Just as the decimal system with its ten digits is a base-ten system, the octal number system with its 8 digits, ‘0’, ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, ‘6’ and ‘7’, is a base-eight system. Table 2-4 shows the weighting for the octal number system up to 3 decimal places before and 2 decimal places after the octal point (.).

 

Weights 82 81 80 . 8-1 8-2

Table 2-4 Octal Weights

Just like the other counting conventions discussed previously, the LSB is begins with zero (0) and is incremented until the maximum digit value is reached. The adjacent bit positions are then filled appropriately as the iterative counting process continues. Thus the counting convention for octal is 0,1,2,3,4,5,7,10,11,12,13,14,15,16,17,20….

 

Conversion from Octal to Decimal

To express the value of a given octal number as its decimal equivalent we just need to sum the digits after each has been multiplied by its associated weight.

Example #1

Convert (237.04) 8 to decimal form.

 

Weights 82 81 80 8-1 8-2

Weight Value

64 8 1 0.125 0.015625

Octal Number

2 3 7 0 4

Decimal Value

128 24 7 0 0.0625 Total (159.0625)10

 

Conversion from Decimal Whole Numbers to Octal

To convert from Decimal whole numbers to Octal we may use the systematic approach called the Repeated-Division-by-8 method shown in the example below.

Converting (359) 10 to Octal

a. Divide the quotient by eight and record the remainder.

b. Repeat step (a) until the quotient is equal to zero (0).

c. The first remainder produced is the LSB in the octal number and the last remainder (R) the MSB. Accordingly, the octal number is then written (from left to right) with the MSB occurring first

8 359

8 44 R 7 (LSB)

8 5 R 4

8 0 R 5 (MSB)

Therefore, (359) 10 = (547) 8

Converting Decimal Fractions to Octal

The techniques used to convert decimal fractions to octal are similar to the methods demonstrated previously to convert decimal fractions to binary numbers. We may either use the sum-of–weights method or the repeated multiplication–by-8 method. In the multiplication–by-8 method we repeatedly multiply the fraction by eight, and record the carry, until the fraction product is zero. The first carry produced is the MSB, while the last carry is the LSM. Remember that the octal point precedes the MSB. To illustrate lets consider the conversion of (0.3125) 10 to octal.

0.3125 * 8 Carry

0.5 * 8 2 (MSB)

0.0 4 (LSB)

thus, (0.3125) 10 = (0.24) 8

 

Converting Octal to Binary

The primary application of octal numbers is representing binary numbers, as it is easier to read large numbers in octal form that in binary form. Because each octal digit can be represented by a three-bit binary number (see Table 2-5), it is very easy to convert

from octal to binary. Simply replace each octal digit with the appropriate three-bit binary number as indicated in the examples below.

 

Octal Digit

Binary Digit

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111

Example #1 Table 2-5 Octal and Binary Numbers

 

13 8 = (001011) 2

(37.12) 8 = (011111. 001010) 2

 

Converting Binary to Octal

Converting binary to octal is also a simple process. Break the binary digits into groups of three starting from the binary point and convert each group into its appropriate octal digit. For whole numbers, it may be necessary to add a zero as the MSB in order to complete a grouping of three bits. Note that this does not change the value of

the binary number. Similarly, when representing fractions, it may be necessary to add a trailing zero in the LSB in order to form a complete grouping of three.

 

Examples:

1. Converting (010111) 2 to Octal

111 = 7 (LSB)

010 = 2 (MSB)

thus, (010111) 2 = (27) 8

2. Converting (0.110101) 2 to Octal

110 = 6 (MSB)

111 = 5 (LSB)

thus, (0.110101) 2 = (0.65) 8