Long division
It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps.
[3] Caldrini (1491) is the earliest printed example of long division, known as the Danda method in medieval Italy,[4] and it became more practical with the introduction of decimal notation for fractions by Pitiscus (1608).
[5] Inexpensive calculators and computers have become the most common way to solve division problems, eliminating a traditional mathematical exercise and decreasing the educational opportunity to show how to do so by paper and pencil techniques.
(Internally, those devices use one of a variety of division algorithms, the faster of which rely on approximations and multiplications to achieve the tasks.)
In North America, long division has been especially targeted for de-emphasis or even elimination from the school curriculum by reform mathematics, though it has been traditionally introduced in the 4th, 5th or even 6th grades.
[8] It developed in the 18th century from an earlier single-line notation separating the dividend from the quotient by a left parenthesis.
[9][10] 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).
A more detailed breakdown of the steps goes as follows: If the last remainder when we ran out of dividend digits had been something other than 0, there would have been two possible courses of action: 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.
A slight variation of presentation requires more writing, and requires that we change, rather than just update, digits of the quotient, but can shed more light on why these steps actually produce the right answer by allowing evaluation of q × m + r at intermediate points in the process.
Specifically, we amend the above basic procedure so that we fill the space after the digits of the quotient under construction with 0's, to at least the 1's place, and include those 0's in the numbers we write below the division bracket.
The 111 is then subtracted from the line above, ignoring all digits to the right: Now the digit from the next smaller place value of the dividend is copied down and appended to the result 15: The process repeats: the greatest multiple of 37 less than or equal to 150 is subtracted.
Then the result of the subtraction is extended by another digit taken from the dividend: The greatest multiple of 37 less than or equal to 22 is 0 × 37 = 0.
Instead, we simply take another digit from the dividend: The process is repeated until 37 divides the last line exactly: For non-decimal currencies (such as the British £sd system before 1971) and measures (such as avoirdupois) mixed mode division must be used.
Consider dividing 50 miles 600 yards into 37 pieces: Each of the four columns is worked in turn.
Long division of the feet gives 1 remainder 29 which is then multiplied by twelve to get 348 inches.
Long division continues with the final remainder of 15 inches being shown on the result line.
When the quotient is not an integer and the division process is extended beyond the decimal point, one of two things can happen: China, Japan, Korea use the same notation as English-speaking nations including India.
In Latin America (except Argentina, Bolivia, Mexico, Colombia, Paraguay, Venezuela, Uruguay and Brazil), the calculation is almost exactly the same, but is written down differently as shown below with the same two examples used above.
and In Mexico, the English-speaking world notation is used, except that only the result of the subtraction is annotated and the calculation is done mentally, as shown below: In Bolivia, Brazil, Paraguay, Venezuela, French-speaking Canada, Colombia, and Peru, the European notation (see below) is used, except that the quotient is not separated by a vertical line, as shown below: Same procedure applies in Mexico, Uruguay and Argentina, only the result of the subtraction is annotated and the calculation is done mentally.
In Spain, Italy, France, Portugal, Lithuania, Romania, Turkey, Greece, Belgium, Belarus, Ukraine, and Russia, the divisor is to the right of the dividend, and separated by a vertical bar.
The division also occurs in the column, but the quotient (result) is written below the divider, and separated by the horizontal line.
In Cyprus, as well as in France, a long vertical bar separates the dividend and subsequent subtractions from the quotient and divisor, as in the example below of 6359 divided by 17, which is 374 with a remainder of 1.
Therefore, if one were dividing 12,7 by 0,4 (commas being used instead of decimal points), the dividend and divisor would first be changed to 127 and 4, and then the division would proceed as above.
In Austria, Germany and Switzerland, the notational form of a normal equation is used.
first section of Latin American countries above, where it's done virtually the same way): The same notation is adopted in Denmark, Norway, Bulgaria, North Macedonia, Poland, Croatia, Slovenia, Hungary, Czech Republic, Slovakia, Vietnam and in Serbia.
In the Netherlands, the following notation is used: In Finland, the Italian method detailed above was replaced by the Anglo-American one in the 1970s.
In the early 2000s, however, some textbooks have adopted the German method as it retains the order between the divisor and the dividend.
