ARITHMETIC CALCULATION WITH DECIMALS

Arithmetic calculation with decimals Addition and subtraction Numbers in decimal notation can easily be added by adding the coefficients of corresponding powers of 10 and then adjusting, by the process known as carrying, the coefficients that exceed 9. A typical example (see Box 5, item 36) would be the addition of the numbers 47.65, 5.473, and 649.8. One first adds coefficients of equal powers of 10; e.g., one adds 5 and 7 to obtain 12 as a coefficient of 10-2. After adjustment of the coefficients (see Box 5, item 37), this sum becomes 702.923. The process of addition is commonly carried out as follows (see Box 5, item 38): Beginning at the right, the sum of the coefficients of 10-3, which is 3, is written below the answer line; the sum of the coefficients of 10-2 is 12, the digit 2 being placed to the left of the 3 in the sum; however, the 10 becomes a 1 in the preceding position and may be placed at the top of the column of coefficients of 10-1, etc. In practice the carrying is usually performed mentally. Subtraction is performed by reversing the above procedure. Thus, to subtract 170.8 from 563.142 (see Box 5, item 39), for example, the smaller number is placed under the larger, with the respective decimal points aligned vertically. Again, starting from the right, since 0.8 = 0.800, 0 subtracted from 2 leaves 2 (which is written below the line), and 0 from 4 leaves 4; but since 8 exceeds 1, a unit is borrowed from the unit's place (which changes the 3 to 2), and thus the 1 in the tenth's place becomes 11; then 8 subtracted from 11 leaves 3, 0 from 2 leaves 2, and so forth. Another way of performing subtractions consists in finding the number, digit by digit, that, when added to the second, yields the first as sum. Multiplication Multiplication of decimal numbers is based upon the distributive law. Thus, the multiplication (see Box 5, item 40) of 25.78 by 7 is accomplished as a sum of four products, each involving 7 as a factor. By a process known as short multiplication, this may be carried out as follows (see Box 5, item 41): Since 7 8 = 56, the 6 is written below the answer line, starting at the right, and the 5 is placed above the 7 in the multiplicand; then 7 7 = 49, to which is added the 5 that was carried, making 54; the 4 is written in the answer, the 5 being carried as before. The numbers written above the top line are usually carried mentally. If the multiplier has more than one digit, long multiplication is used (see Box 5, item 42). This process also is based upon the distributive law. The first partial product is 18.046, which was obtained by multiplying 25.78 by .7 by short multiplication. The second partial product is 25.78 5 = 128.90; the third is 25.78 40 = 1,031.2. The product is the sum of these partial products. It is customary to ignore the decimal point in these partial products but to indent each partial product one more place than the partial product above it. In the product as many digits are pointed off from the right as the sum of the number of digits to the right of the decimal point in the multiplicand and the number of digits to the right of the decimal point in the multiplier.