Long division is a method of dividing two numbers, using repeated multiplications and subtractions in a tableau. Because it is easy to do, it is usually taught in schools. There are other methods which are faster, or easier to program with a computer, but they are more difficult to understand and perform manually. If we have a large or complicated division problem, we can use long division to break it down into a series of easier calculations. Long division can also be done on polynomials as well.

Overview 
As with most division problems we have three numbers: the dividend, our first number; the divisor, the second number we divide it by; and the quotient, which is the result. Long division is a kind of algorithm, which means it helps us to find the solution to a problem by following a set of clearly-defined steps:

 First, we need to split our first number (dividend) into a separate number for each digit. If our dividend is 123, then we would split this into 1, 2 and 3.
 Next, we need to divide each of these digits by our second number (divisor). If our divisor is 8, then we would do 1 / 8, 2 / 8, followed by 3 / 8.
 If the division has a remainder, then the remainder is carried to the next step.
 If the division is less than 0 (for example. when trying to divide a small number by a large number), then the dividend, instead of the remainder, is carried to the next step.
 Once all of the numbers have been processed, every result (quotient) would then be combined into a single number again. For example. if the quotients are 7, 8 and 2, then the final result would be 782.
 Any leftover remainders make up the decimal part of the answer.

Examples

Basic case 
Let's assume we want to divide 780 by 4. To do so with long division, we need to split 780 into digits -- 7 and 8 and 0 -- and then divide each one by 4, carrying any remainders to the next step.

7 / 4 = 1 R 3 -- since we have a remainder of 3, we have to carry this down.
38 / 4 = 9 R 2 -- we carry our remainder of 2 down.
20 / 4 = 5 R 0 -- we have reached the end.

By using long division, we have found that 780 / 4 = 195.

Complex case 
Let's perform a similar calculation where our numbers do not divide easily: 468 / 12

4 / 12 = 0 R 0  -- we cannot perform this division as 4<12, so we have to carry our dividend like a remainder.
46 / 12 = 3 R 10 -- we carry our remainder of 10 down.
108 / 12 = 9 R 0  -- we have reached the end.

By the same process we have found that 468 / 12 = 39.

Decimal Answers 

Let's follow the same process for numbers that give an answer with a decimal because they don't have common factors: 123 / 8

1 / 8 = 0 R 0 -- we cannot perform this division as 1<8, so we have to carry our dividend like a remainder.
12 / 8 = 1 R 4 -- we carry our remainder of 4 down.
43 / 8 = 5 R 3  -- we have reached the end with a remainder, which we have to add to our final answer.

123 / 8 = 15 R 3, which equals 15.375 (15 3/8).

Related pages
 Chunking, a different type of long division done in the UK.
 Divisor, a number which evenly divides another number
 Short division, a faster version of long division done with smaller numbers.
 Synthetic division, an alternate algorithm for polynomial long division

References 

Algorithms
Arithmetics
Mathematics