In arithmetic, long division is a 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] The abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit.
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Today, inexpensive calculators and computers have become the most common way to solve division problems, decreasing the traditional educational imperative to know how to do so by paper and pencil techniques. (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]
In English-speaking countries, long division does not use the slash (/) or obelus (÷) signs, instead displaying the dividend, divisor, and (once it is found) quotient in a tableau.
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.
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)
In the above example, the first step is to find the shortest sequence of digits starting from the left end of the dividend, 500, that the divisor 4 goes into at least once; this shortest sequence in this example is simply the first digit, 5. The largest number that the divisor 4 can be multiplied by without exceeding 5 is 1, so the digit 1 is put above the 5 to start constructing the quotient. Next, the 1 is multiplied by the divisor 4, to obtain the largest whole number (4 in this case) that is a multiple of the divisor 4 without exceeding the 5; this product of 1 times 4 is 4, so 4 is placed underneath the 5. Next the 4 under the 5 is subtracted from the 5 to get the remainder, 1, which is placed under the 4 under the 5. This remainder 1 is necessarily smaller than the divisor 4. Next the first as-yet unused digit in the dividend, in this case the first digit 0 after the 5, is copied directly underneath itself and next to the remainder 1, to form the number 10. At this point the process is repeated enough times to reach a stopping point: The largest number by which the divisor 4 can be multiplied without exceeding 10 is 2, so 2 is written above the 0 that is next to the 5 — that is, directly above the last digit in the 10. Then the latest entry to the quotient, 2, is multiplied by the divisor 4 to get 8, which is the largest multiple of 4 that does not exceed 10; so 8 is written below 10, and the subtraction 10 minus 8 is performed to get the remainder 2, which is placed below the 8. This remainder 2 is necessarily smaller than the divisor 4. The next digit of the dividend (the last 0 in 500) is copied directly below itself and next to the remainder 2, to form 20. Then the largest number by which the divisor 4 can be multiplied without exceeding 20 is ascertained; this number is 5, so 5 is placed above the last dividend digit that was brought down (i.e., above the rightmost 0 in 500). Then this new quotient digit 5 is multiplied by the divisor 4 to get 20, which is written at the bottom below the existing 20. Then 20 is subtracted from 20, yielding 0, which is written below the 20. We know we are done now because two things are true: there are no more digits to bring down from the dividend, and the last subtraction result was 0.
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. (1) We could just stop there and say that the dividend divided by the divisor is the quotient written at the top with the remainder written at the bottom; equivalently we could write the answer as the quotient followed by a fraction that is the remainder divided by the dividend. Or, (2) we could extend the dividend by writing it as, say, 500.000... and continue the process (using a decimal point in the quotient directly above the decimal point in the dividend), in order to get a decimal answer, as in the following example.
31.75
4)127.00
12 (12-12=0 which is written on the following line)
07 (the seven is brought down from the dividend 127)
4
30 (3 is the remainder which is divided by 4 to give 0.75)
28 (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 perform the first step on 127 rather than 12 or 1.
When the quotient is not an integer and the division process is extended beyond the decimal point, one of two things can happen. (1) The process can terminate, which means that a remainder of 0 is reached; or (2) a remainder could be reached that is identical to a previous remainder that occurred after the decimal points were written. In the latter case, continuing the process would be pointless, because from that point onward the same sequence of digits would appear in the quotient over and over. So a bar is drawn over the repeating sequence to indicate that it repeats forever.
China, Japan and India use the same notation as English-speakers. Elsewhere, the same general principles are used, but the figures are often arranged differently.
In Latin America (except Mexico, Colombia, and Brazil), 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
30 (3 is the remainder which is divided by 4 to give 0.75)
28 (7 × 4 = 28)
20 (an additional zero is brought down)
20 (5 × 4 = 20)
0
In Mexico, the US notation is used, except that only the result of the subtraction is annotated and the calculation is done mentally, as shown below:
125 (Explanations)
4)500
10 (5 - 4 = 1)
20 (10 - 8 = 2)
0 (20 - 20 = 0)
In Brazil and Colombia, the European notation (see below) is used, except that the quotient is not separated by a vertical line, as shown below:
127|4
-12 31,75
07
- 4
30
-28
20
-20
0
Same procedure applies in Mexico, only the result of the subtraction is annotated and the calculation is done mentally.
In Spain, Italy, France, Portugal, Romania, Turkey, Greece 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.
127|4
−12 |31,75
07
− 4
30
−28
20
−20
0
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.
| 6 | 3 | 5 | 9 | 17 |
| − 5 | 1 | 374 | ||
| 1 | 2 | 5 | ||
| − 1 | 1 | 9 | ||
| 6 | 9 | |||
| − | 6 | 8 | ||
| 1 |
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. 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 Germany, the notation of a normal equation is used for dividend, divisor and quotient:
127 : 4 = 31,75
−12
07
− 4
30
−28
20
−20
0
The same notation is adopted in Norway, Poland, Croatia, Slovenia, Hungary, Czech Republic, Slovakia and in Bulgaria.
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.
A generalised version of this method called polynomial long division is also used for dividing polynomials (sometimes using a shorthand version called synthetic division).