Figure 1: Multiplication Table
Copyright © 2007 jsd
The usual “textbook” instructions for how to do long division are both unnecessarily laborious and unnecessarily hard to understand. There’s another way to organize the calculation that is much less mysterious and much less laborious (especially when long multi-digit numbers are involved).
Note that as discussed above, keeping things lined up in columns is critical. It may help to use grid-ruled paper, or at least to sketch in some guidelines.
For some basic, general-purpose hints on how to do math, see reference 1. For a discussion of long multiplication, see reference 2.
Suppose the assignment is to divide 7527 by 13. After writing down the statement of the problem, and before doing any actual dividing, it helps to make a crib, as shown in figure 1. This is just a multiplication table, showing all multiples of the assigned divisor, which is 13 in our example.
It is super-easy to construct such a table, since no multiplication is required. Successive addition will do the job. We need all the multiples from ×1 to ×9, but you might as well calculate the ×10 row, by adding 117+13. This serves as a valuable check on all the additions that have gone before.
We now begin the main part of the algorithm. The first step is to consider the leading digit of the dividend (which in this case is a 7). This is less than the divisor (13). There is no way we can subtract any nonzero multiple of 13 from 7. Therefore we leave a blank space above the 7 and proceed to the next digit. We could equally write a zero above the 7, as shown in gray in figure 3, but that’s just extra work and confers no lasting advantage, since leading zeros are meaningless.
Now consider the first two digits together, namely the 7 and the 5. Look at the crib to find the largest entry less than or equal to 75. It is 65, as in 5×13=65. Copy this entry to the appropriate row of the division problem, namely row b. Align it with the 75 on the previous row. Since this is 5 times the divisor, write a 5 on the answer line, aligned with the 75 and the 65, as shown. Check the work, to see that 5 (on the answer line) times 13 (the divisor, on line a), equals 65 (on line b).
Now do the subtraction, namely 75−65=10, and write the result on line c as shown.
When we take place value into account, the 5 in the quotient is short for 500, and the 65 that we just subtracted is short for 6500. Normally we would need to write the zeros explicitly as part of these numbers, to indicate place value. However, in this situation we are using columns to keep track of place value. If we keep everything properly lined up, we don’t need to write the zeros. In fact, the zeros would just get in the way (especially in the quotient) so we are better off not writing them.
We now shift attention to the situation shown in figure 4. Bring down the 2 from the dividend, as shown by the red arrow. So now the number we are working on is 102, on line c.
The steps from now on are a repeat of earlier steps.
Look at the crib to find the largest entry less than or equal to 102. It is 91, as in 7×13=91. Write copy this entry to the division problem, on row d, directly under the 102. Since this came from row 7 of the crib, write a 7 on the answer line, aligned with the 102 and the 91, as shown. Check the work, to see that 7 (on the answer line) times 13 (the divisor, on line a), equals 91 (on line d). Do the subtraction.
As a check on the work, when doing this subtraction, the result should never be less than zero, and should never be greater than or equal to the divisor. Otherwise you’ve used the wrong line from the crib, or made an arithmetic error.
We now shift attention to the situation shown in figure 5. Bring down the 7, look in the crib to find the largest entry less than or equal to 117, which is in fact 117, as in 9×13=117. Since this came from line 9 of the crib, write a 9 on the answer line, properly aligned.
The final subtraction yields the remainder on line g. The remainder is zero in this example, because 13 divides 7527 evenly.
The crib plays several important roles.
Perhaps the crib’s most important advantage, especially when people are first learning long division, is that the crib removes the mystery and the guesswork from the long division process. This is in contrast to the “trial” method, where you have to guess a quotient digit, and you might guess wrong. Using the crib means we never need to do a short division or trial division; all we need to do is skim the table to find the desired row.
We have replaced trial division by multiplication and table-lookup. Actually we didn’t even need to do any multiplication, so it would be better to say we have replaced trial division by addition.
Another advantage is efficiency, especially when the dividend has many digits. That’s because you only need to construct the crib once (for any given divisor), but then you get to use it again and again, once for each digit if the dividend. For long dividends, this saves a tremendous amount of work. (This is not a good selling point when kids are just learning long division, because they are afraid of big multi-digit dividends.) Setting up the crib is so fast that you’ve got almost nothing to lose, even for small dividends.
Another advantage is that it is easy to check the correctness of the crib. It’s just sitting there begging to be checked.
When bringing down a digit, you may optionally bring down all the digits. For instance, in figure 4, if you bring down all the digits you get 1027 on row c. One advantage is that 1027 is a meaningful number, formed by the expression 7527−13×5×100. This shows how the steps of the algorithm (and the intermediate results) actually have mathematical meaning; we are not not blindly following some mystical mumbo-jumbo incantation. I recommend that if you are trying to understand the algorithm, you should bring down all the digits a few times, at least until you see how everything works.A small disadvantage is that bringing down all the digits requires more copying. The countervailing small advantage is that it may help keep the digits lined up in their proper columns. Whether the advantages outweigh the cost is open to question, and probably boils down to a question of personal preference.
Another remark: Division is the “inverse function” of multiplication. In a profound sense, for any function that can be tabulated, you can construct the inverse function – if it exists – by switching columns in the table. That is, we interchange abscissa and ordinate: (x,y)↔(y,x). That’s why we are able to perform division using what looks like a multiplication table; we just use the table backwards.
Copyright © 2007 jsd
