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Long division is the pencil and paper technique for dividing two numbers to arbitrary precision when multiple steps are required. For example, 114/3, or 114 ÷ 3, can be written as:
___
3|114
or generally:
________
divisor|dividend
Basic Long Division
The technique is to work left to right, dividing the dividend by the divisor one piece at a time in the smallest possible chunks. In this case, we start by dividing 11 by 3, because that is the smallest piece of 114 that can be divided by 3 with a result greater than 1. 3 × 3 = 9, which leaves a remainder of 2 from 11. Ignoring the leftover 4 on the right side of 114 for now, we record our progress thusly:
3
3|114
-9
2
We continue the problem by dropping the next element from 114, the last 4, down next to our remainder, and then proceed as before. In this case, we are left with 24 ÷ 3, which is 8.
3
3|114
-9
24
38
3|114
-9
24
-24
0
Now that we have run out of digits in our dividend, the problem is finished. Our remainder is 0, so 3 divides into 114 evenly. The quotient (result of division, or our answer) is 38.
Long Division with non-Integers
In practice, most results will not divide evenly. Take for example 3 ÷ 4:
_
4|3
In order to do this problem, we need to introduce at least one decimal place, like this:
___
4|3.0
For now, we will ignore the decimal place and treat the problem as if it were 30 ÷ 4. We'll come back to that later.
7
4|3.0
-2 8
2
In order to continue, we need to introduce another 0 after the decimal.
75
4|3.00
-2 8
20
-20
0
Now that we have a zero remainder, we can stop. All that remains is to account for that decimal point we ignored earlier. Simply push it up to the answer.
0.75
4|3.00
-2 8
20
-20
0
And we see our answer is 0.75.
Arbitrary Precision
By continuing to add zeroes after the decimal point, we can take an answer out to any arbitrary precision.
3.1428571
7|22.0000000
-21
1 0
-7
30
-28
20
-14
60
-56
40
-35
50
-49
10
-7
3
In this case, we can see that we run into a case of a repeating decimal. If we were to continue the division out, the pattern beginning at 10-7=3 and ending at 50-49=1 would continue repeating indefinitely. This is written:
______
3.142857
The overline signifies that 142857 is repeating.
Alternately, we could say:
3
7|22
-21
1 R
This is written "3 remainder 1", which means the answer is "three and one-seventh", or 31/7 when written as a fraction.
Variables and Polynomials
Long division can even be done with variables included in the problem. For example:
____________
x-3|2x2 + x - 21
The basic idea is the same. 2x2+x is the smallest part we can divide by x-3, so we start there.
2x
x-3|2x2 + 1x - 21
-2x2 + 6x
7x
Note the x term has been changed to 1x for clarity's sake. If the x term were missing altogether, we would add it in as 0x, so that we have a placeholder for that term. Otherwise there would be no x term to subtract from.
Notice also that the -3 in the divisor means that we subtract 2x2 but we add 6x. Continuing:
2x + 7
x-3|2x2 + 1x - 21
-2x2 + 6x
7x - 21
-7x + 21
0
Our answer is 2x+7.
Swap and themanwho tell me that the format I have provided is the one used in most English-speaking countries. Below are the alternate formats I have found, preceded by the Anglo-American format for comparison. themanwho points out that the Anglo-American format has an advantage in that the empty space available to the right of the dividend allows you to keep adding 0's if you need extra precision. I also like the fact that it puts everything (except the divisor) in a neat, easy to use column format.
38
3|114 114:3 = 38 114|3 114,3
-9 -9 -9 |38 -9 38
24 24 24| 24
-24 -24 -24| -24
0 0 0| 0 | 
