In this article we will discuss about:- 1. Binary Operation of a System 2. Octal Number System 3. Binary Representation of Octal Number 4. Conversion of Octal Number to Hexadecimal Number and Vice Versa 5. Conversion of Real Number 6. Conversion of Decimal Fraction to Binary Fraction 7. Advantages of Hexadecimal System over Octal System 8. Advantages of Octal Number System over the Decimal System.
Contents:
- Binary Operation of a System
- Octal Number System
- Binary Representation of Octal Number
- Conversion of Octal Number to Hexadecimal Number and Vice Versa
- Conversion of Real Number
- Conversion of Decimal Fraction to Binary Fraction
- Advantages of Hexadecimal System over Octal System
- Advantages of Octal Number System over the Decimal System
1. Binary Operation of a System:
The binary system of arithmetic has only two possible states to represent all quantities. These two states are either 0 (zero) or 1(one). Since 0 and 1 can be easily represented by two-state digital circuits, the binary operation finds wide use in computer technology. Thus, a hole in a card can represent a binary 1 while no hole is a binary 0.
ADVERTISEMENTS:
Similarly, the magnetic flux of a magnetic material can represent 1 in one direction and 0 in the opposite direction. In a computer, to transmit digits from one part to another, a binary 1 can be represented by a positive level and a binary 0 by a negative level or vice versa. Various designations are used, in general, for the two quantized states of a binary system, the most common of which are given in Table 5.8.
In logic a statement may be either true or false which the notation under 1 is. Similarly, a switch may be either closed or open as shown under notation 2 and so on.
A binary digit (either 0 or 1) is called a bit and a group of bits having a significance is called a code, word or byte.
ADVERTISEMENTS:
For explaining the binary numbers, let us start with the familiar decimal system based on the use of 10 digits.
Consider, as an example, the meaning of the decimal number 634 which can be written as follows:
It is seen, therefore, that in a number the individual digits represent coefficients for the increasing powers of 10. The farthest digit to the right is the coefficient of the 0th power, the next is the coefficient of the 1st power and so on. In general, any decimal number can be represented by the equation-
Where Y is the entire number, dn is the nth digit from the point and r is the radix or base. This equation can also be applied to the binary system.
Since only two digits (0 and 1) are available in a binary system, the individual digits represent the coefficients of powers of 2 rather than 10 as in the decimal system.
The binary number 10111, as an example, may be solved as follows:
This means the binary number 10111 represents the same quantity as the decimal number 23. A short list of decimal-binary equivalents is given in Table 5.9.
The order of a binary number is designated as 1, 2, 4, 8, 16, 32, 64, 128, 256, etc., and not as unit, tens, hundreds, thousands, etc., as in the decimal system. There are two methods for converting a decimal to a binary number.
ADVERTISEMENTS:
In the first method a table of powers of 2 is utilized. The decimal number to be evaluated is taken and the largest power of 2 from the table is noted without exceeding the original number. Then subtract the table-oriented number from the original number and repeat the process for the remainder until it becomes zero.
In the second method of converting decimal to binary, the number is successively divided by 2 and its remainders recorded. The final binary result is obtained by reading the remainders from the bottom to the top.
Addition of binary numbers is very simple. The addition of two binary numbers A and B can be done by using the Table 5.11. The Table shows all the possible combinations of A and B.
A digit, in the fourth entry of Table 5.10, is carried by shifting it to the next position to the left.
Binary numbers can also be subtracted, multiplied and divided using similar rules as applied for addition.
2. Octal Number System
:
The base of the octal number system is 8. The digits from 0 to 7 are same as those for the decimal number system. The number 8 of the decimal number system is represented by 10 in octal number system, similarly, 9 by 11 and so on. The octal representation of decimal number is given in Table 5.12.
To convert a decimal number to octal number similar process is applied as decimal to binary conversion but in this case the decimal number is to be divided by 8.
The following two examples illustrate the conversion of octal to decimal number:
3. Binary Representation of Octal Number
:
For binary representation of octal number, each octal number is represented by 3 binary bits. For example, the octal number 5 is represented by 101 while the octal 45 by 100101. The binary representation of octal numbers is shown in Table 5.13.
Conversion of Binary Number to Octal Number:
We know that the base of an octal number is 8. As 8 = 23, so for a conversion of binary number to octal number groups of 3 bits are formed in the binary number from right. After forming the groups each 3-bit binary group is replaced by its octal equivalent.
Conversion of Octal Number to Binary Number:
For the conversion each digit of the given octal number is converted to its 3-bit binary equivalent.
4. Conversion of Octal Number to Hexadecimal Number and Vice Versa
:
The conversion of an octal number to hexadecimal number and vice versa can be done through the binary method.
This is illustrated below by the following examples:
5. Conversion of Real Number
:
Example:
Convert the binary real number 1011.011 to its decimal equivalent.
The given binary real number has two parts—an integer part and a fraction. The decimal equivalent is to be expressed as both the integer and the fraction, and thereby they are to be added to convert the binary one to the decimal real number.
Now we can write,
6. Conversion of Decimal Fraction to Binary Fraction
:
It is obtained by using the technique of successive multiplication by 2. In this technique the integer part is noted down after each multiplication and the new fraction obtained is used for further multiplication.
Example:
Convert 50.7 (decimal) to its binary equivalent.
For this decimal real number its binary equivalent is obtained separately for the integer and the fraction.
We first convert the decimal number 50 to binary number in the following way:
We find that the fraction has not become zero and the process will continue. So, we have taken an approximation in this case. We have taken here 7 binary bits after the binary point.
The result is-
0.7 (decimal) = 0.1011001 (binary), combining the above two steps, we have-
50.7 (decimal) = 110010.1011001 (binary).
7. Advantages of Hexadecimal System over Octal System
:
The advantages are:
i. For a long computer word length the binary representation of hexadecimal number is shorter.
ii. For the word length which is divisible by 4 and not by 3, the hexadecimal system is a convenient one. The octal number system needs the extension of zero bits ahead of the most significant bit in this case.
8. Advantages of Octal Number System over the Decimal System
:
The advantages are:
i. Conversion of octal number to decimal number is easier as the octal number contains only digits. During the process of conversion, the letters used in hexadecimal numbers are to be converted into equivalent decimal numbers.
ii. In hexadecimal number system letters are used to represent decimal numbers 10 to 15. The numbers using letters resemble with words and consequently the use of letters in hexadecimal numbers make confusion. In octal system no such confusion arises as the octal number system uses digits 1 to 7 only.