long division

For the album by Rustic Overtones, see Long Division.

For the album by Low, see Long Division (Low album).

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.^[1]

Education

Today our 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). In the United States, 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, 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 continue to be taught ^[2].

An abbreviated form of long division is called short division.

Notation in America, Canada, the UK, and Japan

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)

The process is begun by dividing the left-most digit of the dividend by the divisor. The quotient (rounded down to an integer) becomes the first digit of the result, and the remainder is calculated (this step is notated as a subtraction). This remainder carries forward when the process is repeated on the following digit of the dividend (notated as 'bringing down' the next digit to the remainder). When all digits have been processed and no remainder is left, the process is complete.

Here is an example of the process not producing an integer result:

      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)
       2.8      (7 × 4 = 28)
         20     (an additional zero is brought down)
         20     (5 × 4 = 20)
          0

In this example, the decimal part of the result is calculated by continuing the process beyond the units digit, 'bringing down' zeros as being the decimal part of the dividend.

This example also illustrates that, at the beginning of the process, a step that produces a zero can be omitted. Since the first digit 1 is less than the divisor 4, the first step is instead performed on the first two digits 12. Similarly, if the divisor were 13, one would start by trying to divide it by 127 rather than 12 or 1.

Notation in non-English-speaking parts of the world

In Europe, Latin America and French-speaking Africa, students learn a different notation. The calculation is almost exactly the same, but is written down differently as shown below with the same two examples used above. Usually the quotient is written under a bar drawn under the divisor. A long vertical line is sometimes drawn to the right of the calculations.

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

and

     127 ÷ 4 = 31.75
     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)
       2 8      (7 × 4 = 28)
         20     (an additional zero is brought down)
         20     (5 × 4 = 20)
          0

Unlike the English notation, decimal numbers are not divided directly. Instead the dividend and divisor are multiplied by a power of ten so that the division involves two whole numbers.

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).

Hints for learning long division

1. When dividing two numbers -- for example, n divided by m -- n is the dividend and m is the divisor; the answer is the quotient.
2. Confirm the spot of all decimals in the dividend and divisor.
3. If necessary, simplify the long division by moving the decimals of the divisor and dividend by the same number of decimal places, to the right, so that the decimal of the divisor is to right of the last digit.
4. When doing long division, be sure to keep the numbers lined up straight from top to bottom under the tableau.
5. After each step, be sure the remainder for that step is less than the divisor. If it is not, there are three possible problems: the multiplication is wrong, the subtraction is wrong, or a greater quotient is needed.
6. In the end, the remainder, r, is added to the growing quotient as a fraction, r/m.

See also

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

References

External links

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

• Algorithm for long division and C++ implementation: Implementing Multiple-Precision Arithmetic, Part 2

• Long Division Algorithm Long Division Algorithm