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.
| First the division problem is set up with the divisor on the left side and the dividend on the right side of the staircase. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Will 17 go into 14? No. Now you bring down the next digit of the dividend, which is 9. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Will 17 go into 13? No We bring down the next digit of the dividend, which is 6. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 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. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Long division calculator
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: a^{n}:a^{p}=a^{n-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 =
10^{3.4983105537896004} : 10^{1.1760912590556813} =
10^{3.4983105537896004 - 1.1760912590556813} =
10^{2.3222192947339191}
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.