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[change | change source]
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[change | change source]
Basic case[change | change source]
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[change | change source]
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[change | change source]
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[change | change source]
- 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[change | change source]
- "The Definitive Higher Math Guide to Long Division and Its Variants — for Integers". Math Vault. 2019-02-24. Retrieved 2020-08-26.
- Weisstein, Eric W. "Long Division". mathworld.wolfram.com. Retrieved 2020-08-26.
- "Long Division". www.mathsisfun.com. Retrieved 2020-08-26.