Octal Numbering System

Posted By a.p.r.pillai On 23 Sep 2010. Under How Tos Tags: Octal Numbers

This is the continuation of earlier Articles:

Please go through the earlier Articles before continuing. . .

We will bring forward the result of the Decimal 255 to Binary conversion for a closer look at these two numbers: 11111111.

Decimal Number 255 needs only 3 digits to write this quantity but when we convert it into binary it needs 8 binary digits or bits. Earlier computer programs were written using binary instructions. Look at the example Code given below:

Source: www.wikipedia.org

Later programming language like Assembly Language was developed using Mnemonics (8 bit based binary instructions) like ADD, MOV, POP etc. Present day Compilers for high-level languages are developed using Assembly Language. A new numbering system was devised to write binary numbers in short form.

This numbering system has number 8 as its base and is known as Octal Numbering System. Based on the general rule that we have learned, Octal Numbering System have digits 0 to 7 (one less than the base value 8) to write numerical quantities. There is no 8 or 9 digits in this numbering system. Octal Numbering System has been devised to write Binary instructions into short form and to write program codes easily.

For example an 8 bit Binary instruction looks like the following:

00010111 (instruction in Octal form 027), ADD B,A (Assembly Language)

The first two bits (00) represent the operation code ADD, next three bits (010) represents CPU Register B and next three bits (111) represents CPU Register A, the 8 bit binary instruction says add the contents of register A to register B. If the instruction must be changed to (ADD A,B) add register B to A then the last six bits must be altered as 00,111,010. This can be easily understood if it is written in Octal 027 to 072 rather than Binary 00111010.

So, Octal (Base-8) Numbering System was devised to write binary based instructions in short form. Coming back to the Octal Numbering System, let us see how we can work with these numbers. First, we will create a table similar to the decimal/binary numbering Systems.

8⁵	8⁴	8³	8²	8¹	8⁰
32768	4096	512	64	8	1

We will use the same methods we have used for Binary to convert Decimal to Octal Numbers.

Example: Converting 255 into Octal Number?

For Binary conversion we could take the highest integer value from the table above and subtract it from 255. But here we cannot do that. By looking at the table above we can see that 512 is more than 255, so we must see next number 64 goes how many times into 255?

Method-1:

255/64= Quotient=3, Remainder=63 (Here, we have to take the Quotient as the Octal Digit).

In this method we must take Quotient 3 ( 64 x 3 = 192) for our result value and the balance is 63 (i.e. 255 - 192)

8⁵	8⁴	8³	8²	8¹	8⁰
32768	4096	512	64	8	1
			3

63/8 = Quotient = 7, Remainder=7

8⁵	8⁴	8³	8²	8¹	8⁰
32768	4096	512	64	8	1
			3	7

7 is not divisible by 8, hence 7 goes into the Units position

8⁵	8⁴	8³	8²	8¹	8⁰
32768	4096	512	64	8	1
			3	7	7

Method-2:

255/8 = Quotient = 31, Remainder=7

8⁵	8⁴	8³	8²	8¹	8⁰
32768	4096	512	64	8	1
					7

31/8 = Quotient = 3, Remainder=7

8⁵	8⁴	8³	8²	8¹	8⁰
32768	4096	512	64	8	1
				7	7

3 is not divisible by 8, hence it is taken to the third digit position.

8⁵	8⁴	8³	8²	8¹	8⁰
32768	4096	512	64	8	1
			3	7	7

As I mentioned earlier that Octal Numbering System has been devised to write Binary Numbers into short form. Let us see how we can do this and convert binary number easily into Octal Number.

The Decimal Number 255 when converted into Binary we get 11111111. To convert it into Octal Numbers set the binary digits into groups of three bits (011,111,111) from right to left and add up binary values of each group and write the Octal value.

011 = 3

111 = 7

You may try converting few more numbers of your own to understand this Numbering System better. First, convert some Decimal Numbers to Binary and group the Binary Digits in sets of 3 bits and add up the values of the group as if they are the first three bits of binary number.

Since, Octal Numbers are written with digits 0 to 7, it looks like decimal numbers to both man and machine. There must be a way to distinguish Octal Numbers from decimal numbers. Therefore it is written with the prefix characters &O. When Octal numbers are written alone or used in expressions it is written with the prefix characters &O (the letter O, not case sensitive) or &0 (digit zero), like &O0377 or &O377 or &0377.

You can try this by typing the above number in the Debug Window in the VBA Editing Screen of Microsoft Access or Excel.

Examples:

? &O0377

result: 255

? &0377

result: 255

? &0377 * 2

result: 510

Next we will learn Base-16 (Hexadecimal) Numbering System.

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