Number Systems - Octal

Base-8 (octal)

Octal is the least frequently use of these alternative number systems, but it has some interesting properties. One common usage is for dealing with file permissions in the Unix filesystem.

Theory

• Uses the symbols 0, 1, 2, 3, 4, 5, 6, and 7
• Digits carry over to the next place when 7 becomes 0
• One digit can represent eight unique numbers
• Two digits can represent 64 unique numbers
• Moving right to left, positions represent:
  • 8^0 = 1
  • 8^1 = 8
  • 8^2 = 64
  • 8^3 = 512

Comprehension: How many more unique numbers can be represented in four decimal digits versus four octal digits?

Counting

In your notebook write a header Octal and create a table with two columns. In the left column put the decimal numbers 0 through 20. In the right column record the octal equivalents that you find through the counting process below.

Follow these steps/rules with your groupmates:

1. Start with all zeros across the viewer
2. Increment the rightmost strip
3. If the rightmost hits the 8, move the next strip to the left up one and move the rightmost back to 0.
4. If that second strip hits the 8, use the same method to increment the third strip and move the second back to 0.
5. Do the same for the third and fourth strips
6. Record the “output” number you have after completing 2-5, then repeat until your table is full.

Do you get it? Try each of these counts:

• Count up three decimal values from 627
• Count up eight decimal values from 767
• Count down four decimal values from 123
• Count down six decimal values from 604

Conversions

Now let’s practice direct conversions in your notebook. You can choose to collaborate or do this solo:

From Octal to Decimal

To convert octal to decimal:

• Start from the right
• Multiply the digit in that place by the power of 8 corresponding to that place
• Add the results together

For example, say you have 1267:

7 * 8^0 = 7
6 * 8^1 = 48
2 * 8^2 = 128
1 * 8^3 = 512

Total = 7 + 48 + 128 + 512 = 695 in decimal

From Decimal to Octal

• Take the whole decimal number
• Divide by eight
• Note the quotient and the remainder
• Divide the quotient by 8, noting the new quotient and remainder
• Repeat until your quotient reaches zero
• Record the remainders from bottom to top

For example, say you have 1258:

1258 / 8 = 157 remainder 2
 157 / 8 =  19 remainder 5
  19 / 8 =   2 remainder 3
   2 / 8 =   0 remainder 2

Bottom to top it's 2352 in octal

Exercises - Conversion

1. Convert 712 decimal to octal
2. Convert 777 octal to decimal
3. Convert 1024 decimal to octal
4. Convert 3041 octal to decimal

Addition & Subtraction

You can convert back and forth to decimal and do your normal decimal addition/subtraction, but doing them right in octal is actually straight-forward.

You use the same rules as when doing addition in decimal:

• Start from the right
• Add the two digits together
• If you get a summed value less than eight, write it down
• If you get a summed value greater than eight, write down the value minus eight and carry a 1 to the next column to the left

For example, let’s add 456 and 153 like this:

  456
+ 153
-----

The rightmost 6 and 3 add together, exceed 8, so we carry a one and write the remainder 1.

   1
  456
+ 153
-----
    1

In the second column the previously carried 1 adds with the existing 5 and 5 to give eleven. Carry one to the next column and bring down the remaining 3:

  1
  456
+ 153
-----
   31

Then the leftmost 1, 4, and 1 add to 6:

  456
+ 153
-----
  631

Subtraction works the same way where you borrow from the left (so 1 becomes 10 aka eight in decimal).

Exercises - Addition & Subtraction

1. Add 777 to 111
2. Add 4531 to 3275
3. Subtract 131 from 765
4. Subtract 654 from 1421