View source for UTP-monosaccharide-1-phosphate uridylyltransferase

In [[arithmetic]], '''short division''' is a [[division algorithm]] which breaks down a [[division (mathematics)|division]] problem into a series of easy steps. As in all division problems, one number, called the [[division (mathematics)|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&nbsp;– typically less than 10.

Short division enables computations involving arbitrarily large dividends to be performed by following a series of simple steps.<ref>{{cite book |title=A Practical Arithmetic |author=G.P Quackenbos, LL.D. |chapter=Chapter VII: Division |publisher=D. Appleton & Company |year=1874}}</ref>

==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 [[Long division#Method|tableau]]. An example is shown below, representing the division of 500 by 4, with a result of 125.
 
:<math>
\begin{matrix}
\quad 125\\
4\overline{)500}\\
\end{matrix}
</math>

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

<ol>
<li>The dividend and divisor are written in the short division tableau:

:<math>
\begin{matrix}
4 \overline{)950} \ 
\end{matrix}
</math>

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.

<li>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).

:<math>
\begin{matrix}
2\\
4\overline{)9^150}
\end{matrix}
</math>

<li>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.)

:<math>
\begin{matrix}
\,\,23\\
4\overline{)9^15^30}\\
\end{matrix}
</math>

<li>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).

:<math>
\begin{matrix}
\quad 237\\
4\overline{)9^15^30^2}\\
\end{matrix}
</math>

<li>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:

:<math>
\begin{matrix}
\quad 237.5\\
4\overline{)9^15^30.^20}\\
\end{matrix}
</math>
</ol>

==Modulo division==
{{importance-section|see talk|date=December 2012}}{{unreferenced-section|date=December 2012}}
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_operation|modulo]] calculation, or as a [[Divisibility_rule|divisibility test]].

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

:<math>
\begin{matrix}
7)16^27^66^32^41^60^49^0
\end{matrix}
</math>

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.

==See also==
*[[Chunking (division)]]
*[[Long division]]
*[[Fourier division]]
*[[Elementary arithmetic]]
*[[Division algorithm]]
*[[Arbitrary-precision arithmetic]]
*[[Polynomial long division]]
*[[Synthetic division]]

==References==
{{reflist}}

==External links==
*Alternative Division Algorithms: [http://www.doubledivision.org Double Division], [http://www.math.nyu.edu/~braams/links/em-arith.html Partial Quotients & Column Division], [http://mb.msdpt.k12.in.us/Math/PartialQuotients.wmv Partial Quotients Movie]

*Lesson in Short Division: [http://www.themathpage.com/arith/divide-whole-numbers.htm TheMathPage.com]

[[Category:Division]]