How to Convert from Octal to Decimal?
To confirm that our decimal-to-octal conversion was correct, we’ll now take the octal number we obtained earlier and convert it back to decimal using the steps we just outlined.
Example: Let’s convert \(\mathsf{145.063_{8}}\) into base-10.
To begin the conversion, we separate the octal number into its whole and fractional parts. The digits before the octal point form the whole number part, while those after it make up the fractional part. Hence \(\mathsf{145_8}\) represents the whole number part, and \(\mathsf{063_8}\) is the fractional part.
The whole number conversion simply involves assigning index values to each digit, multiplying each digit by \(\mathsf{8}\) raised to its index, and then summing all the resulting products.
This gives us \(\mathsf{101_{10}}\), the decimal equivalent of \(\mathsf{145_{8}}\).
For the fractional part, the method is the same, except the indexing starts from the left with the first digit assigned an index of \(\mathsf{-1}\), decreasing further to the right.
The decimal equivalent of the fractional part, computed through this method, is \(\mathsf{0.0996_{10}}\).
After separately converting both the whole and fractional components using their respective methods, we combine the results to form the complete decimal representation. Thus, the decimal number \(\mathsf{145.063_{8}}\) is equivalently expressed in decimal as \(\mathsf{101.0996_{10}}\).
Did we Get it Right?
Now, if you remember, we originally got this octal number by converting \(\mathsf{101.101_{10}}\) from decimal. When we reversed the process, the result was close—but not exactly the same. That’s because that both the decimal and octal version had \(\mathsf{3}\) digits of precision in the fractional part. Since octal only uses eight symbols (\(\mathsf{0}\) through \(\mathsf{7}\)), it needs more digits to represent the same level of detail. So even with \(\mathsf{3}\) octal digits in the fractional part, we couldn’t fully capture the precision of \(\mathsf{3}\) decimal digits for this particular example.
However, we were able to capture more precision in \(\mathsf{3}\) digits of octal when compared with \(\mathsf{7}\) digits of binary here.