You are correct. The complete number is actually
.2568 = .3398437510.
Now, recall that we converted octal whole numbers to decimal whole numbers by expansion. It is possible to convert octal fractions into decimal fractions by expansion too, but the decimals get so long they become a nuisance. For example,
.2568 =
2(8 − 1) + 5(8 − 2) + 6(8 − 3)
=
2⎛⎝(1)/(8)⎞⎠ + 5⎛⎝(1)/(64)⎞⎠ + 6⎛⎝(1)/(512)⎞⎠
=
2(.125) + 5(.015625) + 6(.001953125)
=
.3398437510.
It’s easier to learn to multiply by 128, as we did in this lesson, than it is to juggle decimals such as .001953125.
You have seen that to add octal digits having a sum of 78 or less, we use the same combinations we learned in school arithmetic. You remember that you memorized additions of single-digit decimal numbers such as 2 + 2 = 4, 2 + 3 = 5, 2 + 4 = 6, and so on. We can use these combinations for octal sums up to 78 because numbers up to seven are written the same way in both systems.
To add single-digit octal numbers having a sum greater than
78 but not exceeding
178, we developed the rule: Add the digits as decimal digits then add 2 to get the digits of the octal sum. Thus
68 + 78 = (6 + 7 = 13, 13 + 2 = 15) = 158.
Remember, the octal counting sequence goes: 18, 28, 38, 48, 58, 68, 78, 108, 118, 128, 138, 148, 158, 168, 178, 208, 218, and so on.
The addition of two multiple-digit octal numbers can be accomplished in accordance with the same rule of adding 2 to the decimal sum. That is, since no two single octal digits can result in a sum greater than 168, the addition of a carry of 1 increases the sum to 178, which is within the limits of the application of the rule.
Octal multiplication, like decimal multiplication, is basically a series of additions:
28 × 68 =
68 + 68 = 148;
58 × 58 =
58 + 58 + 58 + 58 + 58
= 58
+ 58
128
+ 58
178
+ 58
248
+ 58
318
The most important thing to understand about multiplication in octal is multiplication by 128, since this is the multiplier used to convert octal fractions to decimals. You remember that we multiplied by 128 by using the octal multiplication table for reference, multiplying first by 2 and then by 1 and adding the partial products.
Multiplication by 128 can also be considered, of course, as multiplying first by 108 and then by 28 and adding the products of these separate multiplications. We sometimes use this method in decimal arithmetic.
Methods of converting octals to decimals and decimals to octals can be summed up as follows:
|
CONVERSION
|
METHOD
|
|
Octal to decimal whole numbers
|
Expansion
|
|
Octal to decimal fractions
|
Repeated multiplication by 128
|
|
Decimal to octal whole numbers
|
Repeated division by 8
|
|
Decimal to octal fractions
|
Repeated multiplication by 8
|
If you feel a little uncertain about any of these methods, you had better put a bookmark here and work through a few more examples before going on.
When you are ready, turn to the Self-Test Questions for Lesson 8 in the Appendix.
