This document discusses numbering systems and their importance in computer science. It covers topics like positional notation, binary and hexadecimal numbering, arithmetic in different bases, and how data types like numbers, characters, images and sound are represented in computers using binary. Understanding numbering systems is essential as the computer uses binary at its core, and knowledge of hexadecimal helps read error messages and understand machine level operations.
The document discusses various number systems including binary, decimal, octal, and hexadecimal. It provides explanations of how numbers are represented and converted between these number systems. The key points covered include:
- Definitions of different number systems and their bases
- How positional notation works in number systems
- Steps for converting numbers between decimal, binary, octal, and hexadecimal bases
- Examples of converting specific numbers between these number 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 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 introduces number systems and provides examples of converting between different number systems. It discusses decimal, binary, octal, and hexadecimal number systems. Conversion between these systems can be done directly by dividing and taking remainders or via shortcuts by grouping digits. Understanding number systems is important for IT professionals as computers use binary to represent all data internally.
This document discusses different number systems used in computing including binary, decimal, octal, and hexadecimal. It explains that computers use binary to represent numbers internally and provides tables to demonstrate how to convert between the different bases. Examples are given of converting decimal numbers to their binary, octal, and hexadecimal equivalents through repeated division or grouping of bits. Readers are provided exercises to practice base conversion between the number systems.
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 provides information about the Digital System Design course offered at Government Engineering College Raipur. The course code is B000313(028) and it is a 4 credit course taught over 3 lectures and 1 tutorial per week. The course aims to teach students to design, analyze, and interpret combinational and sequential circuits. It covers topics like Boolean algebra, minimization techniques, combinational circuits, sequential circuits, and digital logic families. The document lists 5 expected learning outcomes and provides a brief overview of the topics to be covered in each of the 5 units. It also mentions the relevant textbooks.
The document discusses various number systems including binary, decimal, octal, and hexadecimal. It provides explanations of how numbers are represented and converted between these number systems. The key points covered include:
- Definitions of different number systems and their bases
- How positional notation works in number systems
- Steps for converting numbers between decimal, binary, octal, and hexadecimal bases
- Examples of converting specific numbers between these number 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 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 introduces number systems and provides examples of converting between different number systems. It discusses decimal, binary, octal, and hexadecimal number systems. Conversion between these systems can be done directly by dividing and taking remainders or via shortcuts by grouping digits. Understanding number systems is important for IT professionals as computers use binary to represent all data internally.
This document discusses different number systems used in computing including binary, decimal, octal, and hexadecimal. It explains that computers use binary to represent numbers internally and provides tables to demonstrate how to convert between the different bases. Examples are given of converting decimal numbers to their binary, octal, and hexadecimal equivalents through repeated division or grouping of bits. Readers are provided exercises to practice base conversion between the number systems.
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 provides information about the Digital System Design course offered at Government Engineering College Raipur. The course code is B000313(028) and it is a 4 credit course taught over 3 lectures and 1 tutorial per week. The course aims to teach students to design, analyze, and interpret combinational and sequential circuits. It covers topics like Boolean algebra, minimization techniques, combinational circuits, sequential circuits, and digital logic families. The document lists 5 expected learning outcomes and provides a brief overview of the topics to be covered in each of the 5 units. It also mentions the relevant textbooks.
The document discusses different number systems used in programmable logic controllers (PLCs), including binary, decimal, hexadecimal, and BCD. It explains that PLCs use binary numbers to represent on/off conditions as 1s and 0s. Binary numbers can be converted to decimal by multiplying each bit by its place value weight and summing the results. Hexadecimal represents 4 binary bits with a single hexadecimal digit for compact representation of data. Conversion between number systems involves dividing by the base and multiplying by the place value weights.
The document provides an introduction to digital systems and numerical representations. It discusses:
- Analog vs. digital representations and conversions between them using ADCs and DACs.
- Different number systems including binary, decimal, octal and hexadecimal. Methods to convert between these systems are described.
- Digital electronics uses discrete voltage levels (0V and 5V) to represent binary digits (0 and 1). Timing diagrams show the relationship between digital signals over time.
This document discusses number systems used in programmable logic controllers (PLCs), including binary, decimal, hexadecimal, and BCD. It explains that PLCs use binary numbers to represent on and off conditions, with each bit representing a 1 or 0. It also describes how to convert between binary, decimal, hexadecimal, and BCD numbering systems.
The document discusses number systems. It defines a number system as a system for writing and representing numbers using digits or symbols in a consistent manner. It allows for arithmetic operations and provides a unique representation for every number. The four most common number systems are decimal, binary, octal, and hexadecimal. Binary uses only two digits, 0 and 1, and is used to represent electrical signals in computers. Decimal uses base 10 with digits 0-9 in place values. [END SUMMARY]
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 a syllabus for a course on Digital Logic Design (DLD). It includes:
- The course instructor's name and details.
- A list of textbooks and online resources for the course.
- The course outcomes, which are to identify digital logic concepts, simplify Boolean expressions, design data processing circuits, and design sequential circuits.
- An outline of the course units, which cover basic logic circuits, number systems, Boolean algebra, circuit implementation, and sequential circuits.
The document provides an overview of the topics, resources, and goals of the DLD course.
This is my notes on simplified version of slides from DTU course 02135 Introduction to cyber systems. The lectures and slides are originally given by Prof. Pezzarossa.
Number-Systems presentation of the mathematicsshivas379526
The document discusses different number systems including decimal, binary, hexadecimal, and their importance. It provides the following key points:
- Decimal is base-10 as it uses 10 digits (0-9). Binary is base-2 as it uses two digits, 0 and 1. Hexadecimal is base-16 as it uses 16 symbols (0-9 and A-F).
- Different number systems are important because computers use binary to simplify calculations and reduce circuitry/costs. Larger systems like hexadecimal are used to represent large memory addresses.
- Converting between systems involves placing the remainder of successive divisions by the base in each position. For example, converting 42 to binary is 101010 by dividing 42
The document provides information about different number systems used in computers, including binary, octal, hexadecimal, and decimal. It explains the characteristics of each system such as the base and digits used. Methods for converting between number systems like binary to decimal and vice versa are presented. Shortcut methods for direct conversions between binary, octal, and hexadecimal are also described. Binary arithmetic and binary-coded decimal number representation are discussed.
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.
- Decimal, binary, octal, and hexadecimal are different number systems used to represent numeric values.
- Decimal uses 10 digits (0-9), binary uses two digits (0-1), octal uses 8 digits (0-7), and hexadecimal uses 16 digits (0-9 and A-F).
- Each system has a base or radix - the number of unique digits used. Decimal is base 10, binary base 2, octal base 8, and hexadecimal base 16.
- Numbers can be converted between these systems using division and multiplication operations that take into account the place value of each digit based on the system's base.
The document provides information about number systems used in computers. It discusses binary system which uses two digits (0 and 1) to represent ON and OFF states of switches in a computer. It explains how to convert between binary, decimal, octal and hexadecimal number systems using different methods. The document also covers signed binary numbers, binary codes like ASCII and Gray codes. Finally, it discusses binary logic, truth tables, Boolean expressions and logic gates used in switching circuits.
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.
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.
The document outlines a lesson plan covering number systems. It includes converting between decimal, binary, octal, and hexadecimal number systems. The key concepts covered are the different number systems used in computing, including binary, octal, hexadecimal, and their bases. Conversion between these systems involves multiplying digits by place values to get the value in another base. The skills practiced are computational thinking and step-wise thinking. Values reinforced include awareness of computer technology development and patience.
This document discusses different number systems including binary, decimal, octal, and hexadecimal. It provides definitions and characteristics of each system. The key points covered are:
- Binary uses digits 0 and 1, octal uses 0-7, hexadecimal uses 0-9 and A-F, and decimal uses 0-9.
- Each system has a base (2, 8, 16, 10 respectively) that determines the value of each digit position.
- Methods for converting between number systems are presented, including using division or multiplying by the place value to change between decimal, binary, octal, and hexadecimal.
This document discusses different number systems including binary, decimal, octal, and hexadecimal. It provides details on each system such as their number bases and allowed digits. The document also describes how to convert between these different number systems using methods like dividing numbers by the target base or grouping binary digits into sets of four for hexadecimal conversion. The goal is to understand representation of numbers in computing systems which commonly use binary and hexadecimal formats.
The document discusses different number systems including positional and non-positional systems. It describes the decimal, binary, octal and hexadecimal number systems. Decimal uses base 10 with digits 0-9. Binary uses base 2 with digits 0-1. Octal uses base 8 with digits 0-7. Hexadecimal uses base 16 with digits 0-9 and A-F. It provides examples of converting between these number systems using direct conversion or shortcuts involving grouping digits. Fractional numbers are also represented in these positional systems.
The document describes the syllabus for the course EEE365 Digital Electronics. The course covers topics such as number systems, Boolean algebra, combinational and sequential logic circuit design, memory devices, and digital signal conversion. Reference books for the course include titles on digital logic, digital systems, and digital design principles.
This document provides an introduction to computers and their components. It defines a computer as a machine that can carry out arithmetic and logical operations automatically according to programming instructions. It then lists and describes the major internal and external components of a computer, including the central processing unit (CPU), motherboard, memory (RAM), hard drive, graphics processing unit (GPU), power supply, input/output devices, and optical drives. It also discusses computer data representation systems such as binary, octal, decimal, and hexadecimal numbering.
This document discusses the objectives and limitations of computer science and outlines innovations that may shape the future of computing. It covers topics like the limitations of von Neumann architecture, new computing paradigms like light and biological computers, innovations in storage technologies, and the development of artificial intelligence through techniques such as neural networks and machine learning. The document emphasizes that computing hardware and paradigms are constantly changing and that future computers may be integrated with other devices and accessible through more natural human interaction.
This document outlines the key objectives and concepts covered in the Connecting with Computer Science course. The objectives include learning about computer hacking, system intruders, malicious code, social engineering, types of attacks, security measures, passwords, encryption, firewalls, and computer crime laws. The document also discusses ethics and privacy in computing.
The document discusses different number systems used in programmable logic controllers (PLCs), including binary, decimal, hexadecimal, and BCD. It explains that PLCs use binary numbers to represent on/off conditions as 1s and 0s. Binary numbers can be converted to decimal by multiplying each bit by its place value weight and summing the results. Hexadecimal represents 4 binary bits with a single hexadecimal digit for compact representation of data. Conversion between number systems involves dividing by the base and multiplying by the place value weights.
The document provides an introduction to digital systems and numerical representations. It discusses:
- Analog vs. digital representations and conversions between them using ADCs and DACs.
- Different number systems including binary, decimal, octal and hexadecimal. Methods to convert between these systems are described.
- Digital electronics uses discrete voltage levels (0V and 5V) to represent binary digits (0 and 1). Timing diagrams show the relationship between digital signals over time.
This document discusses number systems used in programmable logic controllers (PLCs), including binary, decimal, hexadecimal, and BCD. It explains that PLCs use binary numbers to represent on and off conditions, with each bit representing a 1 or 0. It also describes how to convert between binary, decimal, hexadecimal, and BCD numbering systems.
The document discusses number systems. It defines a number system as a system for writing and representing numbers using digits or symbols in a consistent manner. It allows for arithmetic operations and provides a unique representation for every number. The four most common number systems are decimal, binary, octal, and hexadecimal. Binary uses only two digits, 0 and 1, and is used to represent electrical signals in computers. Decimal uses base 10 with digits 0-9 in place values. [END SUMMARY]
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 a syllabus for a course on Digital Logic Design (DLD). It includes:
- The course instructor's name and details.
- A list of textbooks and online resources for the course.
- The course outcomes, which are to identify digital logic concepts, simplify Boolean expressions, design data processing circuits, and design sequential circuits.
- An outline of the course units, which cover basic logic circuits, number systems, Boolean algebra, circuit implementation, and sequential circuits.
The document provides an overview of the topics, resources, and goals of the DLD course.
This is my notes on simplified version of slides from DTU course 02135 Introduction to cyber systems. The lectures and slides are originally given by Prof. Pezzarossa.
Number-Systems presentation of the mathematicsshivas379526
The document discusses different number systems including decimal, binary, hexadecimal, and their importance. It provides the following key points:
- Decimal is base-10 as it uses 10 digits (0-9). Binary is base-2 as it uses two digits, 0 and 1. Hexadecimal is base-16 as it uses 16 symbols (0-9 and A-F).
- Different number systems are important because computers use binary to simplify calculations and reduce circuitry/costs. Larger systems like hexadecimal are used to represent large memory addresses.
- Converting between systems involves placing the remainder of successive divisions by the base in each position. For example, converting 42 to binary is 101010 by dividing 42
The document provides information about different number systems used in computers, including binary, octal, hexadecimal, and decimal. It explains the characteristics of each system such as the base and digits used. Methods for converting between number systems like binary to decimal and vice versa are presented. Shortcut methods for direct conversions between binary, octal, and hexadecimal are also described. Binary arithmetic and binary-coded decimal number representation are discussed.
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.
- Decimal, binary, octal, and hexadecimal are different number systems used to represent numeric values.
- Decimal uses 10 digits (0-9), binary uses two digits (0-1), octal uses 8 digits (0-7), and hexadecimal uses 16 digits (0-9 and A-F).
- Each system has a base or radix - the number of unique digits used. Decimal is base 10, binary base 2, octal base 8, and hexadecimal base 16.
- Numbers can be converted between these systems using division and multiplication operations that take into account the place value of each digit based on the system's base.
The document provides information about number systems used in computers. It discusses binary system which uses two digits (0 and 1) to represent ON and OFF states of switches in a computer. It explains how to convert between binary, decimal, octal and hexadecimal number systems using different methods. The document also covers signed binary numbers, binary codes like ASCII and Gray codes. Finally, it discusses binary logic, truth tables, Boolean expressions and logic gates used in switching circuits.
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.
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.
The document outlines a lesson plan covering number systems. It includes converting between decimal, binary, octal, and hexadecimal number systems. The key concepts covered are the different number systems used in computing, including binary, octal, hexadecimal, and their bases. Conversion between these systems involves multiplying digits by place values to get the value in another base. The skills practiced are computational thinking and step-wise thinking. Values reinforced include awareness of computer technology development and patience.
This document discusses different number systems including binary, decimal, octal, and hexadecimal. It provides definitions and characteristics of each system. The key points covered are:
- Binary uses digits 0 and 1, octal uses 0-7, hexadecimal uses 0-9 and A-F, and decimal uses 0-9.
- Each system has a base (2, 8, 16, 10 respectively) that determines the value of each digit position.
- Methods for converting between number systems are presented, including using division or multiplying by the place value to change between decimal, binary, octal, and hexadecimal.
This document discusses different number systems including binary, decimal, octal, and hexadecimal. It provides details on each system such as their number bases and allowed digits. The document also describes how to convert between these different number systems using methods like dividing numbers by the target base or grouping binary digits into sets of four for hexadecimal conversion. The goal is to understand representation of numbers in computing systems which commonly use binary and hexadecimal formats.
The document discusses different number systems including positional and non-positional systems. It describes the decimal, binary, octal and hexadecimal number systems. Decimal uses base 10 with digits 0-9. Binary uses base 2 with digits 0-1. Octal uses base 8 with digits 0-7. Hexadecimal uses base 16 with digits 0-9 and A-F. It provides examples of converting between these number systems using direct conversion or shortcuts involving grouping digits. Fractional numbers are also represented in these positional systems.
The document describes the syllabus for the course EEE365 Digital Electronics. The course covers topics such as number systems, Boolean algebra, combinational and sequential logic circuit design, memory devices, and digital signal conversion. Reference books for the course include titles on digital logic, digital systems, and digital design principles.
This document provides an introduction to computers and their components. It defines a computer as a machine that can carry out arithmetic and logical operations automatically according to programming instructions. It then lists and describes the major internal and external components of a computer, including the central processing unit (CPU), motherboard, memory (RAM), hard drive, graphics processing unit (GPU), power supply, input/output devices, and optical drives. It also discusses computer data representation systems such as binary, octal, decimal, and hexadecimal numbering.
Similar to Numbering system data representation (20)
This document discusses the objectives and limitations of computer science and outlines innovations that may shape the future of computing. It covers topics like the limitations of von Neumann architecture, new computing paradigms like light and biological computers, innovations in storage technologies, and the development of artificial intelligence through techniques such as neural networks and machine learning. The document emphasizes that computing hardware and paradigms are constantly changing and that future computers may be integrated with other devices and accessible through more natural human interaction.
This document outlines the key objectives and concepts covered in the Connecting with Computer Science course. The objectives include learning about computer hacking, system intruders, malicious code, social engineering, types of attacks, security measures, passwords, encryption, firewalls, and computer crime laws. The document also discusses ethics and privacy in computing.
The document discusses the software engineering process for developing applications. It explains that a design document acts as a blueprint, outlining requirements, diagrams, data structures, and more. The design process involves learning user needs, creating UML diagrams, a data dictionary, and prototypes. An effective team includes roles like project manager, developer, and tester. The goal is to follow a structured process to successfully deliver working software that meets user needs.
This document provides an overview of computer programming concepts including:
- The difference between low-level and high-level programming languages, using Assembly and Java as examples.
- The structure of a program including algorithms, pseudocode, variables, operators, and control structures.
- Programming language basics like data types, variables, operators, and control flow are introduced using Java syntax.
- Key terms are defined like algorithms, pseudocode, compilers, interpreters, variables, data types, and operators.
The document discusses file systems and file organization. It compares the FAT and NTFS file systems, describing their advantages and disadvantages. FAT groups hard drive sectors into clusters and uses a file allocation table to track files. NTFS overcomes FAT limitations with features like security and crash recovery. The document also describes sequential and random access of files, how hashing converts keys into indexes to allow random access, and how hashing algorithms deal with collisions.
chapter09 -Programming Data Structures.pdfsatonaka3
This document discusses various data structures used in computer science including arrays, linked lists, stacks, queues, and trees. It provides examples of each data structure and how they are used to organize data efficiently. Key points covered include how arrays store and access data, how linked lists allow dynamic allocation of memory, common uses of stacks and queues, and searching and sorting algorithms like binary search trees and selection sort.
The document provides an overview of database concepts and the database design process. It discusses the history and applications of databases. The key concepts covered include database fundamentals, database modeling, normalization, relationships, and the six-step design process. The goal is to introduce readers to essential database topics and skills needed for computer science.
The document discusses the fundamentals of how the Internet works, including its architecture, protocols, IP addressing, DNS, and web technologies. It covers topics such as how the TCP/IP protocol suite facilitates communication, how routers direct traffic, and how HTML and URLs enable the functioning of the world wide web. The overall aim is to explain the key components and operations that make the Internet a global information network.
This document provides an overview of computer networking concepts. It covers topics such as transmission media like copper wire and fiber optics, network protocols, the OSI model, common network types (LAN, WAN, WLAN), network devices, and switched networks. The objectives are to learn the basics of how computers connect and communicate over networks.
The document discusses operating systems and their functions. It defines what an operating system is, describes the major types of operating systems, and explains the key functions of operating systems, which include providing a user interface, managing processes and resources, and providing security. It also discusses how to perform basic file management tasks in different operating systems like Windows, UNIX, and DOS.
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2. Connecting with Computer Science 2
Objectives
• Learn why numbering systems are important to
understand
• Refresh your knowledge of powers of numbers
• Learn how numbering systems are used to count
• Understand the significance of positional value in a
numbering system
• Learn the differences and similarities between
numbering system bases
3. Connecting with Computer Science 3
Objectives (continued)
• Learn how to convert numbers between bases
• Learn how to do binary and hexadecimal math
• Learn how data is represented as binary in the
computer
• Learn how images and sounds are stored in the
computer
4. Connecting with Computer Science 4
Why You Need to Know About...
Numbering Systems
• Computers store programs and data in binary code
• Understanding of binary code is key to machine
• Binary number system is point of departure
• Hexadecimal number system
– Provides convenient representation
– Written into error messages
5. Connecting with Computer Science 5
Powers of Numbers - A Refresher
• Raising a number to a positive power (exponent)
– Self-multiply the number by the specified power
– Example: 23 = 2 * 2 * 2 = 8 (asterisk = multiplication)
– Special cases: 0 and 1 as powers
• Any number raised to 0 = 1; e.g, 10,5550 = 1.
• Any number raised to 1 = itself; e.g., 10,5551 = 10,555
6. Connecting with Computer Science 6
Powers of Numbers -A Refresher
(continued)
• Raising a number to a negative power
– Follow same steps for positive power
– Divide result into 1; e.g., 2-3 = 1/ (23) = .125
7. Connecting with Computer Science 7
Counting Things
• Numbers are used to count things
– Base 10 (decimal) most familiar
• The computer uses base 2, called binary
– Base 2 has two unique digits: 0 and 1
8. Connecting with Computer Science 8
Counting Things (continued)
• Hexadecimal system used to represent binary digits
– Base 16 has sixteen unique digits: 0 – 9, A - F
• Counting for all number systems similar
– Count digits defined in number system until exhausted
– Place zero in ones column. Carry one to the left
9. Connecting with Computer Science 9
Positional Value
• Weight assigned digit based on position in number
– Determine positional value of each digit by raising 10
to position within number
– Determine digit’s contribution to overall number by
multiplying digit by positional value
– Consider 5 in 3456.123 (radix = 10 = decimal point)
• Positional value = 101
• Overall contribution = 5 x 101 = 50
11. Connecting with Computer Science 11
Positional Value (continued)
• Number: sum of products of each digit and positional
value
– Example: 3456.123 = 3 x 103 + 4 x 102 + 5 x 101 + 6 x
100 + 1 x 10-1 + 2 x 10-2 + 3 x 10-3
• Numbers in all bases can be defined by position
– Base 2: Multiply each digit by 2 digit position
– Base 16: Multiply each digit by 16 digit position
– Base b: Multiply each digit by b digit position
13. Connecting with Computer Science 13
How Many Things Does
A Number Represent
• Number = sum of each digit x positional value
– Translate number of things to accord with base 10
– e.g.: 10012 is equivalent to nine things = (1 * 20) + (0 *
21) + (0 * 22) + (1 * 23)
• General procedure for evaluating numbers (any base)
1. Calculate the value for each position of the number
by raising the base value to the power of the position
2. Multiply positional value by digit in that position
3. Add each of the calculated values together
14. Connecting with Computer Science 14
Converting Numbers
Between Bases
• Any quantity can be represented by some number
in any base
• Counting process similar for all bases
1. Count until highest digit for base reached
2. Add 1 to next higher position to left
3. Return 0 to current position
• Conversion is a map from one base to another
– Identities can be easily calculated
– Identities may also be obtained by table look-up
16. Connecting with Computer Science 16
Converting To Base 10
• Three methods:
1. Table look-up (more extensive than Table 4-1)
2. Calculator
3. Algorithm for evaluating number in any base
• Example: consider 169AE in base 16
– Identify base: 16
– Map positions to digits: 4 3 2 1 0
– Raise, multiply and add: 169AE = (1 x 164) + (6 x
163) + (9 x 162) + (10 x 161) + (14 x 160) = 92,590
17. Connecting with Computer Science 17
Converting From Base 10
• Three methods:
1. Table look-up (more extensive than Table 4-1)
2. Calculator
18. Connecting with Computer Science 18
Converting From Base 10
(continued)
3. Algorithm for converting from base 10
1. Divide the decimal number by the number of the target
base (for example, 2 or 16)
2. Write down the remainder
3. Divide the quotient of the prior division by the base
again
4. Write the remainder to the left of the last remainder
written
5. Repeat Steps 3 and 4 until the whole number result is 0
19. Connecting with Computer Science 19
Converting From Base 10
(continued)
• Practice conversion algorithm: find hexadecimal
equivalent of decimal 45
– Divide 45 by 16 (base)
– Write down remainder D
– Divide 2 by 16
– Write down remainder 2 to the left of D (2D)
– Stop since reduced quotient = 0
– Check: 2D = (2 x 161) + (13 x 160) = 32 + 13 = 45
20. Connecting with Computer Science 20
Binary And Hexadecimal Math
• Procedure for adding numbers similar in all bases
– Difference lies in carry process
– Value of carry = value of base
– Example: 1011
+1101
11000
– Carry value for above = 102 = (1 x 101 + 0 x 100 ) = 210
• Procedure for subtraction, multiplication, and
division also similar
22. Connecting with Computer Science 22
Data Representation In Binary
• Binary values map to two-state transistors
• Bit: fundamental logical/physical unit (1/0 = on/off)
• Byte: grouping of eight bits (nibble = ½ byte)
• Word: collection of bytes (4 bytes is typical)
• Hexadecimal used as binary shorthand
– Relate each hexadecimal digit to 4-bit binary pattern
– Example: 1111 1010 1100 1110 =
F A C E (see Table 4-1)
23. Connecting with Computer Science 23
Representing Whole Numbers
• Whole numbers stored in fixed number of bits
– 200410 stored as 16-bit integer 0000011111010100
• Signed numbers stored with two’s complement
– Left most bit reserved for sign (1 = neg and 0 = pos)
– If positive, store with leading zeroes to fit field
– If negative, perform two’s complement
• Reverse bit pattern
• Add 1 to number using binary addition
25. Connecting with Computer Science 25
Representing Fractional Numbers
• Computers store fractional numbers (neg and pos)
• Storage technique based on floating-point notation
– Example of floating point number: 1.345 E+5
– 1.345 = mantissa, E = exponent, + 5 moves decimal
• IEEE-754 specification uses binary mantissas and
exponents
• Implementation details part of advanced study
26. Connecting with Computer Science 26
Representing Characters
• Computers store characters according to standards
• ASCII
– Represents characters with 7-bit pattern
– Provides for upper and lowercase English letters,
numeric characters, punctuation, special characters
– Accommodates 128 (27) different characters
• Globalization places upward pressure
– Extended ASCII: allows 8-bit patterns (256 total)
– Unicode: defined for 16 bit patterns (34,168 total)
27. Connecting with Computer Science 27
Representing Images
• Screen image made up of small dots of colored light
– Dot called “pixel” (picture element), smallest unit
– Resolution: # pixels in each row and column
– Each pixel is stored in the computer as a binary pattern
• RGB encoding
– Red, blue, and green assigned to eight of 24 bits
– White represented with 1s, black with 0s
– Color is the amount of red, green, and blue specified in
each of the 8-bit sections
28. Connecting with Computer Science 28
Representing Images (continued)
• Images, such as photos, stored with pixel-based
technologies
• Large image files can be compressed (JPG, GIF
formats)
• Moving images can also be compressed (MPEG,
MOV, WMV)
29. Connecting with Computer Science 29
Representing Sounds
• Sound represented as waveform with
– Amplitude (volume) and
– Frequency (pitch)
• Computer samples sounds at fixed intervals
– Samples given a binary value according to amplitude
– # bits in each sample determines amplitude range
– For CD-quality audio
• Sound must be sampled over 44,000 times a second
• Samples must allow > 65,000 different amplitudes
31. Connecting with Computer Science 31
One Last Thought
• Binary code is the language of the machine
• Knowledge of base 2 and base 16 prerequisite to
knowledge of machine language
• Computer scientists are more effective with binary
and hexadecimal concepts
32. Connecting with Computer Science 32
Summary
• Knowledge of alternative number systems essential
• Machine language based on binary system
• Hexadecimal used to represent binary numbers
• Power rule for numbers defines self-multiplication
• Any number can be represented in any base
33. Connecting with Computer Science 33
Summary (continued)
• Positional value: weight based on digit position
• Counting processes similar for all bases
• Conversion between bases is one-to-one mapping
• Arithmetic defined for all bases
• Data representation: bits, nibbles, bytes, words
34. Connecting with Computer Science 34
Summary (continued)
• Two’s complement: technique for storing signed
numbers
• Floating point notation: system used to represent
fractions and irrationals
• ASCII and Unicode: character set standards
• Image representation: based on binary pixel
• Sound representation: based on amplitude samples