Devices for Multiplication
Suppose that we have to multiply 372 by 465. With the ordinary school method, we write 465 under the 372 and proceed about as follows: 5 times 2 is 10, put down the 0 and carry the 1; 5 times 7 is 35, 35 and 1 is 36, put down the 6 and carry the 3; 5 times 3 is 15, 15 and 3 is 18, put down the 8 and carry the 1;... The method is based mainly on a well-learned subroutine of continually changing steps:
1. Select a multiplicand digit.
2. Select a multiplier digit.
3. Refer to the multiplication table with these digits.
4. Obtain the value of their product, called a partial product.
5. Add the preceding carry.
6. Set down the right-hand digit.
7. Carry the left-hand digit.
We can, however, simplify this subroutine for a machine by delaying the carrying. We collect in one place all the right-hand digits of partial products, collect in another place all the left-hand digits, and delay all addition until the end.
For example, let us multiply 372 by 465 with this method:
| Right-Hand Digits | Left-Hand Digits | Usual Method for Comparison |
|---|---|---|
| 372 | 372 | 372 |
| × 465 | × 465 | × 465 |
| 550 | 131 | 1860 |
| 822 | 141 | 2232 |
| 288 | 120 | 1488 |
| 37570 | 13541 | 172980 |
| Final Addition | ||
| 37570 | ||
| + 13541 | ||
| 172980 | ||
37570 is called the right-hand component of the product. It is convenient to fill in with 0 the space at the end of 13541 and to call 135410 the left-hand component of the product.
This process is called multiplying by right- and left-hand components. It has the great advantage that no carrying is necessary to complete any line of the original multiplications. Some computing machines use this process. Built into the hardware of the machine is a multiplication table up to 9 × 9. The machine, therefore, can find automatically the right-hand digit and the left-hand digit of any partial product. In a computing machine that uses this process, all the left-hand digits are automatically added in one register, and all the right-hand digits are added in another register. The only carrying that is needed is the carrying as the right-hand digits are accumulated and as the left-hand digits are accumulated. At the end of the multiplication, one of the registers is automatically added into the other, giving the product.
Another device used in computing machines for multiplying is to change the multiplier into a set of digits 0 to 5 that are either positive or negative. For example, suppose that we want to multiply 897 by 182. We note that 182 equals 200 minus 20 plus 2, and so we can write it as
222.
The minus over the 2 marks it as a negative digit 2. Then to multiply we have:
897
222
1794
-1794
1794
163254
The middle 1794 is subtracted. This process is usually called short-cut multiplication. Everybody discovers this trick when he decides that multiplying by 99 is too much work, that it is easier to multiply by 100 and subtract once.