Number systems - Efficiency of number system, Decimal, Binary, Octal, Hexadecimalconversion
from one to another- Binary addition, subtraction, multiplication and division,
representation of signed numbers, addition and subtraction using 2’s complement and I’s
complement.
Binary codes - BCD code, Excess 3 code, Gray code, Alphanumeric code, Error detection
codes, Error correcting code.Deepak john,SJCET-Pala
- The document discusses number systems and bases, including binary, decimal, octal, and hexadecimal.
- It explains positional notation and how numbers are represented in different bases using place values that are powers of the base.
- The range of numbers that can be represented depends on the base and number of digits used. More digits allow larger numbers to be represented.
This document provides an overview of Boolean algebra and logic gates. It introduces Boolean logic operations like AND, OR, and NOT. It covers Boolean algebra laws and De Morgan's theorems. It also discusses logic gate types like AND, OR, NOT, NAND, NOR, XOR and XNOR. Karnaugh maps are introduced as a method to simplify Boolean expressions.
This document discusses binary numbers and arithmetic operations on binary numbers. It begins with an introduction to binary numbers, defining them as a numbering system with a base of 2 that uses only the digits 0 and 1. It then explains how addition, subtraction, multiplication, and division are performed on binary numbers, providing examples of each operation. The key methods of binary arithmetic are performing column-by-column addition and subtraction as in decimal, and using bit-wise logic for multiplication and division. Complements are also introduced for simplifying subtraction. In the end, it notes that the binary system has a long history of use prior to its modern application in computers.
The document discusses different methods for representing signed binary numbers:
1) Sign-magnitude notation represents positive and negative numbers by using the most significant bit to indicate the sign (0 for positive, 1 for negative) and the remaining bits for the magnitude.
2) One's complement represents negative numbers by inverting all bits of the positive number.
3) Two's complement, the most common method, represents negative numbers by inverting all bits and adding 1 to the result. This allows simple addition to perform subtraction.
This document summarizes a lecture on linear systems and convolution in continuous time. It discusses how any continuous signal can be represented as the limit of thin, delayed pulses using the sifting property. Convolution for continuous-time linear and time-invariant (LTI) systems is defined by the convolution integral. The convolution integral calculates the output of an LTI system by integrating the product of the input signal and impulse response over all time. Examples are provided to demonstrate calculating the output of an LTI system using convolution integrals.
Digital systems represent information using discrete binary values of 0 and 1 rather than continuous analog values. Binary numbers use a base-2 numbering system with place values that are powers of 2. There are various number systems like decimal, binary, octal and hexadecimal that use different number bases and represent the same number in different ways. Complements are used in binary arithmetic to perform subtraction by adding the 1's or 2's complement of a number. The 1's complement is obtained by inverting all bits, while the 2's complement is obtained by inverting all bits and adding 1.
Richard Hamming developed Hamming codes in the late 1940s to enable error correction in computing. Hamming codes are perfect 1-error correcting codes that use parity checks to detect and correct single bit errors in binary data. The codes work by encoding k message bits into an n-bit codeword with additional parity check bits such that the minimum distance between any two codewords is 3, allowing correction of single bit errors. Hamming codes see widespread use and can be generalized to non-binary alphabets. Extended Hamming codes provide both single-error correction and double-error detection.
Parity bits are used to detect single bit errors during data transmission. There are two common types of parity - even and odd. Even parity means the total number of 1s in the transmitted bits including the parity bit should be even. Odd parity means the total should be odd. The receiving device calculates parity and compares it to the received parity bit to check for errors. While parity can detect single bit errors, it cannot detect errors if an even number of bits are corrupted.
- The document discusses number systems and bases, including binary, decimal, octal, and hexadecimal.
- It explains positional notation and how numbers are represented in different bases using place values that are powers of the base.
- The range of numbers that can be represented depends on the base and number of digits used. More digits allow larger numbers to be represented.
This document provides an overview of Boolean algebra and logic gates. It introduces Boolean logic operations like AND, OR, and NOT. It covers Boolean algebra laws and De Morgan's theorems. It also discusses logic gate types like AND, OR, NOT, NAND, NOR, XOR and XNOR. Karnaugh maps are introduced as a method to simplify Boolean expressions.
This document discusses binary numbers and arithmetic operations on binary numbers. It begins with an introduction to binary numbers, defining them as a numbering system with a base of 2 that uses only the digits 0 and 1. It then explains how addition, subtraction, multiplication, and division are performed on binary numbers, providing examples of each operation. The key methods of binary arithmetic are performing column-by-column addition and subtraction as in decimal, and using bit-wise logic for multiplication and division. Complements are also introduced for simplifying subtraction. In the end, it notes that the binary system has a long history of use prior to its modern application in computers.
The document discusses different methods for representing signed binary numbers:
1) Sign-magnitude notation represents positive and negative numbers by using the most significant bit to indicate the sign (0 for positive, 1 for negative) and the remaining bits for the magnitude.
2) One's complement represents negative numbers by inverting all bits of the positive number.
3) Two's complement, the most common method, represents negative numbers by inverting all bits and adding 1 to the result. This allows simple addition to perform subtraction.
This document summarizes a lecture on linear systems and convolution in continuous time. It discusses how any continuous signal can be represented as the limit of thin, delayed pulses using the sifting property. Convolution for continuous-time linear and time-invariant (LTI) systems is defined by the convolution integral. The convolution integral calculates the output of an LTI system by integrating the product of the input signal and impulse response over all time. Examples are provided to demonstrate calculating the output of an LTI system using convolution integrals.
Digital systems represent information using discrete binary values of 0 and 1 rather than continuous analog values. Binary numbers use a base-2 numbering system with place values that are powers of 2. There are various number systems like decimal, binary, octal and hexadecimal that use different number bases and represent the same number in different ways. Complements are used in binary arithmetic to perform subtraction by adding the 1's or 2's complement of a number. The 1's complement is obtained by inverting all bits, while the 2's complement is obtained by inverting all bits and adding 1.
Richard Hamming developed Hamming codes in the late 1940s to enable error correction in computing. Hamming codes are perfect 1-error correcting codes that use parity checks to detect and correct single bit errors in binary data. The codes work by encoding k message bits into an n-bit codeword with additional parity check bits such that the minimum distance between any two codewords is 3, allowing correction of single bit errors. Hamming codes see widespread use and can be generalized to non-binary alphabets. Extended Hamming codes provide both single-error correction and double-error detection.
Parity bits are used to detect single bit errors during data transmission. There are two common types of parity - even and odd. Even parity means the total number of 1s in the transmitted bits including the parity bit should be even. Odd parity means the total should be odd. The receiving device calculates parity and compares it to the received parity bit to check for errors. While parity can detect single bit errors, it cannot detect errors if an even number of bits are corrupted.
The document discusses the binary number system, explaining that it uses only two digits, 0 and 1, with each column in a binary number being twice the value of the previous column. It covers converting between binary and decimal numbers, with examples, and explains that binary is important because computers represent information using electronic switches that can be either on or off, corresponding to 1 and 0.
This document discusses numerical methods and errors. It introduces that numerical methods provide approximate solutions rather than exact analytical solutions due to errors from measurements, algorithms, and output. Accuracy refers to how close an approximation is to the true value, while precision refers to the reproducibility of results. Significant figures indicate the precision of a number. True error, relative error, and percent error are defined to quantify the error between approximations and true values. Round-off errors from floating point representation on computers are also discussed.
Canonical forms express Boolean functions as a sum of minterms or product of maxterms. Standard forms, like canonical forms, can have variables that do not appear in each term. Canonical forms are not usually minimal, while standard forms are simplified versions of canonical forms. Boolean functions can be expressed in Sum of Products or Product of Sum forms, and minimized using algebraic methods or Karnaugh maps.
The document discusses digital communication systems. It provides examples of digital communication including an email sent to invite team members to a meeting. It then explains the key building blocks of a digital communication system including the input source, source encoder, channel encoder, digital modulator, channel, digital demodulator, channel decoder, source decoder and output transducer. The document also discusses channels used for digital communication, causes of signal loss, and comparisons between digital and analog communication systems.
This document discusses floating point number representation in IEEE-754 format. It explains that floating point numbers consist of a sign bit, exponent, and mantissa. It describes single and double precision formats, which use excess-127 and excess-1023 exponent biases respectively. Examples are given of representing sample numbers in both implicit and explicit normalized forms using single and double precision formats.
Linear differential equation with constant coefficientSanjay Singh
The document discusses linear differential equations with constant coefficients. It defines the order, auxiliary equation, complementary function, particular integral and general solution. It provides examples of determining the complementary function and particular integral for different types of linear differential equations. It also discusses Legendre's linear equations, Cauchy-Euler equations, and solving simultaneous linear differential equations.
The document provides an overview of SONET (Synchronous Optical Networking) and SDH (Synchronous Digital Hierarchy) architectures and technologies. It describes the four layers of SONET (path, line, section, photonic), SONET frame structure including overhead bytes, and how lower-rate STS frames are multiplexed into higher-rate frames. It also discusses different types of SONET networks including linear, ring and mesh, as well as the use of virtual tributaries to transport digital signals of different rates over SONET.
Contents:
1.What is number system?
2.Conversions of number from one radix to another
3.Complements (1's, 2's, 9's, 10's)
4.Binary Arithmetic ( Addition, subtraction, multiplication, division)
The document discusses the principle of mathematical induction and how it can be used to prove statements about natural numbers. It provides examples of using induction to prove statements about sums, products, and divisibility. The principle of induction states that to prove a statement P(n) is true for all natural numbers n, one must show that P(1) is true and that if P(k) is true, then P(k+1) is also true. The document provides examples of direct proofs of P(1) and inductive proofs of P(k+1) to demonstrate applications of the principle.
The document provides information on digital and analog signals, different number systems used in computing including binary, octal, decimal and hexadecimal. It explains:
- Digital signals have discrete amplitude values of 0V and 5V, while analog signals can have any amplitude value.
- Number systems like binary, octal and hexadecimal are used in computing to represent values using discrete digits. Conversion between number systems involves place value weighting.
- Binary uses two digits 0 and 1. Octal uses eight digits 0-7. Hexadecimal uses sixteen digits and letters 0-9 and A-F. Conversion between number systems and decimal is done by successive multiplication or division.
This chapter discusses digital systems and number conversion. Digital systems use discrete values rather than continuous values as in analog systems. They can provide exact outputs. The chapter covers converting between number bases, such as decimal to binary, using division or multiplication. It also addresses representing negative numbers and binary codes. The design of digital systems includes system, logic, and circuit design. Combinational and sequential circuits are introduced.
This document contains instructions for conducting network simulation experiments using the NCTUns simulator. It discusses setting up NCTUns, drawing network topologies, editing node properties, running simulations, and performing post-analysis. Experiment 1 involves simulating a 3-node point-to-point network with duplex links, varying the bandwidth, and measuring the number of dropped packets. The steps provided outline how to draw the topology in NCTUns and configure the nodes before running the simulation.
Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
This document discusses binary coded decimal (BCD). It defines BCD as a numerical code that assigns a 4-bit binary code to each decimal digit from 0 to 9. Numbers larger than 9 are expressed digit by digit in BCD. BCD is used because it is easy to encode/decode decimals and useful for digital systems that display decimal outputs. The document also describes how addition and subtraction are performed in BCD through binary addition rules and handling carries.
Weighted codes assign a positional weight or value to each digit, where the sum of the digit values multiplied by their weights represents the number. Non-weighted codes do not assign positional weights. BCD is a weighted 4-bit code that represents the decimal digits 0-9. It uses weights of 24, 23, 22, 21 from most to least significant bit. The Gray code is a non-weighted code where each number differs from the previous by one bit. Excess-3 code is a non-weighted 4-bit BCD code where 3 is added to each decimal digit before conversion to BCD.
This document discusses different types of number complements that allow subtraction to be performed using addition. It explains that subtraction using borrowing is inefficient for computers, so instead subtraction is done by taking the complement of one of the numbers and then adding. For binary numbers, the 2's complement is commonly used, where the 2's complement of a number is found by flipping all the bits and adding 1. The document provides examples of calculating 1's, 2's, and other bases' complements.
Introduction to Analog and Digital Systems - Basic definition, Representation, Examples and applications of Analog and Digital Systems - Advantages of Digital system over Analog system - Process of conversion from Analog to Digital and Digital to Analog signal - Digitization Examples - Signal representation of voltage and current in terms of Binary values - Representations of Binary quantities using different terminologies - IC Complexity classification - IC Layout - Development of ICs in terms of size
Manchester & Differential Manchester encoding schemeArunabha Saha
The two main variants of biphase encoding techniques are discussed here. Manchester and Differential Manchester encoding scheme are explained with examples. Comparison between several classes of polar encoding techniques are done along with the exposure about the advantages and disadvantages of both schemes.
Floating Point Representation premium.pptxshomikishpa
This document discusses floating point representation of numbers in computers. It explains that there are two types of computer arithmetic: integer arithmetic and real arithmetic. Real arithmetic uses numbers with fractional parts and includes fixed point arithmetic and floating point arithmetic. Fixed point arithmetic represents numbers in binary form with a sign bit, integral part, and fractional part. Floating point representation uses scientific notation and normalized notation to represent numbers with a sign bit, mantissa, and exponent. It allows for a much larger range of numbers than fixed point representation.
This document discusses number systems and data representation in computers. It covers topics like binary, decimal, hexadecimal, and ASCII number systems. Some key points covered include:
- Computers use the binary number system and positional notation to represent data precisely.
- Different number systems have different bases (like binary base-2, decimal base-10, hexadecimal base-16).
- Methods for converting between number systems like binary to decimal and hexadecimal to decimal.
- Signed and unsigned integers, ones' complement, twos' complement representation of negative numbers.
- ASCII encoding of characters and how to convert between character and numeric representations.
This document discusses distributed systems and provides examples of distributed system architectures and components. It defines distributed systems as systems where components located at networked computers communicate and coordinate their actions through message passing. It provides examples of distributed systems including the Internet, intranets, and mobile/ubiquitous computing. It also discusses challenges in distributed systems like heterogeneity, security, scalability, and concurrency control.
The document discusses the binary number system, explaining that it uses only two digits, 0 and 1, with each column in a binary number being twice the value of the previous column. It covers converting between binary and decimal numbers, with examples, and explains that binary is important because computers represent information using electronic switches that can be either on or off, corresponding to 1 and 0.
This document discusses numerical methods and errors. It introduces that numerical methods provide approximate solutions rather than exact analytical solutions due to errors from measurements, algorithms, and output. Accuracy refers to how close an approximation is to the true value, while precision refers to the reproducibility of results. Significant figures indicate the precision of a number. True error, relative error, and percent error are defined to quantify the error between approximations and true values. Round-off errors from floating point representation on computers are also discussed.
Canonical forms express Boolean functions as a sum of minterms or product of maxterms. Standard forms, like canonical forms, can have variables that do not appear in each term. Canonical forms are not usually minimal, while standard forms are simplified versions of canonical forms. Boolean functions can be expressed in Sum of Products or Product of Sum forms, and minimized using algebraic methods or Karnaugh maps.
The document discusses digital communication systems. It provides examples of digital communication including an email sent to invite team members to a meeting. It then explains the key building blocks of a digital communication system including the input source, source encoder, channel encoder, digital modulator, channel, digital demodulator, channel decoder, source decoder and output transducer. The document also discusses channels used for digital communication, causes of signal loss, and comparisons between digital and analog communication systems.
This document discusses floating point number representation in IEEE-754 format. It explains that floating point numbers consist of a sign bit, exponent, and mantissa. It describes single and double precision formats, which use excess-127 and excess-1023 exponent biases respectively. Examples are given of representing sample numbers in both implicit and explicit normalized forms using single and double precision formats.
Linear differential equation with constant coefficientSanjay Singh
The document discusses linear differential equations with constant coefficients. It defines the order, auxiliary equation, complementary function, particular integral and general solution. It provides examples of determining the complementary function and particular integral for different types of linear differential equations. It also discusses Legendre's linear equations, Cauchy-Euler equations, and solving simultaneous linear differential equations.
The document provides an overview of SONET (Synchronous Optical Networking) and SDH (Synchronous Digital Hierarchy) architectures and technologies. It describes the four layers of SONET (path, line, section, photonic), SONET frame structure including overhead bytes, and how lower-rate STS frames are multiplexed into higher-rate frames. It also discusses different types of SONET networks including linear, ring and mesh, as well as the use of virtual tributaries to transport digital signals of different rates over SONET.
Contents:
1.What is number system?
2.Conversions of number from one radix to another
3.Complements (1's, 2's, 9's, 10's)
4.Binary Arithmetic ( Addition, subtraction, multiplication, division)
The document discusses the principle of mathematical induction and how it can be used to prove statements about natural numbers. It provides examples of using induction to prove statements about sums, products, and divisibility. The principle of induction states that to prove a statement P(n) is true for all natural numbers n, one must show that P(1) is true and that if P(k) is true, then P(k+1) is also true. The document provides examples of direct proofs of P(1) and inductive proofs of P(k+1) to demonstrate applications of the principle.
The document provides information on digital and analog signals, different number systems used in computing including binary, octal, decimal and hexadecimal. It explains:
- Digital signals have discrete amplitude values of 0V and 5V, while analog signals can have any amplitude value.
- Number systems like binary, octal and hexadecimal are used in computing to represent values using discrete digits. Conversion between number systems involves place value weighting.
- Binary uses two digits 0 and 1. Octal uses eight digits 0-7. Hexadecimal uses sixteen digits and letters 0-9 and A-F. Conversion between number systems and decimal is done by successive multiplication or division.
This chapter discusses digital systems and number conversion. Digital systems use discrete values rather than continuous values as in analog systems. They can provide exact outputs. The chapter covers converting between number bases, such as decimal to binary, using division or multiplication. It also addresses representing negative numbers and binary codes. The design of digital systems includes system, logic, and circuit design. Combinational and sequential circuits are introduced.
This document contains instructions for conducting network simulation experiments using the NCTUns simulator. It discusses setting up NCTUns, drawing network topologies, editing node properties, running simulations, and performing post-analysis. Experiment 1 involves simulating a 3-node point-to-point network with duplex links, varying the bandwidth, and measuring the number of dropped packets. The steps provided outline how to draw the topology in NCTUns and configure the nodes before running the simulation.
Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
This document discusses binary coded decimal (BCD). It defines BCD as a numerical code that assigns a 4-bit binary code to each decimal digit from 0 to 9. Numbers larger than 9 are expressed digit by digit in BCD. BCD is used because it is easy to encode/decode decimals and useful for digital systems that display decimal outputs. The document also describes how addition and subtraction are performed in BCD through binary addition rules and handling carries.
Weighted codes assign a positional weight or value to each digit, where the sum of the digit values multiplied by their weights represents the number. Non-weighted codes do not assign positional weights. BCD is a weighted 4-bit code that represents the decimal digits 0-9. It uses weights of 24, 23, 22, 21 from most to least significant bit. The Gray code is a non-weighted code where each number differs from the previous by one bit. Excess-3 code is a non-weighted 4-bit BCD code where 3 is added to each decimal digit before conversion to BCD.
This document discusses different types of number complements that allow subtraction to be performed using addition. It explains that subtraction using borrowing is inefficient for computers, so instead subtraction is done by taking the complement of one of the numbers and then adding. For binary numbers, the 2's complement is commonly used, where the 2's complement of a number is found by flipping all the bits and adding 1. The document provides examples of calculating 1's, 2's, and other bases' complements.
Introduction to Analog and Digital Systems - Basic definition, Representation, Examples and applications of Analog and Digital Systems - Advantages of Digital system over Analog system - Process of conversion from Analog to Digital and Digital to Analog signal - Digitization Examples - Signal representation of voltage and current in terms of Binary values - Representations of Binary quantities using different terminologies - IC Complexity classification - IC Layout - Development of ICs in terms of size
Manchester & Differential Manchester encoding schemeArunabha Saha
The two main variants of biphase encoding techniques are discussed here. Manchester and Differential Manchester encoding scheme are explained with examples. Comparison between several classes of polar encoding techniques are done along with the exposure about the advantages and disadvantages of both schemes.
Floating Point Representation premium.pptxshomikishpa
This document discusses floating point representation of numbers in computers. It explains that there are two types of computer arithmetic: integer arithmetic and real arithmetic. Real arithmetic uses numbers with fractional parts and includes fixed point arithmetic and floating point arithmetic. Fixed point arithmetic represents numbers in binary form with a sign bit, integral part, and fractional part. Floating point representation uses scientific notation and normalized notation to represent numbers with a sign bit, mantissa, and exponent. It allows for a much larger range of numbers than fixed point representation.
This document discusses number systems and data representation in computers. It covers topics like binary, decimal, hexadecimal, and ASCII number systems. Some key points covered include:
- Computers use the binary number system and positional notation to represent data precisely.
- Different number systems have different bases (like binary base-2, decimal base-10, hexadecimal base-16).
- Methods for converting between number systems like binary to decimal and hexadecimal to decimal.
- Signed and unsigned integers, ones' complement, twos' complement representation of negative numbers.
- ASCII encoding of characters and how to convert between character and numeric representations.
This document discusses distributed systems and provides examples of distributed system architectures and components. It defines distributed systems as systems where components located at networked computers communicate and coordinate their actions through message passing. It provides examples of distributed systems including the Internet, intranets, and mobile/ubiquitous computing. It also discusses challenges in distributed systems like heterogeneity, security, scalability, and concurrency control.
Introduction: OSI Security Architecture, Security attacks, ,Security Services, Security
Mechanisms, Model for Network Security, Fundamentals of Abstract Algebra : Groups, Rings,
Fields, Modular Arithmetic, Euclidean Algorithm, Finite Fields of the form GF(p),Polynomial
Arithmetic, Finite Fields of the form GF(2n),Classical Encryption techniques, Block Ciphers and
Data Encryption Standard.
Machine Language Instruction Formats – Instruction Set of 8086-Data transfer
instructions,Arithmetic and Logic instructions,Branch instructions,Loop instructions,Processor
Control instructions,Flag Manipulation instructions,Shift and Rotate instructions,String
instructions, Assembler Directives and operators,Example Programs,Introduction to Stack,
STACK Structure of 8086, Interrupts and Interrupt Service Routines, Interrupt Cycle of 8086,
Non-Maskable and Maskable Interrupts, Interrupt Programming, MACROS.
This document discusses various topics related to the internet including what the internet is, why we need it, the world wide web, how to access and bookmark websites, search engines, and specialized search engines. It defines the internet as the largest network connecting computer networks around the world. It explains that the world wide web is a system of hyperlinked documents accessed via the internet and the most important service it provides. It also discusses how to search the internet using search engines and save pages or images, and lists some specialized search engines for specific topics like companies, people, images, jobs, games, health, and education.
Advanced Encryption Standard, Multiple Encryption and Triple DES, Block Cipher Modes of
operation, Stream Ciphers and RC4, Confidentiality using Symmetric Encryption, Introduction
to Number Theory: Prime Numbers, Fermat’s and Euler’s Theorems, Testing for Primality, The
Chinese Remainder Theorem, Discrete Logarithms, Public-Key Cryptography and RSA
The document discusses input/output (I/O) organization and buses. It describes how I/O devices connect to the processor and memory via a shared bus. It discusses different I/O access methods like memory-mapped I/O and special I/O instructions. It also covers I/O interfaces, interrupts, direct memory access (DMA), and bus arbitration methods.
This document discusses logic gates and Boolean algebra. It begins by defining basic logic gates like AND, OR, and NOT. It then covers more advanced gates like NAND, NOR, XOR, and XNOR and provides their truth tables. The document explains how to implement logic functions using gates. It also covers Boolean algebra topics like Boolean functions, minterms, maxterms, SOP, POS, Karnaugh maps, and their use in minimizing logic expressions. Worked examples are provided for implementing functions with gates and simplifying expressions using K-maps.
Analysis and design of algorithms part2Deepak John
Analysis of searching and sorting. Insertion sort, Quick sort, Merge sort and Heap sort. Binomial Heaps and Fibonacci Heaps, Lower bounds for sorting by comparison of keys. Comparison of sorting algorithms. Amortized Time Analysis. Red-Black Trees – Insertion & Deletion.
The document discusses the memory system in computers including main memory, cache memory, and different types of memory chips. It provides details on the following key points in 3 sentences:
The document discusses the different levels of memory hierarchy including main memory, cache memory, and auxiliary memory. It describes the basic concepts of memory including addressing schemes, memory access time, and memory cycle time. Examples of different types of memory chips are discussed such as SRAM, DRAM, ROM, and cache memory organization and mapping techniques.
Key Management, Diffie-Hellman Key Exchange, Elliptic Curve Arithmetic, Elliptic Curve
Cryptography, Message Authentication and Hash Functions, Hash and MAC Algorithms
Digital Signatures and Authentication Protocols
1. Real-time systems are systems where the correctness depends on both the logical result and the time at which the results are produced.
2. Real-time systems have performance deadlines where computations and actions must be completed. Deadlines can be time-driven or event-driven.
3. Real-time systems are classified as hard, firm, or soft depending on how critical meeting deadlines are. They are used in applications like medical equipment, automotive systems, and avionics.
Registers are groups of flip-flops that store binary data. Shift registers can transfer data in serial or parallel formats. There are four basic modes of shift registers: serial-in serial-out, serial-in parallel-out, parallel-in serial-out, and parallel-in parallel-out. Counters are circuits made of flip-flops that count clock pulses and can be asynchronous, synchronous, decade, up/down, or cascaded to achieve different counts.
Network Security: Authentication Applications, Electronic Mail Security, IP Security, Web
Security, System Security: Intruders, Malicious Software, Firewalls
Analysis and design of algorithms part 4Deepak John
Complexity Theory - Introduction. P and NP. NP-Complete problems. Approximation algorithms. Bin packing, Graph coloring. Traveling salesperson Problem.
Distributed operating systems present users with an integrated computing platform that hides individual computers. They control all nodes in a network and allocate resources without user involvement. Distributed OS examples include cluster computer systems, V system, and Sprite. Middleware implements network-wide programming abstractions like RPC, event distribution, and resource discovery. The core OS functionality distributed OSs should provide for middleware includes encapsulation, protection, concurrent processing, and invocation mechanisms.
The document provides information about different number systems used in computers, including binary, octal, hexadecimal, and decimal. It explains the characteristics of each system such as the base and digits used. Methods for converting between number systems like binary to decimal and vice versa are presented. Shortcut methods for direct conversions between binary, octal, and hexadecimal are also described. Binary arithmetic and binary-coded decimal number representation are discussed.
FYBSC IT Digital Electronics Unit I Chapter I Number System and Binary Arithm...Arti Parab Academics
Number System:
Analog System, digital system, numbering system, binary number
system, octal number system, hexadecimal number system, conversion
from one number system to another, floating point numbers, weighted
codes binary coded decimal, non-weighted codes Excess – 3 code, Gray
code, Alphanumeric codes – ASCII Code, EBCDIC, ISCII Code,
Hollerith Code, Morse Code, Teletypewriter (TTY), Error detection
and correction, Universal Product Code, Code conversion.
In digital computers, data is stored and represented using binary digits (bits) of 1s and 0s. There are different number systems that can represent numeric values, including binary, decimal, octal and hexadecimal. Each system has a base or radix, with binary having a base of 2, decimal 10, octal 8 and hexadecimal 16. Numbers can be converted between these systems using division and multiplication by the radix at each place value.
The document provides information about digital electronics and digital systems. It introduces digital logic and how digital systems represent information using discrete binary values of 0 and 1. Digital computers are able to manipulate this discrete digital data through programs. Common number systems like binary, octal, hexadecimal and their conversions to decimal are explained. Signed and unsigned binary numbers are also discussed.
Digital computers represent data by means of an easily identified symbol called a digit. The data may
contain digits, alphabets or special character, which are converted to bits, understandable by the computer.
In Digital Computer, data and instructions are stored in computer memory using binary code (or
machine code) represented by Binary digIT’s 1 and 0 called BIT’s.
The number system uses well-defined symbols called digits.
Number systems are classified into two types:
o Non-positional number system
o Positional number system
This document discusses various data representation systems used in computers, including:
- Binary, decimal, hexadecimal, and octal number systems. Binary uses two digits (0,1) while other systems use bases of 10, 16, and 8 respectively.
- Units of data representation such as bits, bytes, kilobytes, megabytes and gigabytes which are used to measure computer storage.
- Methods for converting between number systems, including dividing numbers into place values and multiplying digits by their place values.
- Special codes like Binary Coded Decimal (BCD) which represents each decimal digit with 4 binary bits.
- Binary arithmetic operations and how addition works the same in any number system by following
This document discusses different number systems used in digital computers and their conversions. It begins with an introduction to digital number systems and then describes the decimal, binary, octal and hexadecimal number systems. It explains how to represent integers and real numbers in binary. The document also covers number conversions between these systems using different methods like repeated division. Finally, it discusses various ways of representing integers in binary like sign-magnitude, one's complement and two's complement representations.
Logic Circuits Design - "Chapter 1: Digital Systems and Information"Ra'Fat Al-Msie'deen
Logic Circuits Design: This material is based on chapter 1 of “Logic and Computer Design Fundamentals” by M. Morris Mano, Charles R. Kime and Tom Martin
The document discusses different number systems used in computers such as binary, decimal, octal and hexadecimal. It provides examples and techniques for converting between these number systems. The key number systems covered are binary, which uses two digits (0 and 1), and is used in computers, decimal which uses 10 digits and is used in everyday life, octal which uses 8 digits, and hexadecimal which uses 16 digits and letters A-F. The document also discusses techniques for converting fractions between decimal and binary.
This document provides an introduction to a digital design course. It discusses the recommended textbook, course description, grading breakdown, and course outline. The course focuses on fundamental digital concepts like number systems, Boolean algebra, logic gates, combinational and sequential logic. It will cover topics such as binary numbers, Boolean functions, logic gate minimization, adders/subtractors, multiplexers, flip-flops, and finite state machines. Students are expected to attend every lecture and participate in classroom discussions. Grades will be based on projects, midterm exams, and quizzes/assignments.
This document provides an overview of Boolean algebra and logic gates. It begins with reviewing binary number systems, binary arithmetic, and binary codes. It then covers Boolean algebra, truth tables, canonical and standard forms. It also discusses logic operations and logic gates like Karnaugh maps up to 6 variables including don't care conditions. Finally, it discusses sum of products and products of sum representations.
Introduction
Number Systems
Types of Number systems
Inter conversion of number systems
Binary addition ,subtraction, multiplication and
division
Complements of binary number(1’s and 2’s
complement)
Grey code, ASCII, Ex
3,BCD
This document summarizes different number systems used in computing including binary, octal, decimal, and hexadecimal. It explains how to convert between these number systems using theorems about their bases. Key topics covered include binary arithmetic, signed and unsigned integer representation, and how floating point numbers and characters are stored in binary format. Conversion charts are provided for binary to octal and hexadecimal. Representations of integers, characters, and floating point numbers in binary are also summarized.
1. The document discusses different number systems including binary, decimal, octal, and hexadecimal.
2. It provides methods for converting between these number systems, which involve repeatedly dividing or multiplying by the base and taking remainders or carry values.
3. Examples are given of converting decimal numbers to and from binary, octal, and hexadecimal representations, as well as converting between these number systems.
This document discusses different number systems used in computers such as binary, decimal, octal and hexadecimal. It provides examples of converting between these number systems. The key points are:
- Computers use the binary number system and understand numbers as sequences of 0s and 1s.
- Other common number systems include decimal, octal and hexadecimal, which use different bases.
- Methods for converting between number systems include dividing the number by the new base or using shortcuts that group digits in specific ways.
The document outlines a lesson plan covering number systems. It includes converting between decimal, binary, octal, and hexadecimal number systems. The key concepts covered are the different number systems used in computing, including binary, octal, hexadecimal, and their bases. Conversion between these systems involves multiplying digits by place values to get the value in another base. The skills practiced are computational thinking and step-wise thinking. Values reinforced include awareness of computer technology development and patience.
This document discusses different number systems including decimal, binary, octal, and hexadecimal. It explains how numbers are represented in each system using positional notation. Conversion between these number systems is demonstrated through examples. The document also covers signed integer representation methods like sign-and-magnitude, one's complement, and two's complement. Finally, it briefly introduces representation of characters using coding standards.
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3) Data backup creates additional copies of data that can be used to restore files after data loss or corruption. Backup methods include full
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Module 1 number systems and code1
1. MODULE 1 MCA-102 DIGITAL SYSTEMS & LOGIC DESIGN ADMN 2014-‘17
Dept. of Computer Science And Applications, SJCET, Palai Page 1
NUMBER SYSTEMS
"A set of values used to represent different quantities is known as Number System". For example, a
number system can be used to represent the number of students in a class or number of viewers watching a
certain TV program etc. The digital computer represents all kinds of data and information in binary
numbers. It includes audio, graphics, video, text and numbers. The total number of digits used in a number
system is called its base or radix. A value of each digit in a number can be determined using
The digit
The position of the digit in the number
The base of the number system (where base is defined as the total number of digits available in the
number system).
Efficiency of Number System
Number systems provide the basis for all operations in information processing systems.
In a number system the information is divided into a group of symbols
A group of bits which is used to represent the discrete elements of information is a symbol.
The most common are the decimal, binary, octal, and hexadecimal systems.
Bits and Bytes
• A binary digit is a single numeral in a binary number.
• Each 1 and 0 in the number below is a binary digit:
– 1 0 0 1 0 1 0 1
• The term “binary digit” is commonly called a “bit.”
• Eight bits grouped together is called a “byte.”
Positional Number Systems
We are all familiar with the traditional base-10 number system, which uses 10 digits (0 to 9) and in
which the positions increase in value by powers of 10 from right to left. For example, the number
487 is interpreted as 4 hundreds plus 8 tens plus 7 ones.
Other number systems work similarly, using different numbers for their bases. In computer science
we are particularly interested in binary, octal, and hexadecimal systems, which are base 2, 8, and 16
respectively.
2. MODULE 1 MCA-102 DIGITAL SYSTEMS & LOGIC DESIGN ADMN 2014-‘17
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The relationship between these number systems can be seen in the following table, where the zero
position refers to the rightmost digit of a number, and the ^ symbol represents exponentiation:
Position
Value in base:
10 2 8 16
0 10^0 = 1 2^0 = 1 8^0 = 1 16^0 = 1
1 10^1 = 10 2^1 = 2 8^1 = 8 16^1 = 16
2 10^2 = 100 2^2 = 4 8^2 = 64 16^2 = 256
3 10^3 = 1000 2^3 = 8 8^3 = 512 16^3 = 4,096
4 10^4 = 10,000 2^4 = 16 8^4 = 4,096 16^4 = 65,536
5 10^5 = 100,000 2^5 = 32 8^5 = 32,768 16^5 = 1,048,576
6 10^6 = 1,000,000 2^6 = 64 8^6 = 262,144 16^6 = 16,777,216
7 10^7 = 10,000,000 2^7 = 128 8^7 = 2,097152 16^7 = 268,435,456
Computer Number Systems
1. Binary Numbers
2. Decimal Numbers
3. Octal Numbers
4. Hexadecimal Numbers
1. Binary Numbers
Digital computer represents all kinds of data and information in the binary system. Binary Number
System consists of two digits 0 and 1. Its base is 2. Each digit or bit in binary number system can be 0 or 1.
A combination of binary numbers may be used to represent different quantities like 1001. The positional
value of each digit in binary number is twice the place value or face value of the digit of its right side. The
weight of each position is a power of 2.
Numbering of the digits
msb lsb
n-1 0
Where n is the number of digits in the number. (msb stands for most significant bit, lsb stands for least
significant bit).
Example
3. MODULE 1 MCA-102 DIGITAL SYSTEMS & LOGIC DESIGN ADMN 2014-‘17
Dept. of Computer Science And Applications, SJCET, Palai Page 3
Binary Number: 101012
2. Decimal Number
The Decimal Number System consists of ten digits from 0 to 9. These digits can be used to represent any
numeric value. The base of decimal number system is 10. It is the most widely used number system. The
value represented by individual digit depends on weight and position of the digit.
The value of the number is determined by multiplying the digits with the weight of their position and
adding the results. This method is known as expansion method. The rightmost digit of number has the
lowest weight. This digit is called Least Significant Digit (LSD). The leftmost digit of a number has the
highest weight. This digit is called Most Significant Digit (MSD). For example, the decimal number 1234
consists of the digit 4 in the units position, 3 in the tens position, 2 in the hundreds position, and 1 in the
thousands position, and its value can be written as
(1x1000)+ (2x100) + (3x10) + (4xl)
(1x103
)+ (2x102
) + (3x101
) + (4xl00
)
1000 + 200 + 30 + 4
1234
Decimal to Binary conversion
Step – 1: Divide the decimal number to be converted by the value of the new base. In this case divide it by
2.
Step – 2: Record the remainder from Step – 1 as the rightmost digit.
Step – 3: Divide the quotient of the previous by the new base.
Step – 4: Record the remainder from Step – 3 as the next digit (to the left) of the new base number.
Step – 5: Bottom to top sequence of remainder will be the required converted number. Repeat Step – 3 &
Step – 4, recording remainders from right to left, until the quotient becomes less than the digit of new base
so that it cannot be divided.
Ex:
4. MODULE 1 MCA-102 DIGITAL SYSTEMS & LOGIC DESIGN ADMN 2014-‘17
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The fractional part is again multiplied by 2 and the process repeated.
Example: convert (0.68)10 to binary fraction.
0.68 * 2 = 1.36 integer part is 1
0.36 * 2 = 0.72 integer part is 0
0.72 * 2 = 1.44 integer part is 1
0.44 * 2 = 0.88 integer part is 0
Answer = 0. 1 0 1 0…..
Example: Convert (0.625)10
0.625 x 2 = 1.25(Product) Fractional part=0.25 Carry=1 (MSB)
0.25 x 2 = 0.50(Product) Fractional part = 0.50 Carry = 0
0.50 x 2 = 1.00(Product) Fractional part = 1.00 Carry = 1 (LSB)
The fractional part in the 3rd iteration becomes zero and hence we stop the multiplication iteration.
Carry from the 1st multiplication iteration becomes MSB and carry from 3rd iteration becomes LSB.
Hence, the fractional binary number of the given fractional decimal number (0.625)10 is (0.101)2.
Binary to Decimal Conversion
• Step 1: Multiply each bit by 2n
, where n is the “weight” of the bit
• Step 2: The weight is the position of the bit, starting from 0 on the right
Step 3:Add the results
Ex: Convert the binary number 101012 to decimal.
Step Binary Number Decimal Number
Step 1 101012 ((1 x 24
) + (0 x 23
) + (1 x 22
) + (0 x 21
) + (1 x 20
))10
Step 2 101012 (16 + 0 + 4 + 0 + 1)10
Step 3 101012 2110
1. Convert (101.101)2
↑ ↑
MSB LSB
= 1 x 22
+ 0 x 21
+ 1 x 20
. 1 x 2-1
+ 0 x 2-2
+ 1 x 2-3
= 1 x 4 + 0 x 2 + 1 x 1. 1 x ( 1 / 2 ) + 0 x ( 1 / 4 ) + 1 x ( 1 / 8 )
5. MODULE 1 MCA-102 DIGITAL SYSTEMS & LOGIC DESIGN ADMN 2014-‘17
Dept. of Computer Science And Applications, SJCET, Palai Page 5
= 4 + 0 + 1 . (1 / 2) + 0 + (1 / 8)
= 5. 0.5 + 0.125
= 5. 625
Therefore (1 0 1. 1 0 1)2 = (5.625)10
3. Convert (0.0001)2
= 0 x 20
. 0 x 2-1
+ 0 x 2-2
+ 0 x 2-3
+ 1 x 2-4
= 0 x 1. 0 x ( 1 / 2 ) + 0 x ( 1 / 4 ) + 0 x ( 1 / 8 ) + 1 x ( 1 / 16 )
= 0. 0 + 0 + 0 + (1 / 16)
= 0. 0.0625
= 0. 0625
Therefore (0. 0 0 0 1)2 = (0.0625)10
4. Convert (101010.1111)2
= 1 x 25
+ 0 x 24
+ 1 x 23
+ 0 x 22
+ 1 x 21
+ 0 x 20
. 1 x 2-1
+ 1 x 2-2
+ 1 x 2-3
+ 1 x 2-4
= 1 x 32 + 0 x 16 + 1 x 8 + 0 x 4 + 1 x 2 + 0 x 1. 1 x ( 1 / 2 ) + 1 x ( 1 / 4 ) + 1 x ( 1 / 8 ) + 1 x ( 1 / 16 )
= 32 + 0 + 8 + 0 + 2 + 0 . (1 / 2) + ( 1 / 4 ) + ( 1 / 8 ) + ( 1 / 16 )
= 32 + 8 + 2 . ( 0.5 ) + ( 0.25 ) + ( 0.125 ) + ( 0.0625 )
= 42. 9375
Therefore (1 0 1 0 1 0. 1 1 1 1)2 = (42.9375)10
1. Octal Number System
Base or radix 8 number system.
1 octal digit is equivalent to 3 bits.
Octal numbers are 0 to7.
Numbers are expressed as powers of 8.
For example:
Octal to Binary
Converting from octal to binary is as easy as converting from binary to octal. Simply look up each octal
digit to obtain the equivalent group of three binary digits.
6. MODULE 1 MCA-102 DIGITAL SYSTEMS & LOGIC DESIGN ADMN 2014-‘17
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Octal: 0 1 2 3 4 5 6 7
Binary: 000 001 010 011 100 101 110 111
Octal to decimal
Example: convert (632)8 to decimal
= (6 x 82) + (3 x 81) + (2 x 80)
= (6 x 64) + (3 x 8) + (2 x 1)
= 384 + 24 + 2
= (410)10
Ex: (407.304)8
= 4 x 8 2
+ 0 x 8 1
+ 7 x 8 0
. 3 x 8 -1
+ 0 x 8 -2
+ 4 x 8 -3
= 4 x 8 2
+ 0 x 8 1
+ 7 x 8 0
. 3 x 8 -1
+ 0 x 8 -2
+ 4 x 8 -3
= 4 x 64 + 0 x 8 + 7 x 1.
3 x ( 1 / 8 ) + 0 x ( 1 / 64 ) + 4 x ( 1 / 512 )
= 256 + 0 + 7 . (0. 375) + (0) + (0. 0078125)
= (263 . 3828125)10
Decimal to octal
To convert from decimal to octal, the successive-division procedure or the sum of weights procedure
can be used
Ex: Convert (177)10 to octal
177 / 8 = 22 remainder is 1
22 / 8 = 2 remainder is 6
2 / 8 = 0 remainder is 2
Answer = 2 6 1
Binary to octal
group the binary positions in groups of three
Convert the following binary numbers into octal: a) 10110111 b) 01101100
Solution
10110111 = 010 110 111 = 267
01101100 = 001 101 100 = 154
2. Hexa decimal number system
Base or radix 16 number system.
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1 hex digit is equivalent to 4 bits.
Numbers are 0, 1, 2…..8, 9, A, B, C, D, E, and F.
B is 11, E is 14.
Numbers are expressed as powers of 16.
Hex to binary
replace each hexadecimal number with four equivalent binary numbers even if the number can be
represented by less than four bits
Convert the following hexadecimal number into binary: a) A2E b)60F
Solution:
a) (A2E)16 = 1010 0010 1110
= (101000101110)2
b) (60F)16 = 0110 0000 1111 = (011000001111)2
Binary to hexadecimal
grouping the binary positions in groups of four
Convert the following binary numbers into hexadecimal: a) 10101111 b) 01101100
Solution:
10110111 = 1011 0111 = (B 7)16
01101100 = 0110 1100 = (6 C)16
Hex to decimal
To convert from hexadecimal to decimal, (multiply by weighting factors).
Convert (7AD) 16 to decimal.
Solution:
(7AD)16 = 7 x 162
+ 10 x 161
+ 13 x 160
= (1965)10
Decimal to hex
To convert from decimal to hexadecimal, the successive-division procedure or the sum of weights
procedure can be used.
Convert the following decimal numbers to hexadecimal: a) (596)10 b) (100)10
Solution:
596 ÷ 16 = 37 remainder 4
37 ÷ 16 = 2 remainder 5
2 ÷ 16 = 0 remainder 2
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0 0 1 0 0 0 1 0 1 = 69(base 10)
Note: The rules of binary multiplication are the same as the truths of the AND gate.
Another Method: Binary multiplication is the same as repeated binary addition; add the multiplicand to
itself the multiplier number of times.
For example,
00001000 × 00000011 = 00011000
0 0 0 0 1 0 0 0 = 8(base 10)
0 0 0 0 1 0 0 0 = 8(base 10)
+ 0 0 0 0 1 0 0 0 = 8(base 10)
0 0 0 1 1 0 0 0 = 24(base 10)
Binary Division
For example,
Fig 1.2 example for binary division
REPRESENTATION OF SIGNED NUMBERS
There are three basic ways to represent signed numbers:
Sign-magnitude.
1’s complement.
12. MODULE 1 MCA-102 DIGITAL SYSTEMS & LOGIC DESIGN ADMN 2014-‘17
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2’s complement.
Sign-Magnitude
The number consists of two parts:
the MSB (most significant bit) represents the sign
The other bits represent the magnitude of the number.
If the sign bit is 1 the number is negative and if it is 0 the number is positive
Examples
I. -30 = 1 0011110 (The leftmost 1 indicates that the number is negative. The remaining 7-bits
carry the magnitude of 30).
II. 30 = 0 0011110 (The only difference between –30 and +30 is the sign bit because the
magnitude bits are similar in both numbers.).
III. -121 = 1 1111001.
IV. 99 = 0 1100011
1’s Complement
Negative numbers are represented in 1’s complement format.
positive numbers are represented as the positive sign-magnitude numbers
Examples
30 = 00011110
-30 = 11100001
the number equals the 1’s complement of 30
121 = 01111001
-121 = 10000110
the number equals the 1’s complement of 121
99 = 01100011
2’s Complement
The two’s complement of a binary integer is the 1’s complement of the number plus 1.”
Thus if m is the 2’s complement of n, then: m = n + 1.
Examples:
n = 0101 0100, then m = 1010 1011 + 1 = 1010 1100
n = 0101 1111, then m = 1010 0000 + 1 = 1010 0001
n = 0111 1111, then m = 1000 0000 + 1 = 1000 0001
n = 0000 0001, then m = 1111 1110 + 1 = 1111 1111
13. MODULE 1 MCA-102 DIGITAL SYSTEMS & LOGIC DESIGN ADMN 2014-‘17
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To convert a negative decimal number to 2 To convert a negative decimal number to 2 s’
complement binary: complement binary:
i. Convert the decimal number to a positive binary number.
ii. Take the 1’s complement of that binary number and add 1.
• Ex
– 50: 50 = 0011 0010; 1’s C. = 1100 1101; 2’s C. = 1100 1110.
– 127: 127 = 0111 1111; 1’s C. = 1000 0000; 2’s C. = 1000 0001.
– 1: 1 = 0000 0001; 1’s C. = 1111 1110; 2’s C. =1111 1111
ADDITION AND SUBTRACTION USING 2’S AND I’S COMPLEMENT
1’s and 2’s complement allow the Representation of Negative numbers (-) in binary.
1's complement
The 1's complement of a binary number is found by simply changing all 1s to 0s and all 0s to 1s.
Examples
The 1’s complement of 10001111 = 01110000.
The 1’s complement of 01101100 = 10010011.
The 1’s complement of 00110011 = 11001100.
2's complement
The 2's complement of a binary number is found by adding 1 to the LSB of the 1’s complement.
Another way of obtaining the 2’s complement of a binary number is to start with the LSB (the
rightmost bit) and leave the bits unchanged until you find the first 1. Leave the first 1 unchanged
and complement the rest of the bits (change 0 to 1 and 1 to 0).
Example
The 2’s complement of 10001111 = 01110000 +1 = 01110001
The 2’s complement of 01101100 = 10010011 + 1 =10010100
The 2’s complement of 00110011 = 11001100 + 1 = 11001101
Addition of 2’s complement binary numbers
Subtraction
There are two possibilities
a) M>=N
b) M<N
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If M>=N then follow the given steps
Steps
1: Add the minuend M to the r’s complement of the subtrahend N
2: inspect the result obtained in step 1 for an end carry
a) If an end carry occurs discard it.
b) If an end carry doesn’t occur, take the r’s complement of the number obtained in step 1
and place a negative sign in front.
If M<N then follow the above steps with given additional steps
3. Answer is obtained by taking a r’s compliment of obtained sum and adding a negative sign
Subtraction using 1's complement
The method of binary subtraction becomes very easy with the help of 1's complement. Now let us look at
an example to understand subtraction using 1's complement. –
Suppose A = (0 1 0 1)2
And B = (0 0 1 1)2
And we want to find out A - B
For this first we have to calculate 1's complement of B
1's complement of B = 1 1 0 0
Now we have to add the result with A
Now in the result we can see that there is an overflowing bit which we have to add with the remaining
result
This is the desired result. And when there will not be any overflowing digit the result obtained in the
previous stage will be the answer.
Subtraction using 2’s complement
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One of the most popular applications of 2's complement is the subtraction of binary numbers using 2's
complement method. This method is preferred because here subtraction can be done by doing additions.
With an example we will be able to grasp the method at once.
We want to do 7 - 12
Step1
Taking 2's complement of minuend (12) which is = 1 1 1 1 0 1 0 0
Step 2
Adding the result with binary equivalent of 7
So the subtraction method using 2's complement method is hereby explained.
Binary codes
• Digital data is represented, stored and transmitted as group of binary bits.
• This group is called binary code.
• The binary code can be used for represent the number as well as alphanumeric letters.
1. Weighted Binary Systems- Weighted binary codes are those which obey the positional weighting
principles, each position of the number represents a specific weight.
2. Non Weighted Codes- Non weighted codes are codes that are not positionally weighted. That is,
each position within the binary number is not assigned a fixed value.
Binary Coded Decimal (BCD)
Represent each decimal digit as a 4-bit binary code.
Binary-Coded Decimal is a weighted code because each decimal digit can be obtained from its code
word by assigning a fixed weight to each code-word bit.
The weights for the BCD bits are 8, 4, 2, and 1, and for this reason the code is sometimes called the
8421 code.
Examples:
(234)10 = (0010 0011 0100)BCD
(7093)10 = (0111 0000 1001 0011)BCD
(1000 0)BCD = (86)10
(1001 (1001 0100 0111 0010)BCD = (9472)10
Notes: BCD is not equivalent to binary.
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Example: (234)10 = (11101010)2
GRAY Code
• It is called cyclic or reflected code.
• In this code each code group does not differ from its neighbor in more than one bit.
• This code is used for input and output devices in digital system.
Decimal Binary Gray Code Decimal Binary Gray code
0 0000 0000 8 1000 1100
1 0001 0001 9 1001 1101
2 0010 0011 10 1010 1111
3 0011 0010 11 1011 1110
4 0100 0110 12 1100 1010
5 0101 0111 13 1101 1011
6 0110 0101 14 1110 1001
7 0111 0100 15 1111 1000
Fig 1.2 GRAY code system
Binary into Gary Code
The following Method can be used
1. Write down binary form of the given decimal number.
2. Write MSB as such.
3. Then XOR the binary digit from left to right at the adjacent position.
4. Discard carry if any.
5. Write the digit which comes after addition.
Fig 1.3 binary –gray conversion
Gray – binary
• Method:
1. Write the given grey code.
2. Write the MSB bit as such.
3. XOR this bit to the second left most bit, write the result, and discard carry.
4. Add this result to the next left most bit diagonally.
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Fig 1.4 Gray –Binary conversion
Excess -3 Code
This counting system starts from a count of binary three (0011) then counts up.
Fig 1.5 excess 3 code
Alphanumeric Codes
Apart from numbers, computers also handle textual data.
Character set frequently used includes:
alphabets: ‘A’ .. ‘Z’, and ‘a’ .. ‘z’
digits: ‘0’ .. ‘9’
special symbols: ‘$’, ‘.’, ‘,’, ‘@’, ‘*’, …
Alphanumeric codes are codes used to encode the characters of alphabets in addition to the decimal
digits. They are used primarily for transmitting data between computers and its I/O devices.
Usually, these characters can be represented using 7 or 8 bits.
ASCII
The most commonly used character code is ASCII (the American Standard Code for Information
Interchange).
ASCII represents each character with a 7-bit string, yielding a total of 128 characters.
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The code contains the uppercase and lower case alphabet, numeral, punctuation, and various
nonprinting control characters.
7-bit, plus a parity bit for error detection (odd/even parity).
Character ASCII Code
0 0110000
1 0110001
. . . . . .
9 0111001
: 0111010
A 1000001
B 1000010
. . . . . .
Z 1011010
[ 1011011
1011100
Fig 1.6 ASCII code example
EBCDIC
Extended Binary Coded Decimal Interchange Code (EBCDIC) is an 8-bit character encoding.
Error Detection Codes
Errors can occur data transmission. They should be detected, so that re-transmission can be
requested. With binary numbers, usually single-bit errors occur.
Example: 0010 erroneously transmitted as 0011, or 0000, or 0110, or 1010.
Error-detecting codes normally add extra information to the data. In general, error-detecting codes
contains redundant code. That is a code that uses n-bit strings need not contain 2n
valid code words.
An error-detecting code has the property that corrupting or garbling a code word will likely produce
a bit string that is not a code word. Thus errors in a bit string can be detected by a simple rule - if it
is not a code word it contains an error.
Commonly used error detecting codes area.
a. parity code
b. Checksums
c. Block Parity
a. Parity bit.
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Even parity: additional bit supplied to make total number of ‘1’s even.
Odd parity: additional bit supplied to make total number of ‘1’s odd.
Fig 1.7 Parity Check
Error correcting Code
Hamming code is a set of error-correction code s that can be used to detect and correct bit errors
that can occur when computer data is moved or stored.
The key to the Hamming Code is the use of extra parity bits to allow the identification of a single
error. Create the code word as follows:
1. Mark all bit positions that are powers of two as parity bits. (Positions 1, 2, 4, 8, 16, 32, 64, etc.)
2. All other bit positions are for the data to be encoded. (Positions 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15,
17, etc.)
3. Each parity bit calculates the parity for some of the bits in the code word. The position of the parity
bit determines the sequence of bits that it alternately checks and skips.
Position 1: check 1 bit, skip 1 bit, check 1 bit, skip 1 bit, etc. (1,3,5,7,9,11,13,15,...)
Position 2: check 2 bits, skip 2 bits, check 2 bits, skip 2 bits, etc. (2,3,6,7,10,11,14,15,...)
Position 4: check 4 bits, skip 4 bits, check 4 bits, skip 4 bits, etc.
(4,5,6,7,12,13,14,15,20,21,22,23,...)
Position 8: check 8 bits, skip 8 bits, check 8 bits, skip 8 bits, etc. (8-15,24-31,40-47,...)
Position 16: check 16 bits, skip 16 bits, check 16 bits, skip 16 bits, etc. (16-31,48-63,80-95,...)
Position 32: check 32 bits, skip 32 bits, check 32 bits, skip 32 bits, etc. (32-63,96-127,160-191,...)
etc.
4. Set a parity bit to 1 if the total number of ones in the positions it checks is odd. Set a parity bit to 0
if the total number of ones in the positions it checks is even.
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Here is an example:
A byte of data: 10011010
Create the data word, leaving spaces for the parity bits: _ _ 1 _ 0 0 1 _ 1 0 1 0
Calculate the parity for each parity bit (a ? represents the bit position being set):
Position 1 checks bits 1, 3, 5, 7, 9, 11:
? _ 1 _ 0 0 1 _ 1 0 1 0. Even parity so set position 1 to a 0: 0 _ 1 _ 0 0 1 _ 1 0 1 0
Position 2 checks bits 2, 3, 6, 7, 10, 11:
0 ? 1 _ 0 0 1 _ 1 0 1 0. Odd parity so set position 2 to a 1: 0 1 1 _ 0 0 1 _ 1 0 1 0
Position 4 checks bits 4, 5, 6, 7, 12:
0 1 1 ? 0 0 1 _ 1 0 1 0. Odd parity so set position 4 to a 1: 0 1 1 1 0 0 1 _ 1 0 1 0
Position 8 checks bits 8, 9, 10, 11, 12:
0 1 1 1 0 0 1 ? 1 0 1 0. Even parity so set position 8 to a 0: 0 1 1 1 0 0 1 0 1 0 1 0
Code word: 011100101010.
Finding and fixing a bad bit
The above example created a code word of 011100101010. Suppose the word that was
received was 011100101110 instead. Then the receiver could calculate which bit was wrong and
correct it. The method is to verify each check bit. Write down all the incorrect parity bits. Doing so,
you will discover that parity bits 2 and 8 are incorrect. It is not an accident that 2 + 8 = 10, and that
bit position 10 is the location of the bad bit. In general, check each parity bit, and add the positions
that are wrong, this will give you the location of the bad bit.