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 LEFT-HAND USUAL METHOD, DIGITS DIGITS 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.