Operations on Decimal Fractions 2

1. We multiply each of the numbers with suitable powers of ten, so that they become whole numbers.

   1. 245.45 is multiplied by 100 or 102 & 365.898 is multiplied by 1000 or 103. Notice that the power of each 10 is equal to the number of effective decimal places in the number.

   2. Hence the product has been multiplied by 100 X 1000 or 105. This again is equal to the sum of the decimal places of the individual numbers.

   3. It is very easy to see that 102 X 103 = 10^2+3 by the rule of multiplication of exponentials

2. The product of multiplication is 8980966410

3. We now divide this by 105. This is easy because of the place value notation and means shifting the decimal point to the left by 5 places.

4. Hence 8980966410 converts to 89809.66410

The procedure can be further simplified into

1. Find the total of the decimal places of each number.

2. Multiply both the numbers, ignoring the decimal point.

3. Shift the decimal point to the left as many places as the total arrived at in Step 1.

Treatment of 0 at the right extreme in any of the numbers being multiplied

If the decimal places in a number take into account 0s on the right, then the whole number should retain the 0s.

1. If the decimal places in 245.450 are taken as 3, then the whole number equivalent should be taken as 245450.

2. If the decimal places in 245.450 are taken as 2 (taking the number as 245.45, then the whole number equivalent should be taken as 24545.

Treatment of 0 at the right extreme of the product

In the same example we saw that the product had a 0 in the right extreme. Since the product is a whole number, the 0 cannot be disturbed. Hence while shifting the decimal point 5 places to the left, the 0 was counted as one of the places.

After shifting the decimal point (since it is now a decimal fraction) the trailing 0s can be left out.

Hence 89809.66410 could be written as 89809.6641

Division

The Divisor and the Dividend are multiplied by a suitable power of 10, so that at least the Divisor becomes a whole number. Since both are multiplied by the same number, the result is not affected. Then the problem reduces to that of dividing either a whole number or a decimal fraction by another whole number.