# Short division

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In arithmetic, short division is a division algorithm 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 long division. Because it relies on mental arithmetic, it is suitable for use only if the divisor is a small number – typically less than 10.

Short division enables computations involving arbitrarily large dividends to be performed by following a series of simple steps.[1]

## Notation

Short division does not use the / (slash) or ÷ (obelus) signs. Instead, it displays the dividend, the divisor, and the quotient (once it is found) in a tableau. An example is shown below, representing the division of 500 by 4, with a result of 125.

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

## Example

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

1. The dividend and divisor are written in the short division tableau:
$\begin{matrix} 4 \overline{)950} \ \end{matrix}$
Attempting to divide 950 by 4 in a single step would require knowledge of tables up to 238 times. Instead the division is reduced to small steps. Starting from the left enough digits are selected to form a number between one and ten times the divisor. In this case, that partial dividend is 9.
2. The first number to be divided by the divisor (4) is the partial dividend (9). We write the integer part of the result (2) above the division bar over the leftmost digit of the dividend, and we write the remainder (1) as a small digit (or digits) to the above and to the right of the partial dividend (9).
$\begin{matrix} 2\\ 4\overline{)9^150} \end{matrix}$
3. Next we repeat step 2, using the small digits just written along with the next digit of the dividend to form a new partial dividend (15). Dividing the new partial dividend by the divisor (4), we write the results as before: the quotient above the next digit of the dividend, and the remainder to the right. (Here 15 divided by 4 is 3, with a remainder of 3.)
$\begin{matrix} \,\,23\\ 4\overline{)9^15^30}\\ \end{matrix}$
4. We repeat step 2 until there are no digits remaining in the dividend. (In this example, the next step is to find that 30 divided by 4 is 7, with a remainder of 2.) 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).
$\begin{matrix} \quad 237\\ 4\overline{)9^15^30^2}\\ \end{matrix}$
5. 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:
$\begin{matrix} \quad 237.5\\ 4\overline{)9^15^30.^20}\\ \end{matrix}$

## Modulo division

When one is only interested in the remainder of the division, this variation on short division ignores the quotient and tallies only remainder digits. This can be used for manual modulo calculation, or as a divisibility test.

For example, what is the remainder of 16762109 divided by 7?

$\begin{matrix} 7)16^27^66^32^41^60^49^0 \end{matrix}$

The remainder is zero, so 16762109 is divisible by 7.

At each step, one need not consider the quotient digits. Knowledge of the single-digit multiplication table allows us to think of the nearest multiple of the divisor and subtract without explicitly writing the quotient digit. For example, we know that 27 is 21+6 (or 28-1) so we find the intermediate remainder digit is 6, without needing to write the quotient 3.

## References

1. ^ G.P Quackenbos, LL.D. (1874). "Chapter VII: Division". A Practical Arithmetic. D. Appleton & Company.