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Now consider the two expressions we have been discussing:
1(82) + 3(81) + 2(80),
2(82) + 1(81) + 3(80).
These two expressions are, of course, octal numbers written in expanded form. That is, the numbers are written as sums of multiples of powers of 8. The powers of 8 are arranged in order just as powers of 10 are arranged in order in an expanded decimal number. You have just seen that you can convert expanded octal numbers into decimal numbers by simple arithmetic.
Now, the two expressions above differ only in the coefficients. If we know that the coefficients are to be used to multiply powers of 8, we don’t need to write out the powers of 8, just as we do not write out the powers of 10 in an ordinary decimal number. We can write only the coefficients, just as we do in the decimal system. However, we must identify the coefficients as belonging to an octal number. We do this by writing an 8 halfway below the line following the digits of the octal number as shown below. (To avoid confusion we have identified decimal numbers with a 10 subscript.) So we may write, using our two examples above.
The 1328 and the 2138 are the regular octal numbers which stand for the same quantities as 9010 and 13910, respectively. Just as the digits in 9010 and 13910 are coefficients of powers of 10, the digits in 1328 and 2138 are coefficients of powers of 8.
In the octal number 2138, how do we know which power of 8 the 3 is supposed to multiply?
Answer :
By the power of 8 it is shown multiplying in the number.
