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Long division

To make a long division by hand is the most complicated of the four arithmetic operations.

The first thing you do is to set up the division problem. There are different ways to do that, all of which are quite similar. On this site we use a staircase-like way of setting up the division, where the divisor is on the left side and the dividend is on the right side.

Example of long division

It's easier to understand the method being used, if you take a look at the following example of a long division. The example is a snapshot from the result produced by our calculator further down.

We want to divide 4896 by 17. That is, 17 is divisor and 4896 is dividend.

       
17 4896
  First the division problem is set up with the divisor on the left side and the dividend on the right side of the staircase.


    2  
17 4896
   34  
   14    
  Then you check if the divisor will go into the first digit of the dividend.
Will 17 go into the first digit 4? No.
Now you try with two of the dividend's digits instead. Will 17 go into 48? Yes, 17 is contained 2 times in 48 with the remainder 14.
You write 2 in the result line on top and 34 (because 2 times 17 is 34) below 48, make a horizontal line and write the remainder 14.


    2  
17 4896
   34  
   149   
  Will 17 go into 14? No.
Now you bring down the next digit of the dividend, which is 9.


    28 
17 4896
   34  
   149   
   136 
    13   
  Will 17 go into 149? Yes, 17 is contained 8 times in 149 with the remainder 13.
You write 8 in the result line on top and 136 (because 8 times 17 is 136) below 149, make a horizontal line and write the remainder 13.


    28 
17 4896
   34  
   149   
   136 
    136  
  Will 17 go into 13? No
We bring down the next digit of the dividend, which is 6.


    288
17 4896
   34  
   149   
   136 
    136  
    136
      0  
  Will 17 go into 136? Yes, 17 is contained 8 times in 136 with no remainder.
You write 8 in the result line on top and 136 (because 8 times 17 is 136) below 136, make a horizontal line and write the remainder 0.


Since the calculation ends up with no remainder, 4896 is divided evenly by 17 and the result is 4896:17=288.

Long division calculator

Dividend: 
Divisor: 

Long division with decimals or remainders

If the dividend is not divided evenly into the divisor, and there are no digits of the dividend left to bring down to the remainder, you can add a decimal point followed by 0 to the dividend. It is now possible to bring down 0 to the remainder. You also have to add a decimal point to the number in the result at the top then. All digits you add to the result from now on will be decimals of the results.

Sometimes you want the result to be given as an integer combined with the remainder. The calculation is done the same way, but instead of calculating the decimals you just give the remainder you end up with, when the last digit of the dividend has been brought down as part of the result.

Divide using tables of logarithms

Before calculators and computers were invented, you had to perform long divisions manually.

In order to cope with division of very large numbers, you could convert the numbers to powers of 10, because then you could take advantage of the laws of logarithms, one of which saying that: an:ap=an-p.

That would enable you to transform a difficult division problem into a simple subtraction problem. All you needed was some tables containing the values of the different logarithms.

For example, we want to divide 3150 by 15.
Only extracts from the table of logarithms are presented below (from 1 to 21, from 200 to 219 and from 3140 to 3159), as the entire table is quite comprehensive. It takes up a book.

1 0.0 200 2.3010299956639813 3140 3.496929648073215
2 0.3010299956639812 201 2.303196057420489 3141 3.497067936398505
3 0.47712125471966244 202 2.305351369446624 3142 3.4972061807039547
4 0.6020599913279624 203 2.307496037913213 3143 3.49734438101758
5 0.6989700043360189 204 2.3096301674258988 3144 3.4974825373673704
6 0.7781512503836436 205 2.311753861055754 3145 3.497620649781288
7 0.8450980400142568 206 2.3138672203691533 3146 3.497758718287268
8 0.9030899869919435 207 2.315970345456918 3147 3.49789674291322
9 0.9542425094393249 208 2.3180633349627615 3148 3.498034723687027
10 1.0 209 2.3201462861110542 3149 3.498172660636544
11 1.0413926851582251 210 2.322219294733919 3150 3.4983105537896004
12 1.0791812460476249 211 2.3242824552976926 3151 3.4984484031739997
13 1.1139433523068367 212 2.326335860928751 3152 3.498586208817518
14 1.146128035678238 213 2.3283796034387376 3153 3.4987239707479048
15 1.1760912590556813 214 2.330413773349191 3154 3.498861688992884
16 1.2041199826559248 215 2.3324384599156054 3155 3.498999363580153
17 1.2304489213782739 216 2.3344537511509307 3156 3.4991369945373827
18 1.255272505103306 217 2.3364597338485296 3157 3.4992745818922173
19 1.2787536009528289 218 2.3384564936046046 3158 3.499412125672275
20 1.3010299956639813 219 2.3404441148401185 3159 3.499549625905149


Now we can find the following in the table:
The logarithm of 3150 (highlighted in red) is 3.4983105537896004
The logarithm of 15 (highlighted in blue) is 1.1760912590556813

The division can then be carried out as follows:

3150 : 15 =

103.4983105537896004 : 101.1760912590556813 =

103.4983105537896004 - 1.1760912590556813 =

102.3222192947339191

Now we can use the table in the opposite way to convert the result back into an ordinary number. So we find the number being closest to 2.3222192947339191 (highlighted in green), which is the number in the 210-row.
So 3150 : 15 = 210.