Chapter 2.1 introduction to number systemISMT College
Binary Number System, Decimal Number System, Octal Number System, Hexadecimal Number System, Conversion, Binary Arithmetic, Signed Binary Number Representation, 1's complement, 2's complement, 9's complement, 10's complement
The document discusses data representation in computer systems. It covers different number systems like binary, decimal, hexadecimal and their conversions. It explains how computers use the positional number system to represent numbers. It also discusses signed and unsigned integers, binary arithmetic operations, and character representation using ASCII code.
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.
1. The document discusses different types of codes used to represent digital data including weighted, non-weighted, alphanumeric, error detection, error correction, and binary codes.
2. It describes various binary codes like BCD, Gray, EBCDIC, and ASCII codes explaining how they represent numeric and alphanumeric data.
3. Specific codes discussed in detail include BCD, excess-3, Gray, and ASCII codes explaining their binary representations of decimal numbers and characters.
The document discusses different number systems used in digital electronics and computing. It explains that number systems have different bases and describe the bases of common number systems like decimal, binary, octal and hexadecimal. Decimal uses base 10, binary uses base 2, octal uses base 8 and hexadecimal uses base 16. It provides details on how to convert between these different number systems both for whole numbers and fractions using various techniques like multiplying/dividing by the base, grouping bits or hexadecimal digits. Examples are given to illustrate the conversion methods between the different number systems.
This document discusses data representation in computers. It covers:
- Numbering systems used in computers, including binary and hexadecimal.
- Procedures for converting between decimal, binary, and hexadecimal numbers.
- Signed integer representation, discussing signed magnitude, one's complement, and two's complement notation.
- Examples of adding signed binary integers using signed magnitude representation and how overflow can cause errors.
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
Chapter 2.1 introduction to number systemISMT College
Binary Number System, Decimal Number System, Octal Number System, Hexadecimal Number System, Conversion, Binary Arithmetic, Signed Binary Number Representation, 1's complement, 2's complement, 9's complement, 10's complement
The document discusses data representation in computer systems. It covers different number systems like binary, decimal, hexadecimal and their conversions. It explains how computers use the positional number system to represent numbers. It also discusses signed and unsigned integers, binary arithmetic operations, and character representation using ASCII code.
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.
1. The document discusses different types of codes used to represent digital data including weighted, non-weighted, alphanumeric, error detection, error correction, and binary codes.
2. It describes various binary codes like BCD, Gray, EBCDIC, and ASCII codes explaining how they represent numeric and alphanumeric data.
3. Specific codes discussed in detail include BCD, excess-3, Gray, and ASCII codes explaining their binary representations of decimal numbers and characters.
The document discusses different number systems used in digital electronics and computing. It explains that number systems have different bases and describe the bases of common number systems like decimal, binary, octal and hexadecimal. Decimal uses base 10, binary uses base 2, octal uses base 8 and hexadecimal uses base 16. It provides details on how to convert between these different number systems both for whole numbers and fractions using various techniques like multiplying/dividing by the base, grouping bits or hexadecimal digits. Examples are given to illustrate the conversion methods between the different number systems.
This document discusses data representation in computers. It covers:
- Numbering systems used in computers, including binary and hexadecimal.
- Procedures for converting between decimal, binary, and hexadecimal numbers.
- Signed integer representation, discussing signed magnitude, one's complement, and two's complement notation.
- Examples of adding signed binary integers using signed magnitude representation and how overflow can cause errors.
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
Representation of Signed Numbers - R.D.SivakumarSivakumar R D .
This document discusses the representation of signed numbers in computers. It explains two common methods: sign-magnitude representation and 2's complement representation. For 2's complement, it provides the step-by-step process to convert a positive number to its negative equivalent in binary. It also discusses interpreting numbers as signed or unsigned and how this affects comparisons. Finally, it outlines the different value ranges for unsigned versus signed integers in an n-bit system.
This document discusses different methods of representing data in a computer, including numeric data types, number systems, and encoding schemes. It covers binary, decimal, octal, and hexadecimal number systems. Methods for representing signed and unsigned integers are described, such as signed-magnitude, 1's complement, and 2's complement representations. Floating point number representation with a sign bit, exponent field, and significand is also summarized. Conversion between different number bases and data encodings like binary-coded decimal are explained through examples.
Digital Electronics- Number systems & codes VandanaPagar1
This document covers number systems including decimal, binary, hexadecimal and their representations. It discusses how to convert between different number bases including binary to decimal and hexadecimal to decimal. Binary operations like addition, subtraction and codes like binary coded decimal are explained. Non-weighted codes such as gray code are also introduced. Reference books on digital electronics and number systems are provided.
This document provides an overview of data representation in computers. It discusses binary, decimal, hexadecimal, and floating point number systems. Binary numbers use only two digits, 0 and 1, and can represent values as sums of powers of two. Decimal uses ten digits from 0-9. Hexadecimal uses sixteen values from 0-9 and A-F. Negative binary integers can be represented using ones' complement or twos' complement methods. Twos' complement avoids multiple representations of zero and is commonly used in computers. Converting between number bases involves expressing the value in one base using the digits of another.
Computer architecture data representationAnil Pokhrel
This document discusses various methods of data representation in digital systems, including number systems, data types, and encoding of numeric values. It covers binary, decimal, and floating point representation, as well as techniques for representing negative numbers like signed magnitude, 1's complement, and 2's complement. Error detection codes like parity bits are also introduced as a way to detect errors during data transmission. Key topics include binary conversion of decimal numbers, floating point representation using mantissa and exponent, overflow detection, and even/odd parity generation.
The document discusses data representation in computers, specifically floating point numbers. It explains that floating point representation uses three fields - a sign bit, exponent field, and significand field - to represent numbers in scientific notation. The IEEE 754 standard defines common floating point formats like single and double precision that specify the number of bits used for each field. The document provides examples of how different numbers are represented in a simplified 14-bit floating point format and discusses how operations like addition and multiplication are performed on floating point values.
The document discusses different number systems including binary, octal, decimal, and hexadecimal. It explains that number systems have a radix or base, which determines the set of symbols used and their positional values. The key representations for binary numbers discussed are sign-magnitude, one's complement, and two's complement, which provide different methods for representing positive and negative numbers. The document provides examples of addition, subtraction, multiplication, and division operations in binary.
This document summarizes key concepts in digital systems and binary numbers. It discusses why digital systems are preferred over analog, how to convert between number bases, signed and complement number representations, overflow, binary and decimal codes, BCD addition, Gray code, and parity checks. Digital systems are more cost effective, reliable, programmable and selective compared to analog. Number conversions involve grouping bits or dividing decimals. Signed number systems use complement representations to indicate positive and negative values.
This document presents an overview of different number systems including decimal, binary, octal, and hexadecimal. It defines each system, their bases, and the digits used. Conversion methods between the systems are described, such as repeated division to convert decimal to binary and multiplying place values to convert in the opposite direction. The relationships between the different bases are shown, including that hexadecimal represents groups of 4 binary digits. Examples are provided for conversions between the various number systems.
BCS Certificate Level Examination. Computer and Network Technology (CNT) subject. Fundamentals of Computer Science. Data Representation in Computers. Learn about decimal, binary, octal and hexadecimal number systems and conversion between systems. Learn about binary addition and subtraction. For a complete subject coverage including Information Systems and Software Developments subjects, please visit to https://www.bcsonlinelectures.com/
This is the second lesson of Computer and Network Technology subject of BCS HEQ Certificate Level exam.
Subject: Computer and Network Technology (CNT)
Chapter: Fundamentals
Lesson: Data Representation in Computers
This lesson discuss about how integers, floating point numbers and characters are handled by modern computers.
For more lessons please visit https://www.bcsonlinelectures.com website.
Error detection and correction codes add additional data bits to transmitted messages to detect and possibly correct errors caused by noise during transmission. Error-detecting codes only detect errors while error-correcting codes can detect and correct errors by determining the location of corrupted bits. Common techniques for error detection include parity checks, which use an extra parity bit to make the total number of 1 bits in a message either even or odd.
This document discusses data representation in computer systems. It covers topics like binary number systems, conversion between number bases, signed and unsigned integers, and binary arithmetic. Specifically, it defines basic units like bits and bytes, explains how to convert decimal numbers to binary and other bases, discusses signed integer representation using the sign-magnitude method and issues with overflow, and outlines the basic rules for binary addition, subtraction, multiplication and division.
Binary coded decimal (BCD) is a numerical coding system that uses binary numbers to represent decimal digits. Each decimal digit from 0 to 9 is represented by a unique 4-bit binary code. BCD allows arithmetic operations like addition and subtraction on numbers. For BCD addition, the binary sum is calculated and if it exceeds 9, then 6 is added to obtain a valid BCD result. For BCD subtraction, the 9's complement of the subtrahend is calculated and added to the minuend, with carries propagated to the next group of bits.
Digital Arithmetic: Operations and Circuits discusses binary addition, subtraction, multiplication, and division. It also covers different systems for representing signed numbers, including sign-magnitude, 1's complement, and 2's complement. Key topics include performing arithmetic using the 2's complement system, detecting overflow, and representing decimal values in binary coded decimal. The document provides examples and review questions to illustrate binary arithmetic concepts.
This document discusses different methods for representing data in computers, including numeric and character representations. It covers representing signed and unsigned integers using methods like sign-magnitude, 1's complement, and 2's complement. It also discusses floating point number representation using the IEEE standard. Finally, it discusses character representation using ASCII and Unicode encoding schemes.
The decimal number system is a base-10 positional system with 10 digits (0-9). Binary is a base-2 system using only two digits, 0 and 1, and is preferred for digital systems. Octal uses digits 0-7 as a base-8 system, while hexadecimal uses digits 0-9 and letters A-F as a base-16 system. Converting between number systems involves repeatedly dividing/multiplying by the base and recording remainders/products as the new number in the target system.
FYBSC IT Digital Electronics Unit I Chapter II Number System and Binary Arith...Arti Parab Academics
Binary Arithmetic:
Binary addition, Binary subtraction, Negative number representation,
Subtraction using 1’s complement and 2’s complement, Binary
multiplication and division, Arithmetic in octal number system,
Arithmetic in hexadecimal number system, BCD and Excess – 3
arithmetic.
The document discusses how data is represented in computers using binary numbers. It explains that computers use binary, which represents numbers using only two digits (0 and 1) rather than the decimal system's ten digits. This binary system maps well to the two states of on/off in a computer's electrical circuits. The document provides examples of converting decimal numbers to binary and vice versa. It also discusses how signed integers and floating point numbers are represented using binary.
CDS Fundamentals of digital communication system UNIT 1 AND 2.pdfshubhangisonawane6
The document discusses various number systems including decimal, binary, hexadecimal and their conversions. It explains binary addition and subtraction using 2's complement. Binary coded decimal and gray codes are also covered. The last part discusses ASCII codes for alphanumeric representation. Key points discussed are:
- Decimal, binary and hexadecimal number systems and inter-conversions between them.
- Binary addition and subtraction using 2's complement.
- Binary coded decimal and gray codes for number representation.
- ASCII codes for alphanumeric representation in computers.
Representation of Signed Numbers - R.D.SivakumarSivakumar R D .
This document discusses the representation of signed numbers in computers. It explains two common methods: sign-magnitude representation and 2's complement representation. For 2's complement, it provides the step-by-step process to convert a positive number to its negative equivalent in binary. It also discusses interpreting numbers as signed or unsigned and how this affects comparisons. Finally, it outlines the different value ranges for unsigned versus signed integers in an n-bit system.
This document discusses different methods of representing data in a computer, including numeric data types, number systems, and encoding schemes. It covers binary, decimal, octal, and hexadecimal number systems. Methods for representing signed and unsigned integers are described, such as signed-magnitude, 1's complement, and 2's complement representations. Floating point number representation with a sign bit, exponent field, and significand is also summarized. Conversion between different number bases and data encodings like binary-coded decimal are explained through examples.
Digital Electronics- Number systems & codes VandanaPagar1
This document covers number systems including decimal, binary, hexadecimal and their representations. It discusses how to convert between different number bases including binary to decimal and hexadecimal to decimal. Binary operations like addition, subtraction and codes like binary coded decimal are explained. Non-weighted codes such as gray code are also introduced. Reference books on digital electronics and number systems are provided.
This document provides an overview of data representation in computers. It discusses binary, decimal, hexadecimal, and floating point number systems. Binary numbers use only two digits, 0 and 1, and can represent values as sums of powers of two. Decimal uses ten digits from 0-9. Hexadecimal uses sixteen values from 0-9 and A-F. Negative binary integers can be represented using ones' complement or twos' complement methods. Twos' complement avoids multiple representations of zero and is commonly used in computers. Converting between number bases involves expressing the value in one base using the digits of another.
Computer architecture data representationAnil Pokhrel
This document discusses various methods of data representation in digital systems, including number systems, data types, and encoding of numeric values. It covers binary, decimal, and floating point representation, as well as techniques for representing negative numbers like signed magnitude, 1's complement, and 2's complement. Error detection codes like parity bits are also introduced as a way to detect errors during data transmission. Key topics include binary conversion of decimal numbers, floating point representation using mantissa and exponent, overflow detection, and even/odd parity generation.
The document discusses data representation in computers, specifically floating point numbers. It explains that floating point representation uses three fields - a sign bit, exponent field, and significand field - to represent numbers in scientific notation. The IEEE 754 standard defines common floating point formats like single and double precision that specify the number of bits used for each field. The document provides examples of how different numbers are represented in a simplified 14-bit floating point format and discusses how operations like addition and multiplication are performed on floating point values.
The document discusses different number systems including binary, octal, decimal, and hexadecimal. It explains that number systems have a radix or base, which determines the set of symbols used and their positional values. The key representations for binary numbers discussed are sign-magnitude, one's complement, and two's complement, which provide different methods for representing positive and negative numbers. The document provides examples of addition, subtraction, multiplication, and division operations in binary.
This document summarizes key concepts in digital systems and binary numbers. It discusses why digital systems are preferred over analog, how to convert between number bases, signed and complement number representations, overflow, binary and decimal codes, BCD addition, Gray code, and parity checks. Digital systems are more cost effective, reliable, programmable and selective compared to analog. Number conversions involve grouping bits or dividing decimals. Signed number systems use complement representations to indicate positive and negative values.
This document presents an overview of different number systems including decimal, binary, octal, and hexadecimal. It defines each system, their bases, and the digits used. Conversion methods between the systems are described, such as repeated division to convert decimal to binary and multiplying place values to convert in the opposite direction. The relationships between the different bases are shown, including that hexadecimal represents groups of 4 binary digits. Examples are provided for conversions between the various number systems.
BCS Certificate Level Examination. Computer and Network Technology (CNT) subject. Fundamentals of Computer Science. Data Representation in Computers. Learn about decimal, binary, octal and hexadecimal number systems and conversion between systems. Learn about binary addition and subtraction. For a complete subject coverage including Information Systems and Software Developments subjects, please visit to https://www.bcsonlinelectures.com/
This is the second lesson of Computer and Network Technology subject of BCS HEQ Certificate Level exam.
Subject: Computer and Network Technology (CNT)
Chapter: Fundamentals
Lesson: Data Representation in Computers
This lesson discuss about how integers, floating point numbers and characters are handled by modern computers.
For more lessons please visit https://www.bcsonlinelectures.com website.
Error detection and correction codes add additional data bits to transmitted messages to detect and possibly correct errors caused by noise during transmission. Error-detecting codes only detect errors while error-correcting codes can detect and correct errors by determining the location of corrupted bits. Common techniques for error detection include parity checks, which use an extra parity bit to make the total number of 1 bits in a message either even or odd.
This document discusses data representation in computer systems. It covers topics like binary number systems, conversion between number bases, signed and unsigned integers, and binary arithmetic. Specifically, it defines basic units like bits and bytes, explains how to convert decimal numbers to binary and other bases, discusses signed integer representation using the sign-magnitude method and issues with overflow, and outlines the basic rules for binary addition, subtraction, multiplication and division.
Binary coded decimal (BCD) is a numerical coding system that uses binary numbers to represent decimal digits. Each decimal digit from 0 to 9 is represented by a unique 4-bit binary code. BCD allows arithmetic operations like addition and subtraction on numbers. For BCD addition, the binary sum is calculated and if it exceeds 9, then 6 is added to obtain a valid BCD result. For BCD subtraction, the 9's complement of the subtrahend is calculated and added to the minuend, with carries propagated to the next group of bits.
Digital Arithmetic: Operations and Circuits discusses binary addition, subtraction, multiplication, and division. It also covers different systems for representing signed numbers, including sign-magnitude, 1's complement, and 2's complement. Key topics include performing arithmetic using the 2's complement system, detecting overflow, and representing decimal values in binary coded decimal. The document provides examples and review questions to illustrate binary arithmetic concepts.
This document discusses different methods for representing data in computers, including numeric and character representations. It covers representing signed and unsigned integers using methods like sign-magnitude, 1's complement, and 2's complement. It also discusses floating point number representation using the IEEE standard. Finally, it discusses character representation using ASCII and Unicode encoding schemes.
The decimal number system is a base-10 positional system with 10 digits (0-9). Binary is a base-2 system using only two digits, 0 and 1, and is preferred for digital systems. Octal uses digits 0-7 as a base-8 system, while hexadecimal uses digits 0-9 and letters A-F as a base-16 system. Converting between number systems involves repeatedly dividing/multiplying by the base and recording remainders/products as the new number in the target system.
FYBSC IT Digital Electronics Unit I Chapter II Number System and Binary Arith...Arti Parab Academics
Binary Arithmetic:
Binary addition, Binary subtraction, Negative number representation,
Subtraction using 1’s complement and 2’s complement, Binary
multiplication and division, Arithmetic in octal number system,
Arithmetic in hexadecimal number system, BCD and Excess – 3
arithmetic.
The document discusses how data is represented in computers using binary numbers. It explains that computers use binary, which represents numbers using only two digits (0 and 1) rather than the decimal system's ten digits. This binary system maps well to the two states of on/off in a computer's electrical circuits. The document provides examples of converting decimal numbers to binary and vice versa. It also discusses how signed integers and floating point numbers are represented using binary.
CDS Fundamentals of digital communication system UNIT 1 AND 2.pdfshubhangisonawane6
The document discusses various number systems including decimal, binary, hexadecimal and their conversions. It explains binary addition and subtraction using 2's complement. Binary coded decimal and gray codes are also covered. The last part discusses ASCII codes for alphanumeric representation. Key points discussed are:
- Decimal, binary and hexadecimal number systems and inter-conversions between them.
- Binary addition and subtraction using 2's complement.
- Binary coded decimal and gray codes for number representation.
- ASCII codes for alphanumeric representation in computers.
The document discusses different number systems used in digital computers including binary, decimal, octal, and hexadecimal systems. It describes the characteristics of each system such as the base and digits used. Methods for converting between these different number systems are presented, including using division or grouping bits. The representation of signed integers as binary numbers is also covered, comparing sign-magnitude, one's complement, and two's complement representations. Binary addition is demonstrated with examples.
This document discusses different number systems including binary, decimal, hexadecimal, and octal. It provides details on:
- How each system uses different bases and represents values
- Converting between the different number systems using methods like repeated division, grouping bits, and determining place values
- Binary uses two digits (0,1), decimal uses ten digits (0-9), hexadecimal uses sixteen symbols (0-9 plus A-F), and octal uses eight digits (0-7)
This document discusses different number systems including binary, decimal, hexadecimal, and octal. It provides details on:
- How each system uses different bases and represents values
- Converting between the different number systems using methods like repeated division or adding/removing place values
- Binary uses two digits (0,1), decimal uses ten digits (0-9), hexadecimal uses sixteen symbols (0-9 plus A-F), and octal uses eight digits (0-7)
This document discusses different number systems including binary, decimal, hexadecimal, and octal. It provides details on:
- How each system uses different bases and represents values
- Converting between the different number systems using methods like repeated division, grouping bits, and determining place values
- Binary uses two digits (0,1), decimal uses ten digits (0-9), hexadecimal uses sixteen symbols (0-9 plus A-F), and octal uses eight digits (0-7)
This document discusses different number systems including binary, decimal, hexadecimal, and octal. It provides details on:
- How each system uses different bases and represents values
- Converting between the different number systems using methods like repeated division, grouping bits, and determining place values
- Binary uses two digits (0,1), decimal uses ten digits (0-9), hexadecimal uses sixteen symbols (0-9 plus A-F), and octal uses eight digits (0-7)
This document discusses different number systems including binary, decimal, hexadecimal, and octal. It provides details on:
- How each system uses different bases and represents values
- Converting between the different number systems using methods like repeated division, grouping bits, and determining place values
- Binary uses two digits (0,1), decimal uses ten digits (0-9), hexadecimal uses sixteen symbols (0-9 plus A-F), and octal uses eight digits (0-7)
Lecture 02 - Logic Design(Number Systems).pptxshwan it
This document discusses different number systems including decimal, binary, hexadecimal, and octal. It provides details on:
- The base and place values of each system
- Converting between binary, decimal, hexadecimal, and octal numbers using techniques like summing place values or repeated division/multiplication
- Concepts like most/least significant bits in binary numbers
- How hexadecimal and octal numbers can be used to conveniently represent binary numbers and codes
This document discusses different number systems including binary, decimal, octal, hexadecimal and binary-coded decimal (BCD). It explains how to convert between these number systems using techniques like successive division and multiplying by place values. Floating point numbers can be converted between bases by treating the integer and fractional parts separately and using the remainder method.
This document discusses different number systems used in computing including decimal, binary, hexadecimal, and binary-coded decimal (BCD). It explains how each system uses a positional notation with a radix or base to represent values. Binary uses a base of 2 with digits of 0 and 1. Hexadecimal uses a base of 16 with digits 0-9 and A-F. Conversion between decimal, binary, and hexadecimal is described. Signed and unsigned numbers are also discussed, with two's complement being the most common way of representing signed binary numbers. Fixed precision and the concept of overflow are introduced when numbers are represented with a limited number of bits in computers.
This document discusses different number systems used in computing including decimal, binary, hexadecimal, and binary-coded decimal (BCD). It explains how each system uses a positional notation with a radix or base. Binary uses a base of 2 with digits of 0 and 1. Hexadecimal uses a base of 16 with digits 0-9 and A-F. Conversion between decimal, binary, and hexadecimal is described. Signed numbers can be represented using two's complement where the most significant bit indicates the sign. Fixed precision numbers have a limited range determined by the number of bits used.
This document discusses different number systems used in computing including decimal, binary, hexadecimal, and binary-coded decimal (BCD). It explains how each system uses a positional notation with a base or radix to represent numbers. Decimal uses base 10 with digits 0-9, binary uses base 2 with digits 0-1, hexadecimal uses base 16 with digits 0-9 and A-F, and BCD encodes each decimal digit as a 4-bit binary number. The document describes how to convert between these number systems through division and remainder operations. Hexadecimal is commonly used to efficiently represent large binary numbers with fewer digits.
The document discusses data representation in computer systems. It begins by explaining that computers use the binary system for logic and arithmetic because it is easy to implement in electronics and switches. It then discusses how integers, floating point numbers, and Boolean logic are represented. The document provides details on bits, bytes, words, and how positional numbering systems like binary represent values. It covers converting between decimal and binary, including fractional values. Finally, it discusses signed integer representation using methods like signed magnitude, one's complement, and two's complement.
The document discusses data representation in computer systems. It begins by explaining that computers use the binary system for logic and arithmetic because it is easy to implement in electronics and switches. It then discusses how bits, bytes, words, and other units of data are represented and addressed in memory. Different numbering systems like binary, decimal, hexadecimal and their properties are explained. Methods for converting between these numbering systems like subtraction and division methods are provided with examples. Finally, the document discusses various methods for representing signed integers like signed magnitude, one's complement, and two's complement representations.
The binary number system and digital codes are fundamental to computers and to digital electronics in general. You will learn Binary addition, subtraction, multiplication, and Division.
The document discusses different number systems and digital coding techniques. It describes the decimal, binary, octal and hexadecimal number systems. Conversion methods between these systems are provided, including complement representations. Common codes like binary coded decimal, excess-3, and gray codes are defined along with their properties. NAND and NOR gates are identified as universal gates that can be used to implement any logical function. Methods for constructing common logic gates using only NAND gates are presented.
numbering system binary and decimal hex octalnoor300491
Hexadecimal is a base-16 number system used to compactly represent binary numbers. It uses 16 symbols - 0-9 and A-F. Counting proceeds from F to 10, then 20, etc. Binary numbers can be converted to and from hexadecimal by grouping bits into 4-bit blocks and replacing with the hex symbol. Decimal can also be converted to and from hexadecimal using multiplication/division by 16 or remainders. Hexadecimal addition follows decimal addition rules, carrying when sums exceed 15. Octal is base-8 and uses 0-7 symbols, with binary conversion replacing octal digits with 3-bit groups. Binary Coded Decimal represents each decimal digit with 4 bits for easy decimal interfacing. Gray code changes
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
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A Survey of Techniques for Maximizing LLM Performance.pptx
Introduction to number system
1. Digital Computer
Fundamentals
UNDER THE GUIDENCE: M. JANCY PRIYA, ASST. PROFESSOR
NAME OF THE STUDENT: R. AARTHI,
T. ANJALI,
T. ASWINI
REGISTER NUMBER: CB17S 250336
CB17S 250351
CB17S 250354
SUBJECT CODE: 16SCCCA7
BATCH: 2017-2020
YEAR: 2020
2. Chapter I
Number System and Codes
• introduce different types of number
system
• explain binary arithmetic
• explicate different types of syntax
3. Introduction
Number system is simply the ways to count things. Aim of any number system is to
deal with certain quantities which can be measured, monitored, recorded,
manipulated arithmetically, observed and utilised. Each quantity has to be
represented byits value as efficiently and accurately as is necessary for any
application. The numerical value of a quantity can be basically expressed in either
analog (continuous) or digital (step by step) method of representation. In analog
method, a quantity is expressed by anotherquantity which is proportional to the
first. For example, the voltage output of an amplifier is measured by a
voltmeter. The angular position of the needle of the voltmeter is proportional to the
voltage output of the amplifier.
4. Binary System
• In the binary number system (base of 2), there are only two digits: 0 and 1
and the place values are 20, 21, 22, 23 etc. Binary digits are abbreviated as
bits. For example, 1101 is a binary number of 4 bits (i.e.it is a binary
number containing four binary digits.)
• A binary number may have any number of bits. Consider the number
11001.01 1. Note the binary point (counterpart of decimal point in decimal
number system) in this number.
• Each digit is known as a bit and can take only two values 0 and 1. The left
most bit is the highest-order bit and represents the most significant bit
(MSB) while the lowest-order bit is the least significant bit (LSB).
24 23 2
2
21 20 2-1 2-2 2-3 Positional values or
weight
1
MSB
1 0 0 1 . 0 1
Binary Point
1
LSB
5. Signed numbers
In a signed number, the left most bit is the so called sign bit: 0=positive number
1=negative number.
1’s complement
In this notation, the positive numbers have the same representation as the sign-value
notation, and the negative numbers are obtained by taking the 1’s complement of the
positive correspondents.
2’s complement
The positive numbers have the same representation as the sign-value notation, and
the negative numbers are obtained by taking the 1’s complement of the positive
correspondents.
6. 1’s complement
Binary number
1-complement
1
0
0
1
1
0
1
0
1
0
0
1
1
0
0
1
The 1’s complement of a binary number is obtained just by changing each 0 to 1
and each 1 to 0.
2’s complement
2’s complement = 1’s complement+1
Binary number 1 0 1 1 1 0 1 0
1’s complement 0 1 0 0 0 1 0 1
+ 1
2’s complement 0 1 0 0 0 1 1 0
• Substitute the rest of bits by their 1’s complement.
7. Binary to Decimal Conversion
• Binary number can be converted into its decimal equivalent, by simply
adding the weights of various positions in the binary number which have
bit 1.
• Example 1:
Find the decimal equivalent of the binary number (11111)2
The equivalent decimal number is
=1X24+1X23+1X22+1X21+1X20
=16+8+4+2+1
= (31)10
• To differentiate between numbers represented in different number
systems, either the corresponding number system may be specified along
with the number or a small subscript at the end of the number may be
added signifying the number system. Example (1000)2 represents a binary
number and is not one thousand.
8. Decimal to Binary Conversion
decimal number is converted into its binary equivalent by its repeated
divisions by 2. The division is continued till we get a quotient of
0. Then all the remainders are arranged sequentially with first
remainder taking the position of LSB and the last one taking the
position of MSB. Consider the conversion of 27 into its binary
equivalent as follows
9. Octal Number System
Octal to Decimal Conversion
As has been done in case of binary numbers, an octal number can be converted into
its decimal equivalent by multiplying the octal digit by its positional value. For
example, let us convert 36.48 into decimal number. 36.48 = 3 x 81 + 6 x 80 + 4 x 8-1
= 24 + 6 + 0.5
= (30.5)10
Decimal to Octal Conversion
A decimal number can be converted by repeated division by 8 into equivalent
octal number. This method is similar to that adopted in decimal to binary
conversion. If the decimal number has some digits on the right of the decimal
point, then this part of the number is converted into its octal equivalent by
repeatedly multiplying it by 8. The process is same as has been followed in
binary number system.
10. Hexadecimal Number System
The hexadecimal number system has base 16 that is it has 16 digits (Hexadecimal
means'16'). These digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. The
digits A, B, C, D, E and F have equivalent decimal values 10, 11, 12, 13, 14, and 15
respectively. Each Hex (Hexadecimal is popularly known as hex) digit in a hex
number has a positional value that is some power of 16 depending upon its position
in the number.
Hex digit Decimal equivalent 4-bit Binary equivalent
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0000
0001
0010
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1101
1110
1111
11. 1 Hex to Decimal Conversion
Hex to decimal conversion is done in the same way as in the cases of binary and octal
to decimal conversions. A hex number is converted into its equivalent decimal number by
summing the products of the weights of each digit and their values. This is clear from the
example of conversion of 514.AF16 into its decimal equivalent.
514. AF16 = 5 X 162 + 1 X 161 + 4 X 160 + 10 X 16-1 + 15 X 16-2
= 1280+16+4+0.625+0.0586
= (1300.6836)10
Decimal to Hex Conversion
A decimal number is converted into hex number in the same way as a
decimal number is converted into its equivalent binary and octal numbers. The part of the
number on the left of the decimal point is to be divided repeatedly by 16 and the part an
the right of the decimal point is to be repeatedly multiplied by 16. This will be clear from
the examples of conversion of (579.26)10 into hex equivalent. Split the number into two
parts, 579 and .26. Thus, (579)10 = (2443)16
12. Hex to Binary Conversion
As in octal number system, a hex number is converted into its binary equivalent by
replacing each hex digit by its equivalent 4-bit binary number. This is clear from the
following example:
(BA6)16 = B A 6
= 1011 1010 010
= (101110100110)
Binary to Hex Conversion
By a process that is reverse of the process described in the above section, a binary
number can be converted in to its hex equivalent. Starting from the LSB side, group
the binary number bits into groups of lour bits. If towards the MSB side, the
numbers of bits is less than four, then add zeros on the left of the MSB so that the
group of four is complete. Replace each group by its equivalent hex digit.
13. Hex to Octal Conversion
Each digit of the hex number is first converted into its equivalent four bit binary number. Then thebits
of the equivalent binary number are grouped into groups of three bits. Then each group is replaced by its
equivalent octal digit to get the octal number.
For example:
(5AF)16 = 0101 1010 1111
= 010110101111
=010 110 101 111
= 2 6 5 7
=(2567)8
Octal to Hex Conversion
For octal to hex conversion, just reverse the process described in the section above.
This is clear from the following example:
(5457)8 = 101 100 101 111
= 1011 0010 1111
= B 2 F
= (B2F)16
14. Codes
We had an overview of binary, octal and hexadecimal number system. For any
number system with n base B and digits N0 (LSB), N1 N2...... N10 (MSB), the
decimal equivalent N10 is given by
N10 = Nm X Bm + .... N3 X B3 + N2 < B2 + N1 X B1 + N0Bo
When numbers, letters or words are represented bya specific group of
symbols, it is said that the number, letter or word is being encoded. The
group of symbols is called as the code.
Few codes will be discussed in the following sections.
15. BCD Code
In BCD (BCD stands Binary coded decimal) code, each digit of a decimal number is
converted in to its binary equivalent. The largest decimal digit is 9; therefore the largest
binary equivalent is 1001. This is illustrated as follows
951 10 = 1001 0101 0001
= (100101010001)BCD
ASCII Code
The word ASCII is run acronym of American Standard Code for Information
Interchange. This is the alphanumeric code most widely used in computers. The
alphanumeric code is one that represents alphabets, numerical numbers, punctuation
marks and other special characters recognised by a computer. The ASCII code is a 7-
bit code representing 26 English alphabets, 0 through 9 digits, punctuation marks,
etc. A 7-bit code has 27 = 128 possible code groups which arcquite sufficient.
16. Code Gray
Gray Code is a form of binary that uses a different method of incrementing from one
number to the next. With Gray Code, only one bit changes state from one position to
another. This feature allows a system designer to perform some error checking (i.e., if
more than one bit changes, the data must be incorrect).
Decimal Binary Gray Decimal Binary Gray
0 0000 0000 8 1000 1100
1 0001 0001 9 1001 1101
2 0010 0011 10 1010 1111
3 0011 0010 12 1100 1110
4 0100 0110 13 1101 1010
5 0101 0111 14 1110 1011
6 0110 0101 14 1110 1001
7 0111 0100 15 1111 1000
17. Binary Arithmetic
Addition
Addition of binary numbers can be carried out in a similar way by the column method But
before this, view four simple cases. In the decimal number system, 3 + 6 = 9 symbolizes the
combination of 3 with 6 to get a total of 9.
View the four simple cases.
• Case 1: When nothing is combined with nothing, we get nothing. The binary representation
of this is 0 + 0 = 0.
• Case 2: When nothing is combined with1, we get1. Using binary numbers to denote this
gives 0 + 1 = 1.
• Case 3: Combining.1 with nothing gives 1. The binary equivalent of this is 1 + 0 = 1.
• Case 4: When we combine 1 with 1, the result is 2. Using binary numbers, we symbolize 1
+ 1 = 10
• 0+0 = 0
• 0+1 = 1
• 1+0 = 1
18. Subtraction
Addition has the property of being commutative, that is, a+b = b+a. This is
not true of subtraction. 5 – 3 is not the same as 3 – 5. For this reason, we
must be careful of the order of the operands when subtracting. We call
the first operand, the number which is being diminished, the minuend; the
second operand, the amount to be subtracted from the minuend, is the
subtrahend. The result is called the difference.
51 minuend
– 22 subtrahend
29 difference
19. Multiplication
A simplistic way to perform multiplication is by repeated addition. In the example
below, we could add 42 to the product register 27 times. In fact, some early
computers performed multiplication this way. However, one of our goals is speed,
and we can do much better using the familiar methods we have learned for
multiplying decimal numbers. Recall that the multiplicand is multiplied by each digit
of the multiplier to form a partial product, and then the partial products are added to
form the total product. Each partial product is shifted left to align on the right with its
multiplier digit.
42 multiplicand
x 27 multiplier
294 first partial product
(42 X 7)
84 second partial product (42 X 2)
1134 total product
20. As with the other arithmetic operations, division is based on the paper-and-
pencil approach we learned for decimal arithmetic. We will show an
algorithm for unsigned long division that is essentially similar to the decimal
algorithm we learned in grade school. Let us divide 0110101 (5310) by 0101
(510). Beginning at the left of the dividend, we move to the right one
digit at a time until we have identified a portion of the dividend which is
greater than or equal to the divisor. At this point, a one is placed in
the quotient; all digits of the quotient to the left are assumed to be zero.
The divisor is copied below the partial dividend and subtracted to produce a
partial remainder as shown below.
Division
21. Now digits from the dividend are “brought down” into the partial remainder until the
partial remainder is again greater than or equal to the divisor. Zeroes are placed in the
quotient until the partial remainder is greater than or equal to the divisor, and then a
one is placed in the quotient, as shown below.
This completes the division. The quotient is (1010)2 (1010) and the remainder is
(11)2 (310), which is the expected result. This algorithm works only for unsigned
numbers, but it is possible to extend it to 2’s complement numbers.
As with the other algorithms, it can be implemented using only shifting,
complementation, and addition. Digital computers can perform arithmetic
operations using only binary numbers. And hence the above section of binary
arithmetic is the basic step of digital electronics.