Long division

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For the album by Rustic Overtones, see Long Division.

In arithmetic, long division is the standard procedure suitable for dividing simple or complex multidigit numbers. It 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. Long division is understood by most adults, including teachers and parents, who have been taught elementary arithmetic. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps. ^[1]

1 Education
2 Notation
3 Generalisations
- 3.1 Rational numbers
- 3.2 Polynomials
4 See also
5 References
6 External links

[edit] Education

Today, inexpensive calculators and computers have become the most common way to solve division problems. Long division has been especially targeted for de-emphasis, or even elimination from the school curriculum, by reform mathematics, though traditionally introduced in the 4th or 5th grades. Some curricula such as Everyday Mathematics teach non-standard methods unfamiliar to most adults, or in the case of TERC argue that long division notation is itself no longer in mathematics. However many in the mathematics community have argued that standard arithmetic methods such as long division should be continued to be taught ^[2].

An abbreviated form of long division is called short division.

[edit] Notation

Long 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).

     125     (Explanations)
   4)500
     4        (4 ×  1 = 4)
     10       (5 -  4 = 1)
      8       (4 ×  2 = 8)
      20     (10 -  8 = 2)
      20      (4 ×  5 = 20)
       0     (20 - 20 = 0)

[edit] Generalisations

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