1.Number System :-
1.3 Introduction
The computer that we know of is a digital device, which means that digital signals are used for the functioning of the computer .This property of digital signals proves to be very useful to represent data in the computer.
Why is it so? The answer to this question lies to the fact that there are many electrical and electronic devices which can be in anyone of their two possible states. For example, a simple switch can be either ON or OFF, a bulb or a Light Emitting Diode [LED] can be glowing or not glowing etc. Therefore, representation of data become very easy with these devices. These two values are called bits, which stand for Binary Digits. They take the values 0 and 1. John Von Neumann suggest this convention.
Since there are only two values with which data can be represented in a computer, at the lowest level, two discrete voltage values are used to represent the bits. Thus, digital signals, with their property mentioned above, proved to be ideal for computer systems.
Nearly all of our arithmetic operations are carried out with decimal numbers. The computer on the other hand does not use its numbering system. As mentioned earlier, the computer does its computational jobs with binary numbers, which is based on the number 2, for a number of reasons, The most important of them are the facts that operations with binary numbers are more precise and also the operations are faster because of the fact that for certain binary operations there are hardware circuitry available.
Four common numbering systems are generally used. They are:
a] Decimal Numbering System Base – 10 Digits 0 – 9
b] Binary Numbering System Base – 2 Digits 0 & 1
c] Octal Numbering System Base – 8 Digits 0 – 7
d] Hexadecimal Numbering System Base – 16 Digits 0 – 9 & A – F
We said that numbering system use a base value. What do we understand by a base value? We can say that the base of a number refers to how many digits are required to represent a numerical value in that numbering system.
The Binary Numbering System
Computer use this numbering system for doing its computations. As explained above, it has its own advantages. The computer follows three steps to complete an arithmetic operation:
- it convert the numeric data input to its corresponding binary equivalent
- Performs the desired arithmetic operation in binary
- Convert the result back to its corresponding decimal equivalent and outputs the result
When compared with decimal numbering system, the binary numbering system differs in the number of digits used for a numeric value representation. The decimal system uses ten digits, namely 0 to 9, whereas the binary system uses only two digits, 0 and 1.
Decimal Number |
Binary Equivalent |
Decimal Number |
Binary Equivalent |
0 |
0000 |
5 |
0101 |
1 |
0001 |
6 |
0110 |
2 |
0010 |
7 |
0111 |
3 |
0011 |
8 |
1000 |
4 |
0100 |
9 |
1001 |
Binary Equivalent of decimal number
we have seen the binary equivalent of decimal number.
When the dealing with binary numbers, two terms needs to be understood. these are MSB (Most Significant Bit) and LSB (Least Significant Bit). These two bit play is very important role in many other aspects of computing, such as address calculation and bus optimization.
So how do we define MSB and LSB?
MSB can be defined as the digit that occurs at the leftmost position in a binary number. Similarly, LSB can be defined as the digit that occurs at the rightmost position in a binary number.
1 |
1 |
0 |
1 |
0 |
0 |
0 |
1 |
MSB and LSB in a binary number
The Octal Number System –
The octal number system uses 8 digits, 0 through 7. They are used by some computers and can be employed by using the conversion techniques. There exist a relationship between octal number system and binary number system because the base number 8 is a multiple of base number 2. Since, 8 is 2 raised to 3, the relationship is 3:1, that is, all the digits of octal number system can be represented by a three digit binary number.
Octal |
Binary |
Octal |
Binary |
0 |
000 |
4 |
100 |
1 |
001 |
5 |
101 |
2 |
010 |
6 |
110 |
3 |
011 |
7 |
111 |
Binary and Octal numbers
Example 1: To covert 11000_{2} to its equivalent octal code
The binary number _{ }11000_{2} can be grouped in sets of three digits as _{ }011_{2} and _{ }100_{2}. binary code 001 corresponds to octal digit 3 and binary code 100 corresponds to octal digit 4. Therefore,11000_{2} in octal becomes 34_{8 .}
The Hexadecimal number System :-
The hexadecimal numbering system has a base value of 16. This means that in hexadecimal system, there are 16 digits, which can be used represent any numeric value. The difference between the hexadecimal system and the octal or binary system is that alphabets are used to represent numeric values because our standard numbering system has only 10 digits (0 to 9). A relationship exists between hexadecimal and binary numbering system.
Decimal |
Hexadecimal |
Binary |
Decimal |
Hexadecimal |
Binary |
0 |
0 |
0000 |
8 |
8 |
1000 |
1 |
1 |
0001 |
9 |
9 |
1001 |
2 |
2 |
0010 |
10 |
A |
1010 |
3 |
3 |
0011 |
11 |
B |
1011 |
4 |
4 |
0100 |
12 |
C |
1100 |
5 |
5 |
0101 |
13 |
D |
1101 |
6 |
6 |
0110 |
14 |
E |
1110 |
7 |
7 |
0111 |
15 |
F |
1111 |
Hexadecimal number System
Example – To convert 1101101_{2 }to its hexadecimal equivalent 1101101_{2 }to its hexadecimal equivalent
can be broken down into two groups as 0110_{2} and 1101_{2}. their hexadecimal equivalent is 6 and D respectively. Therefore, the hexadecimal equivalent of 1101101_{2} is 6D_{16.}
The reverse technique can be applied to determine the binary equivalent of a hexadecimal number or octal number.
Conversion Techniques –
There are two methods used most frequently to convert a number in a particular base to any other base. They are called The Remainder Method and Expansion Method.
The Remainder Method –
This method can be used to convert a decimal number to its equivalent value in any other base. The following steps are to be followed to carry out the conversion with the remainder method. Let us assume that the number 14 is to be converted to its binary equivalent. The required base therefore is 2.
- Divide the number by the base and note the remainder.
- Divide the quotient by the base and note the remainder.
- Repeat step 2 until the quotient cannot be divided further. That is, the quotient becomes smaller than the divisor.
The sequence of remainders starting from the last generated one prefixed by the undivided quotient is the converted number.
Example : To convert decimal number 14 to its binary equivalent.
Step 1: 14 divided by 2; Quotient = 7; Remainder = 0
Step 2: 7 divided by 2; Quotient =3; Remainder = 1
Step 3: 3 divided by 2; Quotient = 1; Remainder = 1
The binary number therefore becomes 1110
The Expansion Method –
This method can be applied to convert any number in any base to its equivalent in base 10. To understand how this method is carried out, let us take an example.
Example : Convert a binary number 1001 to its equivalent decimal value.
1001 = 1 x 2 + 0 x 2 + 0 x 2 + 1 x 2
= 8 + 0 + 0 + 1
=9
The following observations are to be made from the above example:
- Each digit in the original number individually precedes the component of expansion. That is, during expansion 1 precedes the expansion component in the left most position followed by the other digits – 0, 0 and 1. This occurrence is exactly according to the digit’s placement in the original binary number.
- During expansion, the base of the number is sequentially raised to a count that starts with 0 and is incremented by one for every digit that occurs in the binary number.
- The result that is obtained applying this method is a decimal number.
Example :
1110_{2} = 1 x 2^{3}+ 1 x 2^{2}+1 x2^{1 }+ 0 x2^{0}
= 8 + 4 + 2 + 0
=14_{10}