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To convert an integer from the octal system to the decimal system, it is necessary only to write the octal number in expanded form; then use ordinary arithmetic to find the sum as you have been doing.
Well, we’ve tried to go easy on the arithmetic in our examples, but we are running out of simple examples. Get out pencil and paper and check the following conversions from octal to decimal. Which group contains a mistake?
Answer :
548
= 4410
2258
= 14910
7108
= 45610
1, 0008
= 51210⎫⎪⎪⎬⎪⎪⎭ This group contains a mistake.
By re-arranging the expanded form of an octal number, we discover a more economical method of conversion. Consider, for example, the octal number 372618. This is, expanded,
3(84) + 7(83) + 2(82) + 6(81) + 1(80),
which may be rewritten in the form
(((3 × 8 + 7) × 8 + 2) × 8 + 6) × 8 + 1.
This is simply evaluated as follows: take the first digit on the left of the octal number to be converted (the most significant digit: in this case 3), multiply by eight and add the next digit (here 7); multiply this result by eight, add in the next digit and continue in this way until the last digit (that on the extreme right, or least significant digit) is added in. Notice that the last operation is always an addition. In the case of 372618, we have:
