In
arithmetic, short division is a
division algorithm which breaks down a
division problem into a series of easier steps. It is an abbreviated form of
long division — whereby the products are omitted and the partial remainders are notated as
superscripts
A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, whil ...
.
As a result, a short division tableau is shorter than its long division counterpart — though sometimes at the expense of relying on
mental arithmetic, which could limit the size of the
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
.
For most people, small integer divisors up to 12 are handled using memorised
multiplication tables
In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.
The decimal multiplication table was traditionally taught as an essen ...
, although the procedure could also be adapted to the larger divisors as well.
As in all division problems, a number called the ''
dividend'' is divided by another, called the ''
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
''. The answer to the problem would be the ''
quotient'', and in the case of
Euclidean division, the
remainder would be included as well.
Using short division, arbitrarily large dividends can be handled.
Tableau
Short division does not use the
slash
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(/) or
division sign (÷) symbols. Instead, it displays the dividend, divisor, and
quotient (when it is found) in a
tableau. An example is shown below, representing the division of 500 by 4. The quotient is 125.
:
Alternatively, the bar may be placed below the number, which means the sum proceeds down the page. This is in distinction to
long division, where the space under the dividend is required for workings:
:
Example
The procedure involves several steps. As an example, consider 950 divided by 4:
Using the alternative layout the final workings would be:
:
As usual, similar steps can also be used to handle the cases with a decimal dividend, or the cases where the divisor involves multiple digits.
Prime factoring
A common requirement is to reduce a number to its prime factors. This is used particularly in working with
vulgar fractions. The dividend is successively divided by prime numbers, repeating where possible:
:
This results in 950 = 2 x 5² x 19
Modulo division
When one is interested only in the
remainder of the division, this procedure (a variation of short division) ignores the quotient and tallies only the remainders. It can be used for manual
modulo calculation or as a
test for even divisibility.
The quotient digits are not written down.
The following shows the solution (using short division) of 16762109 divided by seven.
:
The remainder is zero, so 16762109 is exactly divisible by 7.
See also
*
Arbitrary-precision arithmetic
*
Chunking (division)
*
Division algorithm
*
Elementary arithmetic
*
Fourier division
*
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 generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, bec ...
*
Synthetic division
References
{{reflist
External links
*Alternative Division Algorithms
Double DivisionPartial Quotients & Column DivisionPartial Quotients Movie*Lesson in Short Division
Division (mathematics)