene.*The developers of the Investigations in Numbers, Data, and Space mathematics curriculum.*Technical Education Research Centers*CSIRO Tropical Ecosystems Research Centre...

argue that long division notation is itself no longer in mathematics. However many in the mathematics community have argued that standard arithmetic methods such as long division should continue to be taught.

## Method

In English-speaking countries, long division does not use the slash
Slash (punctuation)
The slash is a sign used as a punctuation mark and for various other purposes. It is now often called a forward slash , and many other alternative names.-History:...

(/) or obelus
Obelus
An obelus is a symbol consisting of a short horizontal line with a dot above and below. It is mainly used to represent the mathematical operation of division. It is therefore commonly referred to as the division sign.- History :The word "obelus" comes from the Greek word for a sharpened stick,...

(÷) signs, instead displaying the dividend, divisor
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer which divides n without leaving a remainder.-Explanation:...

, and (once it is found) quotient
Quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...

in a tableau.

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
Remainder
In arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....

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)

In the above example, the first step is to find the shortest sequence of digits starting from the left end of the dividend, 500, that the divisor 4 goes into at least once; this shortest sequence in this example 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. Next, the 1 is multiplied by the divisor 4, to obtain the largest whole number (4 in this case) that is a multiple of the divisor 4 without exceeding the 5; this product of 1 times 4 is 4, so 4 is placed underneath the 5. Next the 4 under the 5 is subtracted from the 5 to get the remainder, 1, which is placed under the 4 under the 5. This remainder 1 is necessarily smaller than the divisor 4. Next the first as-yet unused digit in the dividend, in this case the first digit 0 after the 5, is copied directly underneath itself and next to the remainder 1, to form the number 10. At this point the process is repeated enough times to reach a stopping point: The largest number by which the divisor 4 can be multiplied without exceeding 10 is 2, so 2 is written above the 0 that is next to the 5 — that is, directly above the last digit in the 10. Then the latest entry to the quotient, 2, is multiplied by the divisor 4 to get 8, which is the largest multiple of 4 that does not exceed 10; so 8 is written below 10, and the subtraction 10 minus 8 is performed to get the remainder 2, which is placed below the 8. This remainder 2 is necessarily smaller than the divisor 4. The next digit of the dividend (the last 0 in 500) is copied directly below itself and next to the remainder 2, to form 20. Then the largest number by which the divisor 4 can be multiplied without exceeding 20 is ascertained; this number is 5, so 5 is placed above the last dividend digit that was brought down (i.e., above the rightmost 0 in 500). Then this new quotient digit 5 is multiplied by the divisor 4 to get 20, which is written at the bottom below the existing 20. Then 20 is subtracted from 20, yielding 0, which is written below the 20. We know we are done now because two things are true: there are no more digits to bring down from the dividend, and the last subtraction result was 0.

If the last remainder when we ran out of dividend digits had been something other than 0, there would have been two possible courses of action. (1) We could just stop there and say that the dividend divided by the divisor is the quotient written at the top with the remainder written at the bottom; equivalently we could write the answer as the quotient followed by a fraction that is the remainder divided by the dividend. Or, (2) we could extend the dividend by writing it as, say, 500.000... and continue the process (using a decimal point in the quotient directly above the decimal point in the dividend), in order to get a decimal answer, as in the following example.

31.75
4)127.00
12 (12-12=0 which is written on the following line)
07 (the seven is brought down from the dividend 127)
4
30 (3 is the remainder which is divided by 4 to give 0.75)
28 (7 × 4 = 28)
20 (an additional zero is brought down)
20 (5 × 4 = 20)
0

In this example, the decimal part of the result is calculated by continuing the process beyond the units digit, "bringing down" zeros as being the decimal part of the dividend.

This example also illustrates that, at the beginning of the process, a step that produces a zero can be omitted. Since the first digit 1 is less than the divisor 4, the first step is instead performed on the first two digits 12. Similarly, if the divisor were 13, one would perform the first step on 127 rather than 12 or 1.

### Basic procedure for long division by longhand

1. When dividing two numbers, for example, n divided by m, n is the dividend and m is the divisor; the answer is the quotient.
2. Find the location of all decimal points in the dividend and divisor.
3. 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.
4. When doing long division, keep the numbers lined up straight from top to bottom under the tableau.
5. 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.
6. In the end, the remainder, r, is added to the growing quotient as a fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...

r/m.

### 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. (1) The process can terminate, which means that a remainder of 0 is reached; or (2) 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.

## Notation in non-English-speaking countries

China, Japan and India use the same notation as English-speakers. Elsewhere, the same general principles are used, but the figures are often arranged differently.

### Latin America

In Latin America
Latin America
Latin America is a region of the Americas where Romance languages  – particularly Spanish and Portuguese, and variably French – are primarily spoken. Latin America has an area of approximately 21,069,500 km² , almost 3.9% of the Earth's surface or 14.1% of its land surface area...

(except Mexico
Mexico
The United Mexican States , commonly known as Mexico , is a federal constitutional republic in North America. It is bordered on the north by the United States; on the south and west by the Pacific Ocean; on the southeast by Guatemala, Belize, and the Caribbean Sea; and on the east by the Gulf of...

, Colombia
Colombia
Colombia, officially the Republic of Colombia , is a unitary constitutional republic comprising thirty-two departments. The country is located in northwestern South America, bordered to the east by Venezuela and Brazil; to the south by Ecuador and Peru; to the north by the Caribbean Sea; to the...

, and Brazil
Brazil
Brazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...

), 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
12 (12-12=0 which is written on the following line)
07 (the seven is brought down from the dividend 127)
4
30 (3 is the remainder which is divided by 4 to give 0.75)
28 (7 × 4 = 28)
20 (an additional zero is brought down)
20 (5 × 4 = 20)
0
In Mexico
Mexico
The United Mexican States , commonly known as Mexico , is a federal constitutional republic in North America. It is bordered on the north by the United States; on the south and west by the Pacific Ocean; on the southeast by Guatemala, Belize, and the Caribbean Sea; and on the east by the Gulf of...

, the US 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)
0 (20 - 20 = 0)

In Brazil
Brazil
Brazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...

and Colombia
Colombia
Colombia, officially the Republic of Colombia , is a unitary constitutional republic comprising thirty-two departments. The country is located in northwestern South America, bordered to the east by Venezuela and Brazil; to the south by Ecuador and Peru; to the north by the Caribbean Sea; to the...

, the European notation (see below) is used, except that the quotient is not separated by a vertical line, as shown below:

127|4
-12 31,75
07
- 4
30
-28
20
-20
0

Same procedure applies in Mexico, only the result of the subtraction is annotated and the calculation is done mentally.

### Europe

In Spain, Italy, France, Germany, Portugal, Romania, Turkey, Greece 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.

127|4
12 |31,75
07
4
30
28
20
20
0

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.
 6 3 5 9 17 − 5 1 374 1 2 5 − 1 1 9 6 9 − 6 8 1

Unlike the English notation, decimal numbers are not divided directly. Instead 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 Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

, the notation of a normal equation is used for dividend, divisor and quotient:

127 : 4 = 31,75
12
07
4
30
28
20
20
0

The same notation is adopted in Norway
Norway
Norway , officially the Kingdom of Norway, is a Nordic unitary constitutional monarchy whose territory comprises the western portion of the Scandinavian Peninsula, Jan Mayen, and the Arctic archipelago of Svalbard and Bouvet Island. Norway has a total area of and a population of about 4.9 million...

, Poland
Poland
Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...

, Croatia
Croatia
Croatia , officially the Republic of Croatia , is a unitary democratic parliamentary republic in Europe at the crossroads of the Mitteleuropa, the Balkans, and the Mediterranean. Its capital and largest city is Zagreb. The country is divided into 20 counties and the city of Zagreb. Croatia covers ...

, Slovenia
Slovenia
Slovenia , officially the Republic of Slovenia , is a country in Central and Southeastern Europe touching the Alps and bordering the Mediterranean. Slovenia borders Italy to the west, Croatia to the south and east, Hungary to the northeast, and Austria to the north, and also has a small portion of...

, Hungary
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...

, Czech Republic
Czech Republic
The Czech Republic is a landlocked country in Central Europe. The country is bordered by Poland to the northeast, Slovakia to the east, Austria to the south, and Germany to the west and northwest....

, Slovakia
Slovakia
The Slovak Republic is a landlocked state in Central Europe. It has a population of over five million and an area of about . Slovakia is bordered by the Czech Republic and Austria to the west, Poland to the north, Ukraine to the east and Hungary to the south...

and in Bulgaria
Bulgaria
Bulgaria , officially the Republic of Bulgaria , is a parliamentary democracy within a unitary constitutional republic in Southeast Europe. The country borders Romania to the north, Serbia and Macedonia to the west, Greece and Turkey to the south, as well as the Black Sea to the east...

.

### Rational numbers

Long division of integers can easily be extended to include non-integer dividends, as long as they are rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...

. This is because every rational number has a recurring decimal expansion. The procedure can also be extended to include divisors which have a finite or terminating decimal
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....

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
Polynomial long division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division...

is also used for dividing polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

s (sometimes using a shorthand version called synthetic division
Synthetic division
Synthetic division is a method of performing polynomial long division, with less writing and fewer calculations. It is mostly taught for division by binomials of the formx - a,\ but the method generalizes to division by any monic polynomial...

).

• Short division
Short division
In arithmetic, short division is a procedure which breaks down a division problem into a series of easy steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. Short division is an abbreviated form of...

• Fourier division
Fourier division
Fourier division or cross division is a pencil-and-paper method of division which helps to simplify the process when the divisor has more than two digits. It was invented by Joseph Fourier.-Method:...

• Elementary arithmetic
Elementary arithmetic
Elementary arithmetic is the simplified portion of arithmetic which is considered necessary and appropriate during primary education. It includes the operations of addition, subtraction, multiplication, and division. It is taught in elementary school....

• Arbitrary-precision arithmetic
Arbitrary-precision arithmetic
In computer science, arbitrary-precision arithmetic indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most ALU hardware, which typically...

• Polynomial long division
Polynomial long division
In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division...