Introduction :

Early computer systems used electrical switches and when electrical switches were replaced by less mechanical devices such as vacuum tubes, than the transistor, the integrated circuit, the concept of switching on and off remained with computers but a representation of the on/off behavior of computers had to be made.

The term, digital, in computing and electronics applies to converting real world information to binary numeric form. The binary (base 2) number system represents two discreet values using two symbols or digits i.e., 0 and 1. The binary number system, where a zero symbolizes no electrical current (OFF) and one represents electrical current exists (ON).

All computer data (alpha-numeric, symbols and characters, audio, graphics and video) are represented or encoded using sequences of binary digits that are interpreted according to appropriate software. Computers are made up of electronic devices and electronic device can exist or either in ON or OFF state. For our convenience an ON State is represented by the code ‘1’ and OFF state is represented by the code‘0’. They are called ‘bits’, an abbreviation for Binary Digit. The numbers represented by bits are known as binary numbers.

Basically number systems are classified in two types :

1) Non – Positional Number System

2) Positional Number System

Non-Positional Number System :

In this number system any number can be represented by arranging symbols in various positions. Each symbol represents a definite value irrespective of the position in which they appear. For example: Roman Number System

Positional Number System :

In a positional number system, a number is represented by a set of symbols. Each symbol represents a particular value, depending on its position. The actual number of symbols used in a position system depends on its base.

Base :

The number of digits or basic symbols used in a positional number system is known as the

BASE or Radix of the system.

1. Decimal Number System (Base 10) :

It is a positional number system with base 10 and thus it uses 10 symbols i.e., 0 to 9. Any number can be represented by arranging symbols in various positions. In the decimal system, each position represents a specific power of 10.

For example : The decimal number 654.52, written as (654.52)₁₀ or 654.52₍₁₀₎ to specify base 10 is represented as follows :

Increasing Powers of 10				Decreasing Powers of 10
	Hundreds	Tens	Units	One Tenth	One Hundredth
Weights	10²	10¹	10⁰	10^-1	10^-2
Digits	6	5	4	5	2

Thus, the expanded notation of

654.52₍₁₀₎

= 6 X 10² + 5 X 10¹ + 4 X 10⁰ + 5 X 10^-1 + 2 X 10^-2

= 6 X 100 + 5 X 10 + 4 X 1 + 5 X 0.1 + 2 X 0.01

= 600 + 50 + 4 + 0.5 + 0.02

= 654.52

Note : In a positional number system all bits or digits to the left of the decimal or binary point have weights that are positive powers of base and those to the right have weights that are negative powers of base. The base is also called Radix and fractional point is called as Radix point.

2. Binary Number System (Base 2) :

The binary system is a positional system to the base 2. It uses two symbols 0 and 1. Each position represents specific power of 2.

For example : The binary number 1 1 0 1 . 1 1 written as 1 1 0 1 . 1 1 ₍₂₎ or (1 1 0 1 . 1 1)₂ to specify base 2, is represented as follows :

Increasing Powers of 2					Decreasing Powers of 2
Weights	2³	2²	2¹	2⁰	2^-1	2^-2
Digits	1	1	0	1	1	1

Thus, the expanded notation of 1 1 0 1 . 1 1 ₍₂₎

= 1 X 2³ + 1 X 2² + 0 X 2¹ + 1 X 2⁰ + 1 X 2^-1 + 1 X 2^-2

3. Octal Number System (Base 8) :

The octal number system is a positional number system with base 8. It uses 8 symbols i.e., 0 to 7. In octal number system, each digit position corresponding to a power of 8.

For example : The octal number 43.12 written as (43.12)₈ or 43.12₍₈₎ to specify base 8, is represented as follows :

Increasing Powers of 8			Decreasing Powers of 8
Weights	8¹	8⁰	8^-1	8^-2
Digits	4	3	1	2

Thus, the expanded notation of 43.12₍₈₎

= 4 X 8¹ + 3 X 8⁰ + 1 X 8^-1 + 2 X 8^-2

4. Hexadecimal Number System (Base 16) :

The hexadecimal number system is a positional number system to the base 16. It uses 16 symbols to represent any number. The first 10 symbols are represented by digits 0 to 9 and the remaining 6 symbols by the letters A to F, representing the decimal values 10 to 15 respectively. Each position represents a specific powers of 16.

For example, the hexadecimal number BA85.12 is written as (BA85.12)₁₆ or BA85.12₍₁₆₎, is represented as follows :

Increasing Powers of 16					Decreasing Powers of 16
Weights	16³	16²	16¹	16⁰	16^-1	16^-2
Digits	B	A	8	5	1	2

Thus, the expanded notation of BA85.12₍₁₆₎

= B X 16³ + A X 16² + 8 X 16¹ + 5 X 16⁰ + 1 X 16^-1 + 2 X 16^-2

INTER CONVERSION OF NUMBER FROM ONE SYSTEM TO ANOTHER INTEGER PART

1. Divide the integer part of the decimal number by the base ‘b’ of the new system. The remainder will give the right most digit of the integer part of the new number.

2. Divide the quotient again by the base ‘b’. The remainder is the next digit from right.

3. Repeat step 2 until a zero quotient is obtained. Last remainder is the left most digit of the new number.

FRACTIONAL PART

1. Multiply the fractional part of the decimal number by the base ‘b’ of the new system. The integer part of the product gives the left most digit of the fractional part of the new number.

2. Multiply the fractional part of the product by the base ‘b’. The integer part of the resultant product is the next digit from left.

3. Repeat step 2 until a zero fractional part or a repeated fractional part or a non-terminating fractional part occurs.

1. Decimal to Binary Conversion :

1) 25. 625 ₍₁₀₎ = ? ₍₂₎

2	25			Product	Integer	Fractional-Part
2	1 2	1	0.625 X 2	1.250	1	0.250
2	6	0	0.250 X 2	0.500	0	0.500
2	3	0	0.500 X 2	1.000	1	0.000
2	1	1
2	0	1	Therefore, 25₍₁₀₎ = 1 1 0 0 1 Thus, the result is 25.625₍₁₀₎		& =	0.625₍₁₀₎ = 1 0 1 1 1 0 0 1 . 1 0 1₍₂₎

2. Binary to Decimal Conversion :

1) 1 1 . 1 0 1 1 ₍₂₎ = ? ₍₁₀₎

=1X 2¹ + 1 X 2⁰ + 1 X 2^-1 + 0 X 2^-2 + 1 X 2^-3 + 1 X 2^-4

=1 X 2 + 1 X 1 + 1 X 0.5 + 0 X 0.25 + 1 X 0.125 + 1 X 0.0625

=2 + 1 + 0.5 + 0 + 0.125 + 0.0625

=3.6575₍₁₀₎

3. Binary to Octal Conversion :

1) 0 1 1 1 0 1 ₍₂₎ = ?₍₈₎

Octal	0	1	2	3	4	5	6	7
Binary	000	001	010	011	100	101	110	111

Note : Place the 3 binary bits into one group from right side and write equivalent octal digit.

0 1 1 1 0 1

3 5

Thus, the result is 0 1 1 1 0 1 ₍₂₎ = 3 5 ₍₈₎

4. Binary to Hexadecimal Conversion :

1) 01101101₍₂₎ = ?₍₁₆₎

Hexa

Decimal

Binary

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

Note : Place the 4 binary bits into one group from right side and write equivalent Hexadecimal digit.

0 1 1 0 1 1 0 1

6 D

Thus, the result is 0 1 1 0 1 1 0 1 ₍₂₎ = 6D ₍₁₆₎

5. Octal to Decimal Conversion :

1) 43.12₍₈₎ = ? ₍₁₀₎

43.12₍₈₎

= 4 X 8¹ + 3 X 8⁰ + 1 X 8^-1 + 2 X 8^-2

= 4 X 8 + 3 X 1 + 1 X 0.125 + 2 X 0.015625

= 32 + 3 + 0.125 + 0.03125

= 35.15625 ₍₁₀₎

6. Octal to Binary Conversion :

Octal	0	1	2	3	4	5	6	7
Binary	000	001	010	011	100	101	110	111

Note : Place the 3 bit binary equivalent of each digit below the number 1) 67.35₍₈₎ = ?₍₂₎

6	7 .	3 5
110	111	011 101

Thus the result is 67.35₍₈₎ = 110111.011101₍₂₎

7. Octal to Hexadecimal Conversion :

(a) First Covert Octal to Decimal 43.12(8) = ? (10)

= 4 X 8¹ + 3 X 8⁰ + 1 X 8^-1 + 2 X 8^-2

= 4 X 8 + 3 X 1 + 1 X 0.125 + 2 X 0.015625

= 32 + 3 + 0.125 + 0.031250

= 35.15625 ₍₁₀₎

(b) Convert Decimal to Hexadecimal

35.15625 ₍₁₀₎ = ? ₍₁₆₎

16	3 5			Product	Integer	Fractional-Part
16	2	3	0.15625 X 16	2.500	2		0.500
	0	2	0.50000 X 16	8.000	0		0.000

Therefore, 35₍₁₀₎ = 2 3 & 0.15625₍₁₀₎ = 2 0

Thus, the result is 35.15625₍₁₀₎ = 2 3 . 2 0 ₍₁₆₎

Representation of Signed Numbers

In Binary System, we represent the sign of a number using an extra bit at the extreme right of the number and is called ‘sign’ bit. By convention the symbol ‘0’ is used to represent the (+) sign and ‘1’ to represent (-) sign.

In the case of binary numbers used in computers, the most significant bit represents the sign

and the remaining bits the magnitude of the number. For example a 6 bit binary equivalent of a decimal number -15 is represented by ( 1 0 1 1 1 1 )₂ and + 15 is represented by ( 0 0 1 1 1 1 )_2.

This method of representing is known as sign magnitude representation.

Representation of Signed numbers are :

1) By sign magnitude representation

2) By using 1’s complement

3) By using 2’s complement

Sign and Magnitude Representation

To represent Positive numbers, the magnitude is represented in the true binary form, and a sign bit 0 is placed in front of the MSB.

For example + 47 represented as follows :

Sign bit True Number

To represent negative number, the magnitude is represented into 2’s complement form, and a sign bit of 1 is placed in front of the MSB.

For example + 47 represented as follows :

Sign bit True Number

1’s Complement

The 1’s complement of a binary number is obtained by complementary all the bits i.e., by changing 1’s to 0’s.

Example : Find the 1’s complement of 1 0 1 1 0 0

Step1 : 1 0 1 1 0 0 given number

Step2 : 0 1 0 0 1 1 complement each bit to form 1’s complement.

Thus, 1’s complement of 1 0 1 1 0 0 = 0 1 0 0 1 1

2’s Complement

The 2’s complement of a binary number is obtained by adding 1 to 1’s complement of a binary number.

Example : Find the 2’s complement of

1 0 1 1 0 0

Step1 : 1 0 1 1 0 0 given number

Step2 : 0 1 0 0 1 1 complement each bit to form 1’s complement. 0 1 0 0 1 1

+ 1

0 1 0 1 0 0

Step3 : 0 1 0 1 0 0 is 2’s complement of original number.

Binary Arithmetic :

Binary Arithmetic is a fundamental code for all digital computers and most other digital systems. It can perform all arithmetic operations like addition, subtraction, multiplication and division.

Binary Addition :

It is performed in the same manner as decimal addition. The following table shows that four basic rules for binary addition.

A		B	Sum	Carry
0	+	0	0	0
0	+	1	1	0
1	+	0	1	0
1	+	1	0	1

Examples :

	1 0 1		1 0 1 1 . 1 0 1
+	0 1 1	+	1 0 1 0 . 1 0 0
	1 0 0 0		1 0 1 1 0 . 0 0 1

Binary Substraction :

Binary subtraction rules are as follows :

A		B	Sum	Carry
0	-	0	0	0
1	-	0	1	0
1	-	1	0	0
0	-	1	1	1

Examples :

	1 0 1		1 0 0 1 0 1 0
-	0 1 1	-	1 0 1 0 1
	0 1 0		0 1 1 0 1 0 1

Subtraction of Binary number using 1’s Complement :

Case 1: Substraction of Smaller number from Larger number

1.Determine 1’s Complement of smaller number

2.Add 1’s compliment to the larger number

3.We get carry. Addition end around carry to the above gives the result of subtraction.

Example :

Subtract 1001 – 1000 using 1’s Complement

1. 1’s complement of 1000 is 0111

2. Add 1’s complement to the larger number 1 0 0 1

0 1 1 1

1 0 0 0 0

3. Add carry 1 to the above result 0 0 0 0

+ 1

0 0 0 1

Therefore,

1 0 0 1 – 1 0 0 0 = 0 0 0 1 ( 9 – 8 = 1 )

Case 2: Substraction of Larger number from Smaller number

1. Determine 1’s Complement of larger number

2. Add 1’s compliment to the smaller number

3. There will be no carry. To get answer take the 1’s complement of the result and put negative sign.

Example :

Subtract 111 – 101 using 1’s Complement

1. 1’s complement of larger number 111 is 000

2. Add 1’s complement to the smaller number 1 0 1

+ 0 0 0

1 0 1

Find 1’s complement of 101 and put negative sign i.e., - 0 1 0

Therefore,1 0 1 – 1 1 1 = - 0 1 0( 5 – 7 = -2 )

Subtraction of Binary number using 2’s Complement :

Case 1: Substraction of Smaller number from Larger number

1. Determine 2’s Complement of smaller number

2. Add 2’s compliment to the larger number

3. We get carry. Neglect the carry and the remaining gives the result of subtraction Example :

Subtract 1001 – 1000 using 2’s Complement

1. 2’s complement of 1000 is

0 1 1 1

+ 1

1 0 0 0

Add 2’s complement to the larger number

1 0 0 1

+ 1 0 0 0

1 0 0 0 1

Neglect the carry i.e., 0 0 0 1

Therefore,1 0 0 1 – 1 0 0 0 = 0 0 0 1(9–8=1 ) Case 2: Substraction of Larger number from Smaller number

1. Determine 2’s Complement of larger number

2. Add 2’s compliment to the smaller number

3. There will be no carry. To get answer take the 2’s complement of the result and put negative sign.

Example :

000

001

Subtract 111 – 101 using 1’s Complement

1. 2’s complement of larger number 111 is i.e.,

2. Add 2’s complement to the smaller number 1 0 1

+ 0 0 1

1 1 0

Find 2’s complement of 110 and put negative sign i.e.,0 0 1

+ 1

0 1 0

i.e., - 0 1 0

Therefore, 1 0 1 – 1 1 1 = - 0 1 0 ( 5 – 7 = -2 )

Computer Codes :

Computer understands everything only in binary. Therefore, when we input numbers, alphabets and other special symbols, they must be represented in the binary format. There are three such coding standards. There are BCD, ASCII and EBCDIC.

1. BCD (Binary Coded Decimal System :

In BCD each digit of a decimal number is independently converted to 4-bit binary number. For example the decimal number 573 would be represented in the 4 bit BCD code as

0 1 0 1 0 1 1 1 0 0 1 1

5 7 3

BCD Coding system is used to represent only decimal numbers, 4-bits are insufficient to represent the various characters used by computer. Therefore 6-bit BCD Code was developed. In this code two more bits called as ‘zero position’ are added. It is possible to represent 64 code groups i.e., 10 decimal digits, 26 alphabets and 28 special characters using 6 bit code.

2. ASCII (American Standard Code for Information Interchange :

It is most widely used alphanumeric code for printers, keyboards and terminals which are interface with computer to represent data. This is 7 bit code and also it has

128 possible code groups i.e., alphabets, numbers, special characters and control character (Enter Key, Escape Key, Space bar etc.,). A is represented in ASCII as 1000001 whose decimal equivalent is 65.

3. EBCDIC (Extended Binary Coded Decimal Interchange Code) :

It is also an alphanumeric code used in IBM computers and mainframe applications. It is an 8 bit code representing 256 different code group. A is represented in EBCDIC code as 11000001.

Digital Logic :

The word Logic is used to describe the circuits which can duplicate specific function of decision making performed by the human mind. In the logic relevant to computers we know that there can be only two states. This logic is thus called two state logic or bivalent logic. Such a logical method was developed by ARISTOTAL for getting at the truth. The method was subsequently developed by mathematician DE-MORGAN and GEORGE BOOLE into a very powerful mathematical tool.

A Gate is a simple electronic circuit or device that performs logical functions. It has one or more inputs and output. Gates are called binary logic gates 1 and 0 are inputs and outputs.

Truth Table

Truth Table is a table which shows all inputs and outputs possibilities of a logical circuits or gate.

Types of Logic Gates

1. AND Gate

2. OR Gate

3. NOT Gate

4. NAND Gate

5. NOR Gate

AND Gate

This is an electronic decision making element with one or more inputs and single outputs. Its function is to implement the AND operation (i.e., Logical Multiplication). The logical symbol for AND Gate is

B Q

Two inputs AND Gate truth table

Input A	Input B	Output Q=A.B
0	0	0
0	1	0
1	0	0
1	1	1

Three inputs AND Gate truth table

Input A	Input B	Input C	Output Q=A.B.C
0	0	0	0
0	0	1	0
0	1	0	0
0	1	1	0
1	0	0	0
1	0	1	0
1	1	0	0
1	1	1	1

AND Gate Rule

1. Logic 1 when all inputs are 1

2. Logic 0 when any input is 0

OR Gate

This is an electronic decision making element with one or more inputs and single output. Its function is to implement the OR operation (i.e., Logical Addition). The logical symbol for OR Gate is

Two inputs OR Gate truth table

Input A	Input B	Output Q=A+B
0	0	0
0	1	1
1	0	1
1	1	1

Three inputs OR Gate truth table

Input A	Input B	Input C	Output Q=A+B+C
0	0	0	0
0	0	1	1
0	1	0	1
0	1	1	1
1	0	0	1
1	0	1	1
1	1	0	1
1	1	1	1

OR Gate Rule

1. Logic 1 when an inputs is 1

2. Logic 0 when all inputs are 0

NOT Gate

This is an electronic decision making element with one input and one output. Its function is to implement the NOT operation (i.e., Inversion or Logical Complement) . The logical symbol for NOT Gate is

Truth table of NOT Gate is

Two inputs OR Gate truth table

Input A	Output Q (Complement of A)
0	1
1	0

NOT Gate Rule is

1. Logic 1 when input is 0

2. Logic 0 when input is 1

NAND Gate

This is combination of NOT and AND Gates. This is like AND Gate but with the output complimented.

The logical symbol for NAND Gate is

Two inputs NAND Gate truth table

Input A	Input B	A.B	Output Q=NOT(A.B)
0	0	0	1
0	1	0	1
1	0	0	1
1	1	1	0

NAND Gate rule is

1. Logic 1 when any input is 0

2. Logic 0 when all inputs are 1

NOR Gate

This is combination of NOT and OR Gates.

This is like OR Gate but with the output complimented.

The logical symbol for NOR Gate is

Two inputs NOR Gate truth table

Input A	Input B	A+B	Output Q=NOT(A+B)
0	0	0	1
0	1	1	0
1	0	1	0
1	1	1	0

NOR Gate rule is

1. Logic 1 when all inputs are 0

2. Logic 0 when any inputs is 1

Universal Gates

NAND and NOR gates are called universal gates because the other gates (i.e., AND, OR and NOT) can realized these individual gates.

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UNIT IV COMPUTER ELECTRONICS

Basically number systems are classified in two types :

Non-Positional Number System :

Positional Number System :

Base :

1. Decimal Number System (Base 10) :

2. Binary Number System (Base 2) :

3. Octal Number System (Base 8) :

4. Hexadecimal Number System (Base 16) :

INTER CONVERSION OF NUMBER FROM ONE SYSTEM TO ANOTHER INTEGER PART

FRACTIONAL PART

1. Decimal to Binary Conversion :

2. Binary to Decimal Conversion :

3. Binary to Octal Conversion :

4. Binary to Hexadecimal Conversion :

5. Octal to Decimal Conversion :

6. Octal to Binary Conversion :

7. Octal to Hexadecimal Conversion :

(b) Convert Decimal to Hexadecimal

Representation of Signed Numbers

Representation of Signed numbers are :

Sign and Magnitude Representation

1’s Complement

2’s Complement

Binary Arithmetic :

Binary Addition :

Binary Substraction :

Subtraction of Binary number using 1’s Complement :

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