System 001

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Conversion: Decimal to Octal?

How do I convert Octal to Decimal, and Decimal to Octal???
Please explain.

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This method will be very clear once you practice with a couple of numbers. I learned how to convert number systems several ways and THIS IS THE EASIEST, THE CLEAREST...so you can remember years later.

First convert octal to decimal:
The octal number system consists of digits that multiply powers of 8. For example, the octal number 172 means 1 X 8² + 7 X 8¹ + 2 X 8º . If you add these up, you have the decimal representation of the octal number 172.

1 X 8² = 1 X 64 = 64 ... ... ... ... ... ... ... ..THIS IS
7 X 8¹ = 7 X 8 = 56 ... ... ... ... ... ... ... ... OCTAL (172)
2 X 8º = 2 X 1 = 2 ... ... ... ... ... ... ... ... ...TO DECIMAL (122)
-----------------------------
... ... ... ... ... ... 122
Add the decimal numbers on the right of the = sign above:
64 + 56 + 2 = ***122***, the decimal representation of octal 172.

Now convert decimal to octal by first converting decimal to binary, then binary to octal, because you can understand it better than plugging into a formula: (To convert between ANY number systems, remember that you can always convert to binary first. This may seem like an extra step or two, but if you practice it a few times, you will always remember it.)

Example: Convert decimal number 122 to octal.
[This means convert 1 X 10² + 2 X 10¹ + 2 X 10º...to...
(1st octal digit) X 8² + (2nd octal digit) X 8¹ + (3rd octal digit) X 8º.]

Solution: Write 122 in binary.
64 32 16 8 4 2 1 <== Binary Heading.
.1... 1...1..1.0.1.0 <== Binary representation of dec #122. *Line 1*
[ Ignore the periods, ... . I use them for spacing in this word processor.]

Now group the binary representation into groups of 3, starting from the right, adding zeros to the left to make three digits if there are less than three, because any octal digit, 0 - 7, can be represented by three binary digits. ( You use groups of three to represent octal digits because the largest digit in the octal system is 7 and that is
1 X 2² + 1 X 2¹ + 1 X 2º = 4 + 2 + 1 = 7 in the binary system. )

001 111 010 <=Binary number from *Line 1* above grouped into 3s.
Now put your little binary heading (4 2 1) above each group of three binary digits, and add up the 3-digit groups:

4 2 1 Binary Heading
0 0 1 = 1

4 2 1 Binary Heading
1 1 1 = 7

4 2 1 Binary Heading
0 1 0 = 2

The digits to the right of the equal signs form the octal number.

172 = Octal number representing decimal 122.<==***Answer***
1 X 8² + 7 X 8¹ + 2 X 8º ...=...1 X 10² + 2 X 10¹ + 2 X 10º
... ... (172 octal)... ... ... ... ... = ... ... (122 decimal)

Practice, practice will make this clear as a bell. /
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