Short division

In arithmetic, short division is a procedure which 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. Short division requires no advanced technology nor mental gymnastics, merely paper and pencil (or any similar means for writing). It is very powerful, enabling computations involving arbitrarily large numbers to be performed by following a series of simple steps.

Short division is an abbreviated form of long division. As short division relies on mental arithmetic, it is only suitable if the divisor is small - typically less than 10.

Notation

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

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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:

:$4\; overline\{)950\}$

Now instead of dividing the whole dividend (950) by the divisor (4), we will take as many digits of the dividend as necessary (starting from the left) to form a number that contains the divisor at least once, but less than ten times. In this case, that partial dividend is 9.

2. The first number to be divided by the divisor (4) is the partial divident (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).

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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.)

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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).

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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:

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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".

ee also

*Long division
*Elementary arithmetic
*Arbitrary-precision arithmetic
*Polynomial long division

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]