Multiplication by Doubling and Halving
There is a fascinating method of multiplication which involves only multiplication and division by two. You are even allowed to ignore remainders. The example below shows the multiplication of 37 by 53.
Put the two numbers to be multiplied at the top of two columns. Halve the numbers on the left until 1 is reached and double the numbers on the right the same number of times. Cross out any even number on the left and the corresponding numbers on the right.
To see how this method works look at the remainders in the halving process. These are 1, 0, 1, 0, 0 and the final 1.
This tells us what 37 is in binary notation.
3710 = 1001012
It follows that 37 = 25 + 22 + 20 = 32 + 4 + 1.
Now add up the numbers on the right which are not asterisked and you should have the answer to 37 × 53.
| 37 | 53 | |
| 18* | | 106 * |
| 9 | 212 | |
| 4* |
| 424 * |
| 2* |
| 848* |
| 1 | 1696 | |
 | ||
| 1961 |
| So 37 × 53 | = 25 × 53 + 22 × 53 + 20 × 53 |
| = 32 × 53 + 4 × 53 + 1 × 53 | |
| = 1696 + 212 + 53 = 1961 |
These are, of course, the numbers added on the right.
Pat Perkins
