Octal Numbers

Octal Number System

The Octal Number System is another type of computer and digital base number system. The Octal Numbering System is very similar in principle to the previous hexadecimal numbering system except that in Octal, a binary number is divided up into groups of only 3 bits, with each group or set of bits having a distinct value of between 000 (0) and 111 ( 4+2+1 = 7 ). Octal numbers therefore have a range of just "8" digits, (0, 1, 2, 3, 4, 5, 6, 7) making them a Base-8 numbering system and therefore, q is equal to "8".

Then the main characteristics of an Octal Numbering System is that there are only 8 distinct counting digits from 0 to 7 with each digit having a weight or value of just 8 starting from the least significant bit (LSB). In the earlier days of computing, octal numbers and the octal numbering system was very popular for counting inputs and outputs because as it works in counts of eight, inputs and outputs were in counts of eight, a byte at a time.

As the base of an Octal Numbers system is 8 (base-8), which also represents the number of individual numbers used in the system, the subscript 8 is used to identify a number expressed in octal. For example, an octal number is expressed as:  237₈

Just like the hexadecimal system, the "octal number system" provides a convenient way of converting large binary numbers into more compact and smaller groups. However, these days the octal numbering system is used less frequently than the more popular hexadecimal numbering system and has almost disappeared as a digital base number system.

Representation of an Octal Number

MSB	Octal Number							LSB
88	87	86	85	84	83	82	81	80
16M	2M	262k	32k	4k	512	64	8	1

As the octal number system uses only eight digits (0 through 7) there are no numbers or letters used above 8, but the conversion from decimal to octal and binary to octal follows the same pattern as we have seen previously for hexadecimal.

To count above 7 in octal we need to add another column and start over again in a similar way to hexadecimal.

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21....etc

Again do not get confused, 10 or 20 is NOT ten or twenty it is 1 + 0 and 2 + 0 in octal exactly the same as for hexadecimal. The relationship between binary and octal numbers is given below.

Octal Numbers

Decimal Number	3-bit Binary Number	Octal Number
0	000	0
1	001	1
2	010	2
3	011	3
4	100	4
5	101	5
6	110	6
7	111	7
8	001 000	10 (1+0)
9	001 001	11 (1+1)
Continuing upwards in groups of three

Then we can see that 1 octal number or digit is equivalent to 3 bits, and with two octal number, 77₈ we can count up to 63 in decimal, with three octal numbers, 777₈ up to 511 in decimal and with four octal numbers, 7777₈ up to 4095 in decimal and so on.

Example No1

Using our previous binary number of 1101010111001111₂ convert this binary number to its octal equivalent, (base-2 to base-8).

Binary Digit Value	001101010111001111
Group the bits into three´s starting from the right hand side	001 101 010 111 001 111
Octal Number form	1 5 2 7 1 78

Thus, 001101010111001111₂ in its Binary form is equivalent to 152717₈ in Octal form or 54,735 in denary.

Example No2

Convert the octal number 2322₈ to its decimal number equivalent, (base-8 to base-10).

Octal Digit Value	23228
In polynomial form	= ( 2x83 ) + ( 3x82 ) + ( 2x81 ) + ( 2x80 )
Add the results	= ( 1024 ) + ( 192 ) + ( 16 ) + ( 2 )
Decimal number form equals:   123410

Then, converting octal to decimal shows that  2322₈ in its Octal form is equivalent to 1234₁₀ in its Decimal form.

While Octal is another type of digital numbering system, it is little used these days instead the more commonly used Hexadecimal Numbering System is used as it is more flexible.