After All (Cher and Peter Cetera song)

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. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps.

Education

Today, inexpensive calculators and computers have become the most common way to solve division problems. (Internally, those devices use one of a variety of division algorithms). 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 .

An abbreviated form of long division is called short division.

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)

another example where there is a remainder:

      31.75     
   4)127
     12       (12-12=0 which is written on the following line)                    
      07      (the seven is brought down from the dividend 127) 
       4       
       3.0      (3 is the remainder which is divided by 4 to give 0.75)

1. First of all look at the first digit of the dividend (127) to see if any 4's (the divisor) can be subtracted from it (in our case there aren't any because we have 1 as the first digit and we aren't allowed to use negative numbers so we can't write -3)
2. If the number in the first digit isn't big enough we take the second one along with it (so in our case we will deal with 12 as our first number)
3. See the maximum number of 4s that can subtracted from it (in our case that will be 3 fours which is 12)
4. Write in the quotient part (above the second digit since that is the last one you used) that 3 you concluded and write under the dividend the 12
5. Subtract the 12 you wrote from the one above it (the result will be of course zero)
6. Repeat the first step again
7. Since the zero won't work bring down the next digit in the dividend (which is 7) next to the zero and you get 07
8. Repeat step 3,4 and 5
9. You will have in the quotient 31 and 3 as a remainder and no more digits in the dividend
10. You can bring down a zero beside the 3 after you put the point and continue the division
11. The result will be 31.75

Generalisations

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.

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).

See also

• Short division
• Elementary arithmetic
• Arbitrary-precision arithmetic
• Polynomial long division

External links

• Alternative Division Algorithms: Double Division, Partial Quotients & Column Division, Partial Quotients Movie

• Step By Step Polynomial Long Division: WebGraphing.com