Computer Arithmetic Operations PDF

Unit - 02 Computer arithmetic operations Lecture-01 Logic Gates  Logic gates are the basic building blocks of any digital system.  It is an electronic circuit having one or more than one input and only one output.  The relationship between...

Unit - 02 Computer arithmetic operations Lecture-01 Logic Gates  Logic gates are the basic building blocks of any digital system.  It is an electronic circuit having one or more than one input and only one output.  The relationship between the input and the output is based on a certain logic.  There are three types logic gates : Logic Gates Basic Logic Gates Universal Logic Gates Special Logic Gates  AND gate,  NAND Gate  Ex-OR Gate  OR gate  NOR Gates  Ex-NOR Gate  NOT gate  Unit - 02 Computer arithmetic operations Lecture-01 AND Gate  A circuit which performs an AND operation is called AND Gate.  It has n input (n >= 2) and one output. Logic diagram Truth Table Logic expression Y = A AND B = A.B = AB  Unit - 02 Computer arithmetic operations Lecture-01 OR Gate  A circuit which performs OR operation is called OR Gate.  It has n input (n >= 2) and one output. Logic diagram Truth Table Logic expression Y = A OR B =A+B  Unit - 02 Computer arithmetic operations Lecture-01 NOT Gate  NOT gate is also known as Inverter.  It has one input A and one output Y. Logic diagram Truth Table Logic expression Y = NOT A = A  Unit - 02 Computer arithmetic operations Lecture-02 Universal Logic Gates  A universal logic gate is a logic gate that can be used to construct all other logic gates.  NAND and NOR gates are known as universal logic gates. NAND Gate  A AND-NOT operation is known as NAND operation.  It has n input (n >= 2) and one output. Logic diagram Truth Table Logic expression Y = AB  Unit - 02 Computer arithmetic operations Lecture-02 NOR Gate  A OR-NOT operation is known as NOR operation.  It has n input (n >= 2) and one output. Logic diagram Truth Table Logic expression Y= A+B  Unit - 02 Computer arithmetic operations Lecture-03 X-OR Gate  XOR gate is a special type of gate.  It can be used in the half adder, full adder and subtractor.  The exclusive-OR gate is also known as EX-OR gate or sometime as X-OR gate.  It has n input (n >= 2) and one output. Truth Table Logic diagram Logic expression Y = A XOR B = A⊕B = AB + BA  Unit - 02 Computer arithmetic operations Lecture-03 X-NOR Gate  XNOR gate is a special type of gate.  It can be used in the half adder, full adder and subtractor.  The exclusive-NOR gate is also known as EX-NOR gate or sometime as X-NOR gate.  It has n input (n >= 2) and one output. Truth Table Logic diagram Logic expression Y = A XNOR B = A⊙B = A.B + AB  Unit - 02 Computer arithmetic operations Lecture-04 Boolean algebra :  Boolean Algebra is used to analyze and simplify the digital logic circuits.  It uses only the binary numbers i.e. 0 and 1.  It is also called as logical Algebra.  Boolean algebra was invented by George Boole in 1854. Rule in Boolean Algebra  Variable used can have only two values. Binary 1 for HIGH and Binary 0 for LOW.  Complement of a variable is represented by an overbar (-). Example : A is represented as A  OR operation of the variables is represented by plus (+) sign between them. Example : ORing of A, B, C is represented as A + B + C.  AND operation of the two or more variable is represented by dot between them such as A.B.C.  Sometime the dot may be omitted like ABC. Unit - 02 Computer arithmetic operations Lecture-04 Boolean Laws There are six types of Boolean Laws. 1) Commutative law 2) Associative law 3) Distributive law 4) AND law 5) OR law 6) Inversion law  Unit - 02 Computer arithmetic operations Lecture-04 Commutative law  Commutative law states that changing the sequence of the variables does not have any effect on the output of a logic circuit. A.B = B.A A+B=B+A Associative law  This law states that the order in which the logic operations are performed is irrelevant as their effect is the same. (A.B).C = A(B.C) (A + B) + C = A + (B + C) Distributive law A.(B + C) = A.B + A.C)  Unit - 02 Computer arithmetic operations Lecture-04 AND law OR law  These laws use the AND operation.  These laws use the OR operation.  Therefore they are called as AND laws.  Therefore they are called as OR laws. 1) A. 0 = 0 1) A + 0 = A 2) A. 1 = A 2) A + 1 = 1 3) A. A = A 3) A + A = A 4) A. A = 0 4) A + A = 1 Inversion law  This law uses the NOT operation.  The inversion law states that double inversion of a variable results in the original variable itself. A =A  Unit - 02 Computer arithmetic operations Lecture-05 De Margan theorem  De Morgan has suggested two theorems which are useful in Boolean Algebra. Theorem 1 A. B = A + B Unit - 02 Computer arithmetic operations Lecture-05 Theorem 2 A + B = A.B  Unit - 02 Computer arithmetic operations Lecture-01 Number System  As we know that for a computer, everything is a number.  For computer alphabets, pictures, sounds, etc., are numbers. There are four types of Number system 1) Binary number system consists of only two values, either 0 or 1 2) Octal number system represents values in 8 digits from 0 to 7. 3) Decimal number system represents values in 10 digits from 0 to 9. 4) Hexadecimal number system represents values in 16 digits from 0 to 9, A, B, C, D, E & F. Unit - 02 Computer arithmetic operations Lecture-01 Data representation : Magnitude representation Complement representation Signed unsigned 1’s Complement 2’s Complement representation representation representation representation  Unit - 02 Computer arithmetic operations Lecture-07 Data representation :  Digital Computers use Binary number system to represent all types of information inside the computers.  There are two major methods to store real numbers : 1) Fixed Point representation 2) Floating Point representation. Fixed-Point Representation :  Unit - 02 Computer arithmetic operations Lecture-07 Floating-Point Representation :  The floating number representation of a number has two part:  First part represents a signed fixed point number called mantissa.  Second part of designates the position of the decimal point and is called the exponent.  Floating -point is always represent a number in the following form: m x be  Unit - 02 Computer arithmetic operations Lecture-07 IEEE 754  The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point computation.  It was established in 1985 by the Institute of Electrical and Electronics Engineers.  IEEE 754 has 3 basic components: 1) Sign 2) exponent 3) Mantissa  Unit - 02 Computer arithmetic operations Lecture-07 Single precision Double precision Unit - 02 Computer arithmetic operations Lecture-07 Unit - 02 Computer arithmetic operations Lecture-07 Unit - 02 Computer arithmetic operations Lecture-07  Unit - 02 Computer arithmetic operations Lecture-01 Complement Arithmetic Binary system complements  As the binary system has base r = 2.  So the two types of complements for the binary system are : 1) 1's complement and 2) 2's complement 1's complement  The 1's complement of a number is found by changing all 1's to 0's and all 0's to 1's.  This is called as taking complement or 1's complement.  Unit - 02 Computer arithmetic operations Lecture-01 2's complement  The 2's complement of binary number is obtained by adding 1 to the Least Significant Bit (LSB) of 1's complement of the number.  2's complement = 1's complement + 1  Unit - 02 Computer arithmetic operations Lecture-01 2's complement multiplication  Unit - 02 Computer arithmetic operations Lecture-07 2's complement multiplication  Unit - 02 Computer arithmetic operations Lecture-08 Combinational Circuits  Combinational circuit is a circuit in which we combine the different logic gates in the circuit.  Example adder, Subtractor, encoder, decoder, multiplexer and demultiplexer etc. Some of the characteristics of combinational circuits are following :  The combinational circuit do not use any memory.  The output of combinational circuit at any instant of time, depends only on the present at input.  The previous state of input does not have any effect on the present state of the circuit.  A combinational circuit can have an n number of inputs and m number of outputs. Block diagram Unit - 02 Computer arithmetic operations Lecture-08 Block diagram  Unit - 02 Computer arithmetic operations Lecture-09 Half Adder  Half adder is a combinational circuit with two inputs and two outputs.  This circuit has two outputs sum and carry.  It does not take any carry.  It is designed to add two single bit binary numbers. Truth Table Circuit Diagram Logical Expression : Sum = A XOR B = A ⊕ B Carry = A AND B = AB  Unit - 02 Computer arithmetic operations Lecture-10 Full Adder  Full adder is developed to overcome the drawback of Half Adder. Truth Table  It can add two one-bit numbers, and carry.  The full adder is a three input and two output combinational circuit. Block diagram Circuit Diagram Logical Expression : SUM = (A XOR B) XOR Cin = (A ⊕ B) ⊕ Cin CARRY-OUT = A.B + Cin(A ⊕ B) = AB + BC + AC  Unit - 02 Computer arithmetic operations Lecture-11 Half Subtractors  Half subtractor is a combination circuit with two inputs and two outputs (difference and borrow).  It produces the difference between the two binary bits and an output (Borrow).  In the subtraction (A-B), A is called as Minuend bit and B is called as Subtrahend bit. Truth Table Circuit Diagram (A ⊕ B) Logical Expression AB Difference = A XOR B Borrow = AB  Unit - 02 Computer arithmetic operations Lecture-12 Full Subtractors  The disadvantage of a half subtractor is overcome by full subtractor.  The full subtractor is a combinational circuit with three inputs and two output Diff. and Borrow'. Circuit Diagram Truth Table DIFFERENCE = (A ⊕ B) ⊕ Cin BORROW = AB + ABin + BBin  Unit - 02 Computer arithmetic operations Lecture-13 Booth's Multiplication/Hardware implementation for multiplication  The booth algorithm is a multiplication algorithm that allows us to multiply the two signed binary integers in 2's complement.  It is also used to speed up the performance of the multiplication process.  Unit - 02 Computer arithmetic operations Lecture-13 Booth's Algorithm/Flow chart to multiplying binary numbers  Unit - 02 Computer arithmetic operations Lecture-13 Multiplying binary numbers Unit - 02 Computer arithmetic operations Lecture-13