How to divide
Division is one of the four basic arithmetic operations, with the other three being multiplication, subtraction, and addition. Division is the inverse of multiplication, and at least for smaller whole numbers, knowing the multiplication chart makes division relatively simple. More complicated division problems can be calculated using long division.
There are a number of ways to express division using different symbols. The following are all examples of the same division problem:
The first two simply use different division symbols, "÷" and "/". The third is a fraction, or a ratio, that can also be looked at as a division problem, and the last is the format used for long division.
Division as the inverse of multiplication
Division being the inverse of multiplication means that knowing all the multiplication facts (the 100 multiplication problems formed by all the combinations of one-digit numbers) makes division of smaller numbers simple. For example, given the division problem 72 ÷ 9, recognizing this as the multiplication fact 8 × 9 = 72 allows us to immediately determine that 72 ÷ 9 = 8.
While we can think of multiplication of whole numbers as repeated addition, we can look at a division problem such as 28 ÷ 7 as breaking a group up into a given number of equal groups. In other words, if the numbers represent oranges, how many groups of 7 oranges can there be if there are a total of 28 oranges?
If you have 28 oranges, and you separate them into groups of 7, as in the figure, you would have 4 equal groups of 7 oranges. Therefore, 28 ÷ 7 = 4. Again, since division is the inverse of multiplication, if we knew the multiplication fact, 7 × 4 = 28, then we could have quickly determined the solution.
For more complicated division problems, we can use long division.
Long division is a division algorithm that can be used to calculate division problems that involve larger numbers, as well as decimal numbers. To perform long division, we first need to understand the long division format and terminology:
The number under the radical symbol, 564, is the dividend; it is the number being divided. The divisor is the number that the dividend is divided by, and is written to the left of the radical. The quotient is the solution to the division problem, and is written above the radical sign, aligned based on their ones places.
Below are the steps for how we arrived at the solution to the problem in the above figure, which can be applied to any long division problem.
- Reading from left to right, we first want to find the smallest sequence of digits (in the dividend) that the divisor can go into at least once. 6 cannot go into the 5 in 564. Since it can't, move on to the next number formed, which is 56. 6 can go into 56 a total of 9 times to equal 54.
- Write the first value of the quotient above the dividend. In this case, write 9 above the 6 in 56 to indicate that 6 goes into 56 a total of 9 times. The position that the number 9 is written in is important. When doing long division, make sure that the numbers align. The starting point of the quotient should be above the last digit (reading from left to right) in the smallest sequence of numbers that the divisor can go into; in this case above the 6 in 56. The next number that forms the quotient should be written directly to the right of the first.
- Write the product, in this case 6 ÷ 9 = 54, below 56, and perform subtraction; in this case there is a remainder of 2.
- Bring the 4 in 564 down next to the remainder to form 24, keeping in mind that alignment is important.
- Repeat the process starting from step 1, treating 24 as the new dividend. Continue this process as long as necessary until the remainder is 0, or we find a repeating pattern. In this case, since 6 × 4 = 24, subtract 24 from 24 to get 0, and the long division is complete. If, on a different problem, the remainder is not 0, continue to step 6.
- If there is still a remainder and there are no more new numbers from the dividend to bring down, add a decimal point and a 0, then bring the 0 down to the remainder and continue the process above (including adding 0's) until there is no longer a remainder, or until a repeating pattern is found.
For the above example, since the remainder is 0, the quotient is therefore 94, meaning that the solution to the problem 564 ÷ 6 is 94.
When the numbers do not divide exactly, we can either say the answer is the quotient and the remainder, or we can take the problem further with some extra steps to determine the solution using decimals. In some cases, we can find an exact solution, but in others, we can only approximate the value if the decimal does not terminate. Using an example similar to the one above, if the dividend were 566 instead of 564:
Following step 6, when we bring 0 down to the remainder of 2, we get 20. 6 goes into 20 a total of 3 times to get 18, resulting in a remainder of 2. No matter how many times we add and bring down a 0, the result will be the same, repeating decimal, so the quotient is 94.3, where the line above the 3 indicates that it repeats indefinitely.