Octal Number System
The Octal Number System is another type of computer and digital numbering system which uses the Base-8 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.
Octal Numbers 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.
Octal Numbers Example No2
Convert the octal number 23228 to its decimal number equivalent, (base-8 to base-10).
| Octal Digit Value | 23228 |
| In polynomial form | = ( 2×83 ) + ( 3×82 ) + ( 2×81 ) + ( 2×80 ) |
| 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.