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You will remember that when we first learned to convert octal numbers to decimal numbers, we did so by expansion (528 = 5(8) + 2(1) = 42). However, octals can also be converted to decimals by the same methods we used for converting decimals to octals. (We used repeated division by 8 for whole numbers, repeated multiplication by 8 for decimal fractions.) But to convert from octal to decimal the arithmetic must be done in octal, since we begin with an octal number. And in converting to a system with a base of 10, we must multiply or divide by 10, which of course is 128 in octal notation.
There’s no particular advantage to converting octal whole numbers to decimals by this method, since dividing by 128 is a clumsy operation. However, multiplying by 128 is not difficult. A decimal value for an octal fraction is easily obtained by repeatedly multiplying by 128.
Let’s begin by converting .2568 to decimal by this repeated multiplication by 128.
There are two partial products, of course, since we are multiplying by a two-digit number. The integral part, 3, of the first complete product is the first digit of the decimal fraction corresponding to .2568. Now, multiply the fractional part only of this first complete product by 128 to get the next digit of the decimal fraction. What is the next digit of the decimal fraction?
