Continued from Last Week's Post.
This is the continuation of earlier Articles:
1. Learn the Binary Number System.
2. Learn Binary Number System-2.
Please refer to the earlier Articles before continuing.
We will bring forward the result of the Decimal (Base 10) Number 255 converted to Binary 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:
Later, programming languages like Assembly Language were 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 number system was devised to write binary numbers in short form.
Octal Number System.
The Octal number system has the number 8 as its base and is known as the Octal Numbers. Based on the general rule that we have learned, Octal Numbers have digits 0 to 7 (one less than the base value 8) to write numerical quantities. Octal Numbers don't have digits 8 or 9. This Number System has been devised to write Binary Instructions for Computers in a shorter 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, the next three bits (010) represent CPU Register B, and the next three bits (111) represent CPU Register A. The 8-bit binary instruction says to add the contents of register A to register B. If the instruction must be changed to (ADD A, B), add the contents of register B to A, then the last six bits must be altered to 00,111,010. This can be easily understood if it is written in Octal form 027 to 072 rather than Binary 00111010.
So, the Octal (Base-8) Number System was devised to write binary-based instructions in short form. Coming back to the Octal Number System, let us consider how we can work with these numbers. First, we will create a table similar to the decimal/binary Number 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 an Octal Number.
For binary-to-decimal conversion, we could simply take the highest integer value from the table and subtract it from 255 to begin the process. However, that approach does not work in this case. Looking at the table, we can see that 512 is greater than 255, so it cannot be used. The next lower value is 64, and our task is to determine how many times 64 can fit 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 Unit's 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 |
Writing Binary to Octal Short Form.
As I mentioned earlier, the Octal Number System has been devised to write Binary Numbers in short form. Let us see how we can do this and convert binary numbers easily into Octal numbers.
The Decimal Number 255, when converted into Binary, we get 11111111. To convert it into Octal Numbers, organize the binary digits into groups of three bits (011,111,111) from right to left and add up the binary values of each group, and write the Octal value.
011 = 1+2 = 3
111 = 1+2+4 = 7
111 = 1+2+4 = 7
Result: = 377 Octal.
To get a better grasp of this Number System, try converting a few more numbers on your own. Start by converting some decimal numbers into binary, then group the binary digits into sets of three bits. Next, calculate the value of each group as though they represent the first three bits of a binary number.
Since octal numbers use digits 0 through 7, they can easily be mistaken for decimal numbers by both humans and machines. To avoid confusion, octal numbers are always written with a prefix. In MS-Access (and VBA), the prefix is &O (the letter O, not case-sensitive) or &0 (digit zero). For example, the octal number 377 can be written as &O0377, &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 the Base-16 (Hexadecimal) Number System.
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