Long division

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For the album by Rustic Overtones, see Long Division.

In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. It requires only the means to write the numbers down, and is simple to perform, even for large dividends. The procedure converts the problem of dividing a divisor into a large dividend into a series of divisions of the divisor into smaller numbers.

1 Notation
2 Example
3 Division algorithm
4 Generalizations
- 4.1 Rational numbers
- 4.2 Polynomials
5 See also
6 External links

[edit] Notation

In long division notation, 500 divided by 4 equals 125 is denoted as follows:

$\begin{matrix} \quad 125\\ 4\overline{)500}\\ \end{matrix}$

[edit] Example

The procedure involves several steps. As an example, consider the problem of 950 divided by 4:

1. Write the dividend and divisor in this form:

$4 \overline{)950} \$

The procedure involves dividing the divisor (4) into a number for each digit of the dividend (950).

2.The first number to be divided by the divisor (4) is the leftmost digit (9) of the dividend. Ignoring any remainder, write the result (2), above the line over the leftmost digit of the dividend. Multiply the divisor by that number (4 times 2) and write the result (8) under the leftmost digit of the dividend.

$\begin{matrix} 2\\ 4\overline{)950}\\ 8 \end{matrix}$

3. Subtract the bottom number (8) from the number immediately above it (9). Write the result (1), under the bottom number (8), then copy the next digit of the dividend (5) to the right of the result of the subtraction.

$\begin{matrix} 2\\ 4\overline{)950}\\ \underline{8}\\ \;\,15 \end{matrix}$

4. Repeat steps 2 and 3, using the newly created bottom number (15) as the number to be divided by the divisor (4), and write the results above and under the next digit of the dividend.

$\begin{matrix} \,\,23\\ 4\overline{)950}\\ \underline{8}\\ \;\,15\\ \ \underline{12}\\ \quad\;\,30 \end{matrix}$

5. Repeat step 4 until there are no digits remaining in the dividend. The number written above the bar (237) is the quotient, and the result of the last subtraction is the remainder for the entire problem (2).

$\begin{matrix} \quad 237\\ 4\overline{)950}\\ \underline{8}\\ \;\,15\\ \ \underline{12}\\ \quad\;\,30\\ \quad\;\,\underline{28}\\ \qquad 2 \end{matrix}$

The answer to the above example is expressed as 237 with remainder 2. Alternatively, one can continue the above procedure to produce a decimal answer. We continue the process by adding a decimal and zeroes as necessary to the right of the dividend, treating each zero as another digit of the dividend. Thus the next step in such a calculation would give the following:

$\begin{matrix} \quad 237.5\\ 4\overline{)950.0}\\ \!\!\!\!\!\underline{8}\\ \!\!15\\ \!\!\underline{12}\\ \ \;\,30\\ \ \;\,\underline{28}\\ \quad\ \;\,20\\ \quad\ \;\,\underline{20}\\ \qquad\ 0 \end{matrix}$

[edit] Division algorithm

The above procedure relies on the division algorithm, which states that given any two integers a and d, with d ≠ 0, there exist unique integers q and r such that a = qd + r and 0 ≤ r < |d |, where |d | denotes the absolute value of d.

[edit] Generalizations

[edit] 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 recurring 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.

[edit] Polynomials

A generalized version of this method called polynomial long division is also used for dividing polynomials (sometimes using a shorthand version called synthetic division).