OCTAL NUMBER SYSTEM

The octal, or base 8, number system is a common system used with computers. Because of
its relationship with the binary system, it is useful in programming some types of
computers.

Look closely at the comparison of binary and octal number systems in table 1-3. You can
see that one octal digit is the equivalent value of three binary digits. The following
examples of the conversion of octal 225_{8} to binary and back again further
illustrate this comparison:

Table 1-3. - Binary and Octal Comparison

Unit and Number

The terms that you learned in the decimal and binary sections are also used with the
octal system.

The unit remains a single object, and the number is still a symbol used to represent
one or more units.

Base (Radix)

As with the other systems, the radix, or base, is the number of symbols used in the
system. The octal system uses eight symbols - 0 through 7. The base, or radix, is
indicated by the subscript 8.

Positional Notation

The octal number system is a positional notation number system. Just as the decimal
system uses powers of 10 and the binary system uses powers of 2, the octal system uses
power of 8 to determine the value of a number's position. The following bar graph shows
the positions and the power of the base:

Remember, that the power, or exponent, indicates the number of times the base is
multiplied by itself. The value of this multiplication is expressed in base 10 as shown
below:

All numbers to the left of the radix point are whole numbers, and those to the right
are fractional numbers.

MSD and LSD

When determining the most and least significant digits in an octal number, use the same
rules that you used with the other number systems. The digit farthest to the left of the
radix point is the MSD, and the one farthest right of the radix point is the LSD.

Example:

If the number is a whole number, the MSD is the nonzero digit farthest to the left of
the radix point and the LSD is the digit immediately to the left of the radix point.
Conversely, if the number is a fraction only, the nonzero digit closest to the radix point
is the MSD and the LSD is the nonzero digit farthest to the right of the radix point.

Addition of Octal Numbers

The addition of octal numbers is not difficult provided you remember that anytime the
sum of two digits exceeds 7, a carry is produced. Compare the two examples shown below:

The octal addition table in table 1-4 will be of benefit to you until you are
accustomed to adding octal numbers. To use the table, simply follow the directions used in
this example:

Add: 6_{8} and 5_{8}

Table 1-4. - Octal Addition Table

Locate the 6 in the *X* column of the figure. Next locate the 5 in the *Y*
column. The point in area *Z* where these two columns intersect is the sum.
Therefore,

If you use the concepts of addition you have already learned, you are ready to add
octal numbers.

Work through the solutions to the following problems:

As was mentioned earlier in this section, each time the sum of a column of numbers
exceeds 7, a carry is produced. More than one carry may be produced if there are three or
more numbers to be added, as in this example:

The sum of the augend and the first addend is 6_{8} with a carry. The sum of 6_{8}
and the second addend is 5_{8} with a carry. You should write down the 5_{8}
and add the two carries and bring them down to the sum, as shown below:

Now let's try some practice problems:

Q.24 Add:

Q.25 Add:

Q.26 Add:

Q.27 Add:

Q.28 Add:

Q.29 Add: