BASIC NUMBERING SYSTEM
Use of digital technique started in 1940 military applications. Digital computer was developed in 1960. Due to the development of semiconductor devices digital computer started becoming small in size. Digital technique is used in all areas of consumer electronic products, communication systems and industrial controls.
Digital technique has many advantages over analog system, such as
1. Reduce the cost
2. Improves the performance
There are two basic types of electronic signals known as analog and digital.
Analog signals are of the most familiar type. It is AC or DC voltage that varies continuously. It does not change abruptly or suddenly or in steps. Most commonly used analog signal is the sine wave. As you know radio signals
& audio tones are sinusoidal waves. Electronic circuits which processes the analog signals are known as linear circuits.
Digital technique has many advantages over analog system, such as
1. Reduce the cost
2. Improves the performance
There are two basic types of electronic signals known as analog and digital.
Analog signals are of the most familiar type. It is AC or DC voltage that varies continuously. It does not change abruptly or suddenly or in steps. Most commonly used analog signal is the sine wave. As you know radio signals
& audio tones are sinusoidal waves. Electronic circuits which processes the analog signals are known as linear circuits.
Digital signals are a series of pulses. These signals are voltages which varies fast between two fixed levels.
NUMBER SYSTEMS
A number system is nothing more than a code.
For each distinct quantity, there is an assigned symbol.
Number System is a representation of number such as
1. Decimal
2. Binary
3. Hexadecimal
DECIMAL NUMBER SYSTEM
The base or radix of a number system refers to the number of digits used. In decimal system 10 is used as base or radix because digits from 0 to 9 are used.
Each digit in the number has certain positions and that position determines its weight. Positions, weights are from right to left, units, tens, hundreds, thousands
Note :
1. The number of digits used in the system is equal to the radix or base.
2. The largest digit is one less than the base i.e. 9
3. In the sum represented by a number each digit is multiplied by the base raised to the appropriate power for digit position.
BINARY NUMBER SYSTEM
In the binary system the base is two and only two numbers 0 and 1 are used.
In binary system, only two signal levels are needed.
Group of four bits make a nibble. eg., 1111, 1100, 1101
A string of 8 bits make a byte. eg., 10001111, 11111111, 00011111
A byte is a basic unit of data in computers. Most computers process data in strings of 8 bits or 16, 24, 32 & so on.
CONVERSION OF NUMBER SYSTEM
As there are two number systems existing, there should be a procedure for converting a number from one system into other.
We now see the methods of conversions.
Binary to Decimal
Decimal to Binary
BINARY TO DECIMAL
This conversion is accomplished in a very simple way. The procedure is
as follows :
1. Write the binary number
2. Multiply each bit with the weight of each digit, (bit) 20, 21, 22, 23, etc 1,2,4,8, 16... from right to left.
3. Cancel the weights, where the bit is zero.
4. Add the remaining digits to get a decimal number.
The method for conversion of decimal fractions in binary form is same.
The weights of the numbers on the left hand side of decimal point increase from 20, 21, 22... etc.
If this continues from the decimal point to the right side, the weights are 2-1, 2-2, 2-3........etc.
Naturally, for mixed numbers i.e. the numbers with integer as well as fraction, each part is solved separately and then joined. The weight for a mixed number are : .....2^4 2^3 2^2 2^1 2^0 . 2^ -1 2^-2 2^-3
DECIMAL TO BINARY
One way to convert the given decimal number into binary is the reverse of the process seen above.
The number is expressed as a sum of powers of two, and then 1's and 0's are written at appropriate positions.
The second method called as Double-Dabble method is easy and more
frequently used. The procedure is as follows.
1. Divide the given number successively by 2.
2. Write down the quotients directly below the given number.
3. Write down the remainders on the right side.
4. The remainders taken in reverse order from bottom to top, form the number.
The last quotient obtained by dividing 2 by 2 is 1. This 1 is not divisible by 2.
Hence the next quotient is 0 and 1 is transferred to remainders.
For the conversion of decimal fractions into binary fractions, the procedure is as follows :
1. Multiply the fraction by 2.
2. The integer from the multiplication is kept aside.
3. The remaining fraction is again multiplied.
4. The integers obtained by this procedure form the binary number.
5. As the process is unending, it can be continued till the answer is sufficiently accurate.
Note that, here the integers are taken in the same order as they are obtained.
HEXADECIMAL NUMBER SYSTEM
Hexadecimal numbers are extensively used in microprocessor work.
To begin with they are much shorter than binary numbers.
Hexadecimal means 16. The hexadecimal number system has a base or radix of 16.
This means that it uses 16 digits to represent all numbers. The digits are 0 through 9, and A through F as follows: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.
Hexadecimal numbers are strings of these digits like 8A5, 4CF7 and
EC58.
Table below shows the equivalences between hexadecimal, binary and decimal digits.
