Positional number systems represent numbers as a sequence of digits with each digit position having an associated weight. In positional systems, the value of a number is the sum of the digit values multiplied by their weights. Common positional systems include decimal, binary, octal, and hexadecimal. Converting between these systems involves grouping digits and mapping groups to new bases. Negative numbers are represented using sign-magnitude or two's complement systems.
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
This document provides information on different number systems including decimal, binary, octal, and hexadecimal. It discusses techniques for converting between these number systems. The key points covered are:
- Decimal uses base 10 with digits 0-9 and places values on units, tens, hundreds etc.
- Binary, octal and hexadecimal use bits or digits to represent values with weights based on their position from right to left.
- Conversion between number systems involves multiplying the place value of each bit/digit and adding the results.
- Techniques are demonstrated for converting binary to decimal, octal to decimal, hexadecimal to decimal, and between other bases.
Digital and Logic Design Chapter 1 binary_systemsImran Waris
This document discusses binary number systems and digital computing. It covers binary numbers, number base conversions between decimal, binary, octal and hexadecimal. It also discusses binary coding techniques like binary-coded decimal, signed magnitude representation, one's complement and two's complement representations for negative numbers.
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.
The document discusses number representation in computers. It begins by introducing different number systems like decimal, binary, and hexadecimal. It then discusses how numeric data is stored in memory, including how integers, floats, characters and strings are represented. It also covers binary operations like addition, subtraction, multiplication and division. Finally, it discusses signed number representation using sign-magnitude, one's complement and two's complement methods.
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.
This document discusses different number systems including positional and non-positional. It describes the decimal, binary, octal, and hexadecimal number systems. For each system it provides the base, symbols used, an example of a number written in that system and its equivalent decimal value, and explanations of how positional notation works. It also provides steps and examples for converting between decimal, binary, octal, and hexadecimal numbers for both integral and fractional values.
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
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
This document provides information on different number systems including decimal, binary, octal, and hexadecimal. It discusses techniques for converting between these number systems. The key points covered are:
- Decimal uses base 10 with digits 0-9 and places values on units, tens, hundreds etc.
- Binary, octal and hexadecimal use bits or digits to represent values with weights based on their position from right to left.
- Conversion between number systems involves multiplying the place value of each bit/digit and adding the results.
- Techniques are demonstrated for converting binary to decimal, octal to decimal, hexadecimal to decimal, and between other bases.
Digital and Logic Design Chapter 1 binary_systemsImran Waris
This document discusses binary number systems and digital computing. It covers binary numbers, number base conversions between decimal, binary, octal and hexadecimal. It also discusses binary coding techniques like binary-coded decimal, signed magnitude representation, one's complement and two's complement representations for negative numbers.
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.
The document discusses number representation in computers. It begins by introducing different number systems like decimal, binary, and hexadecimal. It then discusses how numeric data is stored in memory, including how integers, floats, characters and strings are represented. It also covers binary operations like addition, subtraction, multiplication and division. Finally, it discusses signed number representation using sign-magnitude, one's complement and two's complement methods.
A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases.
This document discusses different number systems including positional and non-positional. It describes the decimal, binary, octal, and hexadecimal number systems. For each system it provides the base, symbols used, an example of a number written in that system and its equivalent decimal value, and explanations of how positional notation works. It also provides steps and examples for converting between decimal, binary, octal, and hexadecimal numbers for both integral and fractional values.
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 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.
The document discusses divisor functions and partition functions in number theory. It defines the divisor function τ(n) as counting the number of divisors of an integer n. It provides examples of calculating τ(n) and describes properties such as τ(p)=2 for prime p. The partition function P(n) counts the number of ways to write n as a sum of positive integers, ignoring order. Recursion formulas and generating functions are presented for both functions.
The document discusses various number systems including decimal, binary, and signed binary numbers. It provides the following key points:
1) Decimal numbers use ten digits from 0-9 while binary only uses two digits, 0 and 1. Binary numbers represent values through place values determined by powers of two.
2) Conversions can be done between decimal and binary numbers through either summing the place value weights or repeated division/multiplication by two.
3) Binary arithmetic follows simple rules to add, subtract, multiply and divide numbers in binary representation.
4) Signed binary numbers use a sign bit to indicate positive or negative values, with the most common 2's complement form representing negative numbers as the 2's
The document discusses binary numbers and arithmetic. It covers topics like addition, subtraction, multiplication in binary, and different methods for representing signed integers like two's complement. It explains how two's complement works by using bitwise operations to represent negative numbers. For example, it shows that adding two positive 8-bit binary numbers in two's complement is simply the bitwise addition, while subtraction can be performed by adding the number and the two's complement of the subtrahend. The document also discusses issues like carry vs overflow that can occur during binary arithmetic operations.
This document provides an overview of basic theories of information and computer data representation. It discusses how computers use binary digits or bits to represent numeric and character data. Key topics covered include binary, octal, decimal, and hexadecimal number systems; conversion between these systems; binary arithmetic; and representation of signed integers and decimal numbers. The objectives are to understand basic computer data units and data representation concepts focusing on numeric and character representation. Exercises are provided to practice conversions between number systems and binary arithmetic.
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.
This document discusses number systems and binary arithmetic. It covers the following number systems: binary, decimal, octal, hexadecimal and their interconversions. It also discusses binary addition, subtraction, multiplication and division operations. Additionally, it covers binary codes, boolean algebra and various types of binary complements like 1's complement, 2's complement, 9's complement and 10's complement.
This document discusses data representation and number systems in computers. It covers binary, octal, decimal, and hexadecimal number systems. Key points include:
- Data in computers is represented using binary numbers and different number systems allow for more efficient representations.
- Converting between number systems like binary, octal, decimal, and hexadecimal is explained through examples of dividing numbers and grouping bits.
- Signed numbers can be represented using complement representations like one's complement and two's complement, with subtraction implemented through addition of complements. Fast methods for calculating two's complement are described.
Digital logic design deals with digital circuits and how to design digital hardware using logic gates. It involves working with binary and other number systems. Binary represents information using two states (0 and 1) which can be represented electrically using voltage levels. Converting between number systems like binary, decimal, and octal allows digital components to interface. Basic logic operations like addition, subtraction and multiplication can then be performed on binary numbers.
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.
digital logic circuits, digital component floting and fixed pointRai University
This document provides an overview of floating point numbers and their representation. It discusses how floating point numbers are used to represent very large and small numbers with exponents. The IEEE 754 standard for floating point representation is described, including the use of sign-magnitude, biased exponents, normalization, denormalization and special values like infinity and NaN. Single and double precision floating point number formats are defined according to IEEE 754. Methods for converting between decimal and binary floating point values are demonstrated through examples.
This document contains examples and problems related to binary number systems. It begins by listing octal, hexadecimal, and base-12 numbers and converting between number systems. Later problems involve operations like addition, subtraction, multiplication and division using binary, octal and hexadecimal numbers. Conversions between decimal, binary, octal and hexadecimal are demonstrated. Complement representations for signed binary numbers are also covered.
The document discusses different number systems including decimal, binary, octal, and hexadecimal. It explains how to represent numbers in these different bases and how to convert between them. The key techniques covered include multiplying place values to convert to and from decimal, grouping bits into sets of 3 or 4 to convert between binary and octal or hexadecimal, and using binary as an intermediate step to convert between non-binary bases. Examples are provided for adding, multiplying, and converting fractions between decimal and binary representations.
This document introduces some basic concepts in number theory, including primes, least common multiples, greatest common divisors, and modular arithmetic. It then discusses different number systems such as decimal, binary, octal, and hexadecimal. Methods are provided for converting between these different bases, including dividing or grouping bits and multiplying by the place value. Prime numbers are defined as integers greater than 1 that are only divisible by 1 and themselves. The fundamental theorem of arithmetic states that every integer can be written as a unique product of prime factors.
The document discusses different ways of representing numerical data in computing systems, including:
1) Binary representation, which converts decimal numbers to binary by repeatedly dividing by column headings and tracking the remainders as 1s and 0s.
2) Negative numbers can be represented using sign-and-magnitude or two's complement methods.
3) Other number systems like octal and hexadecimal are also discussed which use different column headings but the same representation principles.
4) Floating point representation separates a real number into a mantissa and exponent to store fractional numbers more efficiently in binary.
* Let A's age be x and B's age be y
* Then average of A and B is (x + y)/2 = 20
* So, x + y = 40
* If C replaces A, average is (y + c)/2 = 19
* So, y + c = 38
* If C replaces B, average is (x + c)/2 = 21
* So, x + c = 42
* Solving the three equations, we get:
x = 22, y = 18, c = 20
The ages are:
A = 22
B = 18
C = 20
The answer is option 1.
In this slide we have discussed, different arithmetic operations like addition, subtraction, multiplication and division for binary numbers. Addition and subtraction operation is achieved using one's complement and two's complement number system.
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.
Review of Number systems - Logic gates - Boolean
algebra - Boolean postulates and laws - De-Morgan’s
Theorem, Principle of Duality - Simplification using
Boolean algebra - Canonical forms, Sum of product and
Product of sum - Minimization using Karnaugh map -
NAND and NOR Implementation.
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 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.
The document discusses divisor functions and partition functions in number theory. It defines the divisor function τ(n) as counting the number of divisors of an integer n. It provides examples of calculating τ(n) and describes properties such as τ(p)=2 for prime p. The partition function P(n) counts the number of ways to write n as a sum of positive integers, ignoring order. Recursion formulas and generating functions are presented for both functions.
The document discusses various number systems including decimal, binary, and signed binary numbers. It provides the following key points:
1) Decimal numbers use ten digits from 0-9 while binary only uses two digits, 0 and 1. Binary numbers represent values through place values determined by powers of two.
2) Conversions can be done between decimal and binary numbers through either summing the place value weights or repeated division/multiplication by two.
3) Binary arithmetic follows simple rules to add, subtract, multiply and divide numbers in binary representation.
4) Signed binary numbers use a sign bit to indicate positive or negative values, with the most common 2's complement form representing negative numbers as the 2's
The document discusses binary numbers and arithmetic. It covers topics like addition, subtraction, multiplication in binary, and different methods for representing signed integers like two's complement. It explains how two's complement works by using bitwise operations to represent negative numbers. For example, it shows that adding two positive 8-bit binary numbers in two's complement is simply the bitwise addition, while subtraction can be performed by adding the number and the two's complement of the subtrahend. The document also discusses issues like carry vs overflow that can occur during binary arithmetic operations.
This document provides an overview of basic theories of information and computer data representation. It discusses how computers use binary digits or bits to represent numeric and character data. Key topics covered include binary, octal, decimal, and hexadecimal number systems; conversion between these systems; binary arithmetic; and representation of signed integers and decimal numbers. The objectives are to understand basic computer data units and data representation concepts focusing on numeric and character representation. Exercises are provided to practice conversions between number systems and binary arithmetic.
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.
This document discusses number systems and binary arithmetic. It covers the following number systems: binary, decimal, octal, hexadecimal and their interconversions. It also discusses binary addition, subtraction, multiplication and division operations. Additionally, it covers binary codes, boolean algebra and various types of binary complements like 1's complement, 2's complement, 9's complement and 10's complement.
This document discusses data representation and number systems in computers. It covers binary, octal, decimal, and hexadecimal number systems. Key points include:
- Data in computers is represented using binary numbers and different number systems allow for more efficient representations.
- Converting between number systems like binary, octal, decimal, and hexadecimal is explained through examples of dividing numbers and grouping bits.
- Signed numbers can be represented using complement representations like one's complement and two's complement, with subtraction implemented through addition of complements. Fast methods for calculating two's complement are described.
Digital logic design deals with digital circuits and how to design digital hardware using logic gates. It involves working with binary and other number systems. Binary represents information using two states (0 and 1) which can be represented electrically using voltage levels. Converting between number systems like binary, decimal, and octal allows digital components to interface. Basic logic operations like addition, subtraction and multiplication can then be performed on binary numbers.
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.
digital logic circuits, digital component floting and fixed pointRai University
This document provides an overview of floating point numbers and their representation. It discusses how floating point numbers are used to represent very large and small numbers with exponents. The IEEE 754 standard for floating point representation is described, including the use of sign-magnitude, biased exponents, normalization, denormalization and special values like infinity and NaN. Single and double precision floating point number formats are defined according to IEEE 754. Methods for converting between decimal and binary floating point values are demonstrated through examples.
This document contains examples and problems related to binary number systems. It begins by listing octal, hexadecimal, and base-12 numbers and converting between number systems. Later problems involve operations like addition, subtraction, multiplication and division using binary, octal and hexadecimal numbers. Conversions between decimal, binary, octal and hexadecimal are demonstrated. Complement representations for signed binary numbers are also covered.
The document discusses different number systems including decimal, binary, octal, and hexadecimal. It explains how to represent numbers in these different bases and how to convert between them. The key techniques covered include multiplying place values to convert to and from decimal, grouping bits into sets of 3 or 4 to convert between binary and octal or hexadecimal, and using binary as an intermediate step to convert between non-binary bases. Examples are provided for adding, multiplying, and converting fractions between decimal and binary representations.
This document introduces some basic concepts in number theory, including primes, least common multiples, greatest common divisors, and modular arithmetic. It then discusses different number systems such as decimal, binary, octal, and hexadecimal. Methods are provided for converting between these different bases, including dividing or grouping bits and multiplying by the place value. Prime numbers are defined as integers greater than 1 that are only divisible by 1 and themselves. The fundamental theorem of arithmetic states that every integer can be written as a unique product of prime factors.
The document discusses different ways of representing numerical data in computing systems, including:
1) Binary representation, which converts decimal numbers to binary by repeatedly dividing by column headings and tracking the remainders as 1s and 0s.
2) Negative numbers can be represented using sign-and-magnitude or two's complement methods.
3) Other number systems like octal and hexadecimal are also discussed which use different column headings but the same representation principles.
4) Floating point representation separates a real number into a mantissa and exponent to store fractional numbers more efficiently in binary.
* Let A's age be x and B's age be y
* Then average of A and B is (x + y)/2 = 20
* So, x + y = 40
* If C replaces A, average is (y + c)/2 = 19
* So, y + c = 38
* If C replaces B, average is (x + c)/2 = 21
* So, x + c = 42
* Solving the three equations, we get:
x = 22, y = 18, c = 20
The ages are:
A = 22
B = 18
C = 20
The answer is option 1.
In this slide we have discussed, different arithmetic operations like addition, subtraction, multiplication and division for binary numbers. Addition and subtraction operation is achieved using one's complement and two's complement number system.
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.
Review of Number systems - Logic gates - Boolean
algebra - Boolean postulates and laws - De-Morgan’s
Theorem, Principle of Duality - Simplification using
Boolean algebra - Canonical forms, Sum of product and
Product of sum - Minimization using Karnaugh map -
NAND and NOR Implementation.
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 discusses different number systems including binary, decimal, octal and hexadecimal. It provides examples of converting between these number systems. The key points covered are:
- Binary, decimal, octal and hexadecimal number systems use different bases (2, 10, 8, 16 respectively) and sets of digits.
- Numbers can be converted between these systems through repetitive division or multiplication by the base to determine each place value digit.
- Fractional numbers are represented similarly with place values decreasing as negative powers of the base moving right of the radix point.
This document introduces different number systems including binary, octal, decimal, and hexadecimal. It discusses the base, symbols used, and how to convert between these number systems. Conversion is done by multiplying place values according to the base and adding the results. Common powers that are used in computing are also defined in terms of base 2 rather than base 10. The document concludes with discussions of binary addition, multiplication, complement representation, and how complement allows for subtraction using addition operations.
1) The document discusses finite word length effects in digital filters. It covers fixed point and floating point number representations, different number systems including binary, decimal, octal and hexadecimal.
2) It describes various number representation techniques for digital systems including fixed point representation, floating point representation, and block floating point representation. Fixed point representation uses a fixed binary point position while floating point representation allows the binary point to vary.
3) It also discusses signed number representations including sign-magnitude, one's complement, and two's complement forms. Arithmetic operations like addition, subtraction and multiplication are covered for fixed point numbers along with issues like overflow.
This document discusses digital electronics and information representation in digital systems. It covers topics such as number systems including binary, hexadecimal, octal and decimal systems. It describes sign-magnitude and complement number representations and how to perform addition, subtraction, multiplication and division with binary numbers. It also discusses ASCII and hexadecimal character coding. Finally, it provides examples of converting between binary, decimal, octal and hexadecimal number representations.
This document discusses different number systems used in computing, including decimal, binary, octal, and hexadecimal. It provides information on:
- The base or radix of each number system and the digits used
- How to convert between decimal, binary, octal, and hexadecimal numbers using techniques like dividing or multiplying by the base and collecting remainders or integer parts
- Applications of binary, octal, and hexadecimal in computing as "shorthand" for binary numbers
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.
The document discusses different number systems including decimal, binary, octal, and hexadecimal. It provides details on:
- The base and symbols used in each system
- How the place value of each digit is determined
- Methods for converting between decimal, binary, octal, and hexadecimal
- That binary is used in computers due to transistors being in one of two states
The document discusses different numeral systems used in computing including binary, decimal, octal and hexadecimal. It explains how each system uses a different base and symbol set. Binary uses base-2 with symbols 0-1. Decimal is base-10 with 0-9. Octal is base-8 with 0-7. Hexadecimal is base-16 with 0-9 and A-F. The document also provides examples and methods for converting between these different numeral systems that are commonly used for representing numbers, instructions and other data in computers.
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.
The document discusses digital and analog systems. It explains that digital systems represent information as discrete values using bits, whereas analog systems represent information as continuous values. It provides examples of digital and analog signals and discusses how a continuous analog signal can be converted to a discrete digital signal through sampling and quantization. It also covers binary, octal, and hexadecimal number systems and how to convert between them. Finally, it discusses binary addition and subtraction using complement representations.
The document discusses number representation systems, including decimal, binary, signed numbers, and floating point representation. It provides examples of converting between decimal and binary numbers. It also describes algorithms for binary arithmetic operations like addition, subtraction, multiplication, and division. Restoring and non-restoring division algorithms are explained. Finally, it discusses fixed point and floating point number representation, including the IEEE 754 standard for single and double precision floating point formats.
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
Inductive programming incorporates all approaches which are concerned with learning programs or algorithms from incomplete (formal) specifications. Possible inputs in an IP system are a set of training inputs and corresponding outputs or an output evaluation function, describing the desired behavior of the intended program, traces or action sequences which describe the process of calculating specific outputs, constraints for the program to be induced concerning its time efficiency or its complexity, various kinds of background knowledge such as standard data types, predefined functions to be used, program schemes or templates describing the data flow of the intended program, heuristics for guiding the search for a solution or other biases.
Output of an IP system is a program in some arbitrary programming language containing conditionals and loop or recursive control structures, or any other kind of Turing-complete representation language.
In many applications the output program must be correct with respect to the examples and partial specification, and this leads to the consideration of inductive programming as a special area inside automatic programming or program synthesis, usually opposed to 'deductive' program synthesis, where the specification is usually complete.
In other cases, inductive programming is seen as a more general area where any declarative programming or representation language can be used and we may even have some degree of error in the examples, as in general machine learning, the more specific area of structure mining or the area of symbolic artificial intelligence. A distinctive feature is the number of examples or partial specification needed. Typically, inductive programming techniques can learn from just a few examples.
The diversity of inductive programming usually comes from the applications and the languages that are used: apart from logic programming and functional programming, other programming paradigms and representation languages have been used or suggested in inductive programming, such as functional logic programming, constraint
programming, probabilistic programming
Research on the inductive synthesis of recursive functional programs started in the early 1970s and was brought onto firm theoretical foundations with the seminal THESIS system of Summers[6] and work of Biermann.[7] These approaches were split into two phases: first, input-output examples are transformed into non-recursive programs (traces) using a small set of basic operators; second, regularities in the traces are searched for and used to fold them into a recursive program. The main results until the mid 1980s are surveyed by Smith.[8] Due to
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 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 document discusses data representation and number systems in computing. It covers the following key points in 3 sentences:
Data such as numbers and coded information are represented using bits and bytes which can represent values, characters, or instructions. Common number systems used in computing include binary, decimal, octal, and hexadecimal, which use different radixes or bases to represent quantities with distinct symbols. Methods for converting between number systems involve grouping bits or digits into the appropriate radix and determining the place value of each position to arrive at the value in the target base.
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
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 ...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
1. Positional Number Systems
• A number system consists of an order set of symbols (digits) with relations
defined for +,-,*, /
• The radix (or base) of the number system is the total number of digits
allowed in the the number system.
– Example, for the decimal number system:
• Radix, r = 10, Digits allowed = 0,1, 2, 3, 4, 5, 6, 7, 8, 9
• In positional number systems, a number is represented by a string of digits,
where each digit position has an associated weight.
• The value of a number is the weighted sum of the digits.
• The general representation of an unsigned number D with whole and
fraction portions number in a number system with radix r:
Dr = d p-1 d p-2 ….. d1 d0.d-1 d-2 …. D-n
• The number above has p digits to the left of the radix point and n fraction
digits to the right.
• A digit in position i has as associated weight ri
• The value of the number is the sum of the digits multiplied by the associated
weight ri :
D = ∑i =− n di × r
p −1 i
2. Positional Number Systems
Number: Dr = d p-1 d p-2 ….. d1 d0.d-1 d-2 …. D-n
Value: D = ∑i =− n di × ri
p −1
• For example in the decimal number system:
5185.6810 = 5x103 + 1x102 + 8x101 + 5x100 + 6 x 10-1 + 8 x 10-2
= 5x1000 + 1x100 + 8x10 + 5 x 1 + 6x.1 + 8x.01
• For the binary number system with radix = 2, digits 0, 1
D2 = dp-1 × 2p-1 ….. d1 × 21 + d0 . 20 + d-1 × 2-1 + d-2 × 2-2 …..
• For Example:
100112 = 1 × 16 + 0 × 8 + 0 × 4 + 1 × 2 + 1 × 1 = 1910
| |
MSB LSB (least significant bit)
(most significant bit)
Binary Point
101.0012 = 1x4 + 0x2 + 1x1 + 0x.5 + 0x.25 + 1x.125 = 5.12510
3. Number Systems Used in Computers
Name
Radix Set of Digits Example
of Radix
Decimal r=10 {0,1,2,3,4,5,6,7,8,9} 25510
Binary r=2 {0,1} 11111111 2
Octal r= 8 {0,1,2,3,4,5,6,7} 377 8
Hexadecimal r=16 {0,1,2,3,4,5,6,7,8,9,A, B, C, D, E, F} FF16
Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Hex 0 1 2 3 4 5 6 7 8 9 A B C D E F
Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111
4. Radix-r to Decimal Conversion
• The decimal value of a number in any radix r is found by converting
each digit to its radix 10 equivalent and expanding the value using
radix arithmetic:
D = ∑ip=−−1n di × ri
• Examples:
1101.1012 = 1×23 + 1×22 + 1×20 + 1×2-1 + 1×2-3
= 8 + 4 + 1 + 0.5 + 0.125 = 13.62510
572.68 = 5×82 + 7×81 + 2×80 + 6×8-1
= 320 + 56 + 16 + 0.75 = 392.7510
2A.816 = 2×161 + 10×160 + 8×16-1
= 32 + 10 + 0.5 = 42.510
132.34 = 1×42 + 3×41 + 2×40 + 3×4-1
= 16 + 12 + 2 + 0.75 = 30.7510
341.245 = 3×52 + 4×51 + 1×50 + 2×5-1 + 4×5-2
= 75 + 20 + 1 + 0.4 + 0.16 = 96.5610
5. Decimal-to-Binary Conversion
• Separate the decimal number into whole and fraction portions.
• To convert the whole number portion to binary, use successive
division by 2 until the quotient is 0. The remainders form the
answer, with the first remainder as the least significant bit (LSB) and
the last as the most significant bit (MSB).
• Example: Convert 17910 to binary:
179 / 2 = 89 remainder 1 (LSB)
/ 2 = 44 remainder 1
/ 2 = 22 remainder 0
/ 2 = 11 remainder 0
/ 2 = 5 remainder 1
/ 2 = 2 remainder 1
/ 2 = 1 remainder 0
/ 2 = 0 remainder 1 (MSB)
17910 = 101100112
6. Decimal-to-Binary Conversion
• To convert decimal fractions to binary, repeated multiplication by 2 is
used, until the fractional product is 0 (or until the desired number of
binary places). The whole digits of the multiplication results produce
the answer, with the first as the MSB, and the last as the LSB.
• Example: Convert 0.312510 to binary
Result Digit
.3125 × 2 = 0.625 0 (MSB)
.625 × 2 = 1.25 1
.25 × 2 = 0.50 0
.5 × 2 = 1.0 1 (LSB)
0.312510 = .01012
8. Decimal to Radix-r Conversion
• Separate the decimal number into whole and fraction portions.
• To convert the whole number portion to binary, use successive
division by r until the quotient is 0. The remainders form the
answer, with the first remainder as the least significant digit (LSD)
and the last as the most significant digit (MSD).
• To convert decimal fractions to radix-r, repeated multiplication by r
is used, until the fractional product is 0 (or until the desired number
of binary places). The whole digits of the multiplication results
produce the answer, with the first as the MSD, and the last as the
LSD.
• Example: Convert 46710 to octal
467 / 8 = 58 remainder 3 (LSD)
/ 8 = 7 remainder 2
/ 8 = 0 remainder 7 (MSD)
46710 = 7238
9. Binary to Octal Conversion
• Separate the whole binary number portion into groups of
3 beginning at the binary point and working to the left.
Add leading zeroes as necessary .
• Separate the fraction binary number portion into groups
of 3 beginning at the binary point and working to the
right. Add trailing zeroes as necessary.
• Convert each group of 3 to the equivalent octal digit.
• Example:
3564.87510 = 110 111 101 100.1112
= (6 × 83) + (7 × 82) + (5 × 81)+(4 × 80)+(7 × 8-1)
= 6754.78
10. Binary to Hexadecimal Conversion
• Separate the whole binary number portion into groups of
4 beginning at the binary point and working to the left.
Add leading zeroes as necessary.
• Separate the fraction binary number portion into groups
of 4 beginning at the binary point and working to the
right. Add trailing zeroes as necessary.
• Convert each group of 4 to the equivalent hexadecimal
digit.
• Example:
3564.87510 = 1101 1110 1100.11102
= (D × 162) + (E × 161) + (C × 160)+(E × 16-1)
= DEC.E16
11. Conversion between Number Systems Summary
• Radix-r to decimal:
– Multiply digits with their corresponding weights and add
• Decimal to binary (radix 2)
D = ∑ip=−−1n di × ri
Whole numbers: repeated division by 2
Fractions: repeated multiplication by 2
• Decimal to radix-r
Whole numbers: repeated division by r
Fractions: repeated multiplication by r
• Binary to Octal
Substitute groups of three bits with corresponding octal digit.
• Binary to Hexadecimal
Substitute groups of four bits with corresponding hexadecimal
digit.
12. Binary Arithmetic Operations
Addition
• Similar to decimal number addition, two binary
numbers are added by adding each pair of bits together
with carry propagation.
• Addition Example:
1 0 1 1 1 1 0 0 0 Carry
X 190 1 0 1 1 1 1 1 0
Y + 141 + 1 0 0 0 1 1 0 1
X+Y 331 1 0 1 0 0 1 0 1 1
13. Binary Arithmetic Operations
Subtraction
• Two binary numbers are subtracted by subtracting each
pair of bits together with borrowing, where needed.
• Subtraction Example:
0 0 1 1 1 1 1 0 0 Borrow
X 229 1 1 1 0 0 1 0 1
Y - 46 - 0 0 1 0 1 1 1 0
183 1 0 1 1 0 1 1 1
14. Negative Binary Number Representations
• Signed-Magnitude Representation:
– For an n-bit binary number:
Use the first bit (most significant bit, MSB) position to
represent the sign where 0 is positive and 1 is negative.
Ex. 1 1 1 1 1 1 1 12 = - 12710
Sign Magnitude
– Remaining n-1 bits represent the magnitude which may range
from:
-2(n-1) + 1 to 2(n-1) - 1
– This scheme has two representations for 0; i.e., both positive and
negative 0: for 8 bits: 00000000, 10000000
– Arithmetic under this scheme uses the sign bit to indicate the
nature of the operation and the sign of the result, but the sign bit is
not used as part of the arithmetic.
15. Negative Binary Number Representations
• Two’s complement representation:
• MSB is the sign (MSB = 1 indicates a negative number)
• To negate a number complement all bits and add 1
• ex. 11910 = 01110111 complement bits
10001000
+1 add 1
100010012 = - 11910
16. Properties of Two's Complement Numbers
• X plus the complement of X equals 0.
• There is one unique 0.
• Positive numbers have 0 as their leading bit (MSB);
while negatives have 1 as their MSB.
• The range for an n-bit binary number in 2’s
complement representation is:
from -2(n-1) to 2(n-1) - 1
• The complement of the complement of a number is the
original number.
• Subtraction is done by addition to the 2’s complement of
the number.
17. Value of Two's Complement Numbers
• For an n-bit 2’s complement number the weights of the
bits is the same as for unsigned numbers except of the
MSB or sign bit where the weight is -2n-1, thus the value
of the n-bit 2’s complement number is given by:
D 2’s-complement = dn-1 × -2 n-1 + dn-2 × 2n-2 ….. d 1 × 2 1 + d0
For example:
the value of the 4-bit 2’s complement number 1011 is given by:
value = d3 × -2 3 + d2 × 22 + d1 × 21 + d0
= 1 × -2 3 + 0 × 22 + 1 × 21 + 1
= -8 + 0 + 2 + 1
= - 8 +3 = -5
18. Extending Two's Complement Numbers:
Sign Extension
• An n-bit 2’s complement number can converted to an m-bit
number where m>n by appending m-n copies of the sign
bit to the left of the number. This process is called sign
extension.
• Example: To convert the 4-bit 2’s complement number 1011 to
an 8-bit representation, the sign bit (here = 1) must be extended by
appending four 1’s to left of the number:
1011 4-bit 2’s-complement = 11111011 8-bit 2’s-complement
To verify that the value of the 8-bit number is still -5
value of 8-bit number = -27 + 26 + 25 + 24 + 23 +2 +1
= -128 + 64 + 32 + 16 +8 +2+1
= -128 + 123 = -5
19. Two’ complement addition/subtraction
Examples:
4 0100 -2 1110
+ -7 1001 + -6 1010
-3 1101 -8 1 1000
Ignore carry out from MSB
• Overflow occurs if signs (MSBs) of both operands are
the same and the sign of the result is different.
• Overflow can also be detected if the carry in the sign
position is different from the carry out of the sign
position.
20. Negative Binary Number Representations
• One’s-Complement representation
• MSB is the sign (MSB = 1 indicates a negative number)
• Negative numbers are found by complementing all bits
• ex. 11910 = 01110111
-11910 = 10001000
• The range of values for an n-bit binary number in 1’s
complement representation is:
• from -2(n-1) +1 to 2(n-1) - 1
• One’s-complement addition/subtraction:
If there is a carry out of the sign position add 1
Ex. -2 1101
+ -5 1010
-7 10111
+ 1
1000
21. Value of One's Complement Numbers
• For an n-bit 2’s complement number the weights of the
bits is also the same as for unsigned numbers except of
the MSB or sign bit where the weight is -(2n-1 +1), thus
the value of the n-bit 1’s complement number is given
by:
D 1’s-complement = dn-1 × -(2 n-1 -1) + dn-2 × 2n-2 ….. d1 × 21 + d 0
For example:
the value of the 4-bit 1’s complement number 1011 is given by:
value = d3 × -(2 3 -1) + d2 × 22 + d1 × 21 + d0
= 1 × -(2 3 -1) + 0 × 22 + 1 × 21 + 1
= -7 + 0 + 2 + 1
= - 7 +3 = -4