Digital Logic Lecture 2 Number Systems
|
|
- Myles Collins
- 6 years ago
- Views:
Transcription
1 Digital Logic Lecture 2 Number Systems By Ghada Al-Mashaqbeh The Hashemite University Computer Engineering Department
2 Outline Introduction. Basic definitions. Number systems types. Conversion between different number systems. Examples. The Hashemite University 2
3 Introduction There are 4 numbering systems: Binary: 2 digits (0 and 1). Decimal: 10 digits (0 9). Octal: 8 digits (0 7). Hexadecimal: 16 digits (0 9 and A F). The binary system is what used by computers. The decimal system is what used by humans in nature (also called natural number system). Hexadecimal and Octal are a compromise representation between the binary and the decimal. The Hashemite University 3
4 Basic Definitions I The number of digits used within a number system is called the Radix or the Base. So: Binary Base = 2 Octal Base = 8 Decimal Base = 10 Hexadecimal Base = 16 The point in the number, i.e. fractional number, is called radix point. so, based on the radix or base you call it, e.g. in decimal you call it decimal point and in binary you call it binary point. The used digits in each numbering system are borrowed from the decimal system and completed by the alphabet characters (capital letters). In other words, you are able to use the numerals 0-9 and the alphabets A-Z based on the number of digits you need. Then you pick the wanted number of digits in order. The Hashemite University 4
5 Basic Definitions II For binary, you need two digits, starting from the numerals set (i.e. 0-9) you get 0 and 1 (just two digits). E.g For octal, you need 8 digits, following the same approach you get the digits 0-7. E.g For decimal, you need 10 digits then you get 0-9. E.g For hexadecimal you need 16 digit, starting by the numerals you get 0-9 but still you need an additional 6 digits. For this purpose you move to the alphabet letters and pick 6 letters, so you get A-F. E.g. AD4. In decimal these letters correspond to the following: A 10 B 11 C 12 D 13 E 14 F 15 The Hashemite University 5
6 Basic Definitions III To differentiate between numbers from the different numbering systems use one of the following: Add a subscript of the base to the number. e.g: ( ) 2 Binary (176) 8 Octal (AD23) 16 Hexadecimal Add the following letters at the end of the number: B (Binary), Q (Octal), nothing for Decimal, H (Hexa). E.g B The universal approach to represent hexadecimal numbers is to start it with a 0 and append h to the end of the number. E.g. 0Dh. A less common approach to represent binary and hexadecimal numbers is to proceed the number with a: % for binary $ for hexadecimal e.g. %101001, $AD5 Any number proceeded by 0x is a hexadecimal number. Such representation is most commonly used in programming languages such as C++. The Hashemite University 6
7 Basic Definitions IV Every number in any numbering system can be represented as a weighted sum or a power series of the system radix. The coefficient of each term in the series is the digit value where the power of the base of that term depends on the location of the digit within the number with respect to the radix point (left or right). E.g.: (1890) 10 = 1x x x x10 0 (45.678) 10 = 4x x x x x10-3 (ADF) 16 = 10x x x16 0 Note: the result of this power series is the decimal equivalent of the original number as we will see later. The Hashemite University 7
8 Basic Definitions V In computer hardware the following are used: Bit = 0 or 1 Nibble = 4 bits. Byte = 8 bits Word = 2 byte = 16 bits 2 10 = 1 Kilo = = I Mega 2 30 = 1 Giga 2 40 = 1 Tera 1 keyboard character needs 1 byte to store its value. Every thing in computers is power of 2. The architecture of the computer is determined based on its word length, e.g. 32 bit or 64 bit architecture. The Hashemite University 8
9 Basic Definitions VI The least significant bit (LSB) is the bit or digit that is located on the right most location within a number. The most significant bit (MSB) is the bit that is located on the leftmost position within a number. E.g.: for the number , 1 is the MSB and 9 is the LSB. The Hashemite University 9
10 Why binary system for computers? Computers are comprised basically of switches. Switches have only tow states: either ON or OFF. As we learned, such states can be represented as two values: 1 and 0. So it is binary since there are two values and each value needs one digit or bit to represent and store it. The Hashemite University 10
11 Why decimal system for humans? Created with human history on earth and evolved from nature. Easy to understand. Compact: need small number of digits to represent large values. Also, humans have invented shorthand or abbreviations to simplify large decimal numbers representation to enhance readability, such as million, billion, trillion, etc. The Hashemite University 11
12 Conversion Between Binary and Decimal I From Binary to Decimal: Expand the binary number in a power series and sum the terms as shown previously. The same method is used when there is a radix point in the number. This method can be used to convert from any base to decimal. Examples: the binary value represents : = 1x x x x x x x x2 0 = 1x x64 + 0x32 + 0x16 + 1x8 + 0x4 + 1x2 + 0x1 = =202 The Hashemite University 12
13 Conversion Between Binary and Decimal II From Decimal to Binary: Here there are two cases: integer decimal numbers and fraction (floating point) decimal numbers. For integer decimal numbers, there are two methods: 1-Repeated Division By 2 For this method, divide the decimal number by 2, If the remainder is 0, on the right side write down a 0. If the remainder is 1, write down a 1. Repeat the division with the result (new quotient). This process is repeated till the quotient becomes 0. When performing the division, the remainders which will represent the binary equivalent of the decimal number are written beginning at the least significant digit (right) and each new digit is written to more significant digit (the left) of the previous digit. The Hashemite University 13
14 Integer Decimal to Binary First Method -- Example Example: 57 / 2 = 28, remainder = 1 (binary number will end with 1) 28 / 2 = 14, remainder = 0 14 / 2 = 7, remainder = 0 7 / 2 = 3, remainder = 1 3 / 2 = 1, remainder = 1 1 / 2 = 0, remainder = 1 (binary number will start with 1) Therefore, collecting the remainders, = The Hashemite University 14
15 Integer Decimal to Binary Second Method 2- Sum on the fly List the powers of 2 starting from 0 power where you stop at the maximum power less than the number you want to convert. See which of these powers must be included in the sum that will have the result of the decimal number that you want to convert. These powers will have 1 in its position and the powers that are not used will have 0. This method is suitable for small to mid size numbers. The Hashemite University 15
16 Integer Decimal to Binary Second Method -- Example Example: convert to binary Answer = The Hashemite University 16
17 Fraction Decimal to Binary As for integers, for fraction decimal numbers we use the following method: Repeated Multiplication By 2 For this method, multiply the decimal number by 2, If the integer part of the result is 0, on the right side write down a 0. If the integer part of the result is 1, write down a 1. Then remove the integer part from the result and repeat the multiplication by 2 for the fraction part. This process is repeated till the fraction becomes 0 or till you reach the sufficient accuracy. When performing the multiplication, the integers which will represent the binary equivalent of the decimal fraction are written beginning from the right of the decimal point and each new digit is written to less significant digit (the right) of the previous digit. Could be unending!!!!! The Hashemite University 17
18 Fraction Decimal to Binary -- Example Example: convert to binary. 0.3 * 2 = 0.6 integer part = * 2 = 1.2 integer part = * 2 = 0.4 integer part = * 2 = 0.8 integer part = * 2 = 1.6 integer part = * 2 = 1.2 integer part = 1, etc. Answer = The Hashemite University 18
19 Notes The same methods used to convert from decimal (integer or fraction) to binary are used to convert from decimal to any other base. Just change the radix or base value that you use in the multiplication or in the division. When you face a number that have both integer and fraction parts, e.g or , convert each part alone and then combine them. The Hashemite University 19
20 Why to introduce the hexadecimal and the octal (I)? Middle approach between binary and decimal systems: Binary is a must by computers but it is verbose, i.e. need large number of bits to represent small quantities, e.g needs 8 bits in binary. Decimal is preferred by humans, easy to understand and compact (needs small number of bits to represent values). Conversion between binary and decimal (especially decimal to binary) is not trivial and expensive in terms of the needed resources. The Hashemite University 20
21 Why to introduce the hexadecimal and the octal (II)? To keep commonality with binary system, ease of conversion, and the compact feature we need another number system whose base is a power of 2. Base 4 is not compact, however, base 8 (octal) and base 16 (hex) have this feature where base 16 is the most common. From here the hexadecimal system have been introduced to be a middle approach between binary, where it is easy to convert between hexa and binary, as we will see, and the compactness feature as decimal. Recall that computers store data in binary. However, hexa is commonly used in computers to represent locations (addresses) in data storage, in data access, and in RAM. The Hashemite University 21
22 Conversion Between Binary and Octal The octal has a base = 8 which is 2 raised to power 3. In other words, each digit in the octal system can substitute for 3 bits in the binary system. Such observation is utilized in the conversion process between octal and binary. The Hashemite University 22
23 Binary to Octal Conversion Start from the radix point to the left and to the right and do the following: Group the binary bits found in the binary number into groups consist of 3 bits. Substitute each group with the octal digit that represents the same quantity represented by the binary group. The same process are used with the bits on the right of the radix point (i.e. fraction). The Hashemite University 23
24 Binary to Octal Conversion - Example Convert the following number into octal: Answer = The Hashemite University 24
25 Octal to Binary Conversion Just repeat the inverse of the process used to convert from binary to octal. In other words, substitute each octal digit with its 3 bit binary equivalent. Example: convert to binary Answer = The Hashemite University 25
26 Octal to Decimal Conversion From Octal to Decimal: Use the power series expansion method with the base = 8. Example: convert to decimal. = 6x x x x x8-2 = The Hashemite University 26
27 Integer Decimal to Octal Conversion From Integer Decimal to Octal: Use the repeated division method used in converting from decimal to binary but the difference here is that you divide by 8 (radix value) instead of 2. The Hashemite University 27
28 Integer Decimal to Octal Conversion - Example Convert (44978) 10 to Octal Division Quotient Remainder / / / / / /8 0 1 Answer = The Hashemite University 28
29 Fraction Decimal to Octal Conversion Repeated Multiplication By 8 For this method, multiply the decimal number by 8. Write down the integer part of the result. Then remove the integer part from the result and repeat the multiplication by 8 for the fraction part. This process is repeated till the fraction becomes 0 or till you reach the sufficient accuracy. When performing the multiplication, the integers which will represent the octal equivalent of the decimal fraction are written beginning from the right of the decimal point and each new digit is written to less significant digit (the right) of the previous digit. Could be unending!!!!! The Hashemite University 29
30 Fraction Decimal to Octal Conversion - Example Example: convert to octal * 8 = integer part = * 8 = integer part = * 8 = integer part = * 8 = integer part = * 8 = integer part = * 8 = integer part = 3, etc. Answer = The Hashemite University 30
31 Conversion Between Binary and Hexadecimal The Hexa has a base = 16 which 2 raised to power 4. In other words, each digit in the hexa system can substitute for 4 bits in the binary system. Similar to octal, such observation is utilized in the conversion process between hexa and binary. The Hashemite University 31
32 Binary to Hexa Conversion Start from the radix point to the left and to the right and do the following: Group the binary bits found in the binary number into groups consist of 4 bits. Substitute each group with the Hexa digit that represents the same quantity represented by the binary group. The same process are used with the bits on the right of the radix point (i.e. fraction). The Hashemite University 32
33 Binary to Hexa Conversion - Example Convert the following number into Hexa: A F. B E 16 Answer = 25AF.BE 16 The Hashemite University 33
34 Hexa to Binary Conversion Just repeat the inverse of the process used to convert from binary to hexa. In other words, substitute each hexa digit with its 4 bit binary equivalent. Example: convert 56D to binary. 5 6 D Answer = The Hashemite University 34
35 Hexa to Decimal Conversion From Hexa to Decimal: Use the power series expansion method with the base = 16. Example: convert 67B to decimal. = 6x x x x x16-2 = The Hashemite University 35
36 Integer Decimal to Hexa Conversion From Integer Decimal to Hexa: Use the repeated division method used in converting from decimal to binary but the difference here is that you divide by 16 (radix value) instead of 2. The Hashemite University 36
37 Integer Decimal to Octal Conversion - Example Example: Convert to hexa / 16 = 2202, remainder = 11 B (hex number will end with B) 2202 / 16 = 137, remainder = 10 A 137 / 16 = 8, remainder = 9 8 / 16 = 0, remainder = 8 (hex number will start with 8) Therefore, collecting the remainders, = 89AB 16 The Hashemite University 37
38 Fraction Decimal to Octal Conversion Repeated Multiplication By 16 For this method, multiply the decimal number by 16. Write down the integer part of the result. Then remove the integer part from the result and repeat the multiplication by 8 for the fraction part. This process is repeated till the fraction becomes 0 or till you reach the sufficient accuracy. When performing the multiplication, the integers which will represent the hexa equivalent of the decimal fraction are written beginning from the right of the decimal point and each new digit is written to less significant digit (the right) of the previous digit. Could be unending!!!!! The Hashemite University 38
39 Fraction Decimal to Octal Conversion - Example Example: convert to hexa * 16 = integer part = * 16 = integer part = B * 16 = integer part = * 16 = integer part = * 16 = integer part = D * 16 = integer part = 0, etc. Answer = 0.5B22D The Hashemite University 39
40 Conversion Between Octal and Hexadecimal From octal to hexadecimal: First convert from octal to binary then convert the result from binary to hexadecimal. From hexadecimal to octal: First convert from hexa to binary then convert the result from binary to octal. The Hashemite University 40
41 Additional Notes This lecture covers the following material from the textbook: Chapter 1: Sections The Hashemite University 41
Lecture (01) Digital Systems and Binary Numbers By: Dr. Ahmed ElShafee
١ Lecture (01) Digital Systems and Binary Numbers By: Dr. Ahmed ElShafee Digital systems Digital systems are used in communication, business transactions, traffic control, spacecraft guidance, medical
More informationLecture (01) Introduction Number Systems and Conversion (1)
Lecture (01) Introduction Number Systems and Conversion (1) By: Dr. Ahmed ElShafee ١ Digital systems Digital systems are used in communication, business transactions, traffic control, spacecraft guidance,
More informationOctal & Hexadecimal Number Systems. Digital Electronics
Octal & Hexadecimal Number Systems Digital Electronics What, More Number Systems? Why do we need more number systems? Humans understand decimal Check out my ten digits! Digital electronics (computers)
More informationMoodle WILLINGDON COLLEGE SANGLI. ELECTRONICS (B. Sc.-I) Introduction to Number System
Moodle 1 WILLINGDON COLLEGE SANGLI ELECTRONICS (B. Sc.-I) Introduction to Number System E L E C T R O N I C S Introduction to Number System and Codes Moodle developed By Dr. S. R. Kumbhar Department of
More informationTOPICS. Other Number Systems. Other Number Systems 9/9/2017. Octal Hexadecimal Number conversion
Topic : Introduction To computers Faculty : Department of commerce and Management BY: Prof.Meeta R. Gujarathi E mail: meetargujarathi@gmail.com Octal Hexadecimal Number conversion TOPICS Other Number Systems
More informationCHAPTER V NUMBER SYSTEMS AND ARITHMETIC
CHAPTER V-1 CHAPTER V CHAPTER V NUMBER SYSTEMS AND ARITHMETIC CHAPTER V-2 NUMBER SYSTEMS RADIX-R REPRESENTATION Decimal number expansion 73625 10 = ( 7 10 4 ) + ( 3 10 3 ) + ( 6 10 2 ) + ( 2 10 1 ) +(
More informationMACHINE LEVEL REPRESENTATION OF DATA
MACHINE LEVEL REPRESENTATION OF DATA CHAPTER 2 1 Objectives Understand how integers and fractional numbers are represented in binary Explore the relationship between decimal number system and number systems
More informationNumber System. Introduction. Decimal Numbers
Number System Introduction Number systems provide the basis for all operations in information processing systems. In a number system the information is divided into a group of symbols; for example, 26
More informationBINARY SYSTEM. Binary system is used in digital systems because it is:
CHAPTER 2 CHAPTER CONTENTS 2.1 Binary System 2.2 Binary Arithmetic Operation 2.3 Signed & Unsigned Numbers 2.4 Arithmetic Operations of Signed Numbers 2.5 Hexadecimal Number System 2.6 Octal Number System
More informationLevel ISA3: Information Representation
Level ISA3: Information Representation 1 Information as electrical current At the lowest level, each storage unit in a computer s memory is equipped to contain either a high or low voltage signal Each
More informationCOE 202- Digital Logic. Number Systems II. Dr. Abdulaziz Y. Barnawi COE Department KFUPM. January 23, Abdulaziz Barnawi. COE 202 Logic Design
1 COE 0- Digital Logic Number Systems II Dr. Abdulaziz Y. Barnawi COE Department KFUPM COE 0 Logic Design January 3, 016 Objectives Base Conversion Decimal to other bases Binary to Octal and Hexadecimal
More informationKorea University of Technology and Education
MEC52 디지털공학 Binary Systems Jee-Hwan Ryu School of Mechanical Engineering Binary Numbers a 5 a 4 a 3 a 2 a a.a - a -2 a -3 base or radix = a n r n a n- r n-...a 2 r 2 a ra a - r - a -2 r -2...a -m r -m
More informationDigital Fundamentals
Digital Fundamentals Tenth Edition Floyd Chapter 2 2009 Pearson Education, Upper 2008 Pearson Saddle River, Education NJ 07458. All Rights Reserved Decimal Numbers The position of each digit in a weighted
More informationNumber representations
Number representations Number bases Three number bases are of interest: Binary, Octal and Hexadecimal. We look briefly at conversions among them and between each of them and decimal. Binary Base-two, or
More information1.1. Unit 1. Integer Representation
1.1 Unit 1 Integer Representation 1.2 Skills & Outcomes You should know and be able to apply the following skills with confidence Convert an unsigned binary number to and from decimal Understand the finite
More informationMC1601 Computer Organization
MC1601 Computer Organization Unit 1 : Digital Fundamentals Lesson1 : Number Systems and Conversions (KSB) (MCA) (2009-12/ODD) (2009-10/1 A&B) Coverage - Lesson1 Shows how various data types found in digital
More informationCMPE223/CMSE222 Digital Logic Design. Positional representation
CMPE223/CMSE222 Digital Logic Design Number Representation and Arithmetic Circuits: Number Representation and Unsigned Addition Positional representation First consider integers Begin with positive only
More informationInformation Science 1
Information Science 1 - Representa*on of Data in Memory- Week 03 College of Information Science and Engineering Ritsumeikan University Topics covered l Basic terms and concepts of The Structure of a Computer
More informationITEC 1011 Introduction to Information Technologies
Number Systems Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, 7 No No Hexadecimal 16 0, 1, 9, A, B, F No No Quantities/Counting
More informationCHW 261: Logic Design
CHW 261: Logic Design Instructors: Prof. Hala Zayed Dr. Ahmed Shalaby http://www.bu.edu.eg/staff/halazayed14 http://bu.edu.eg/staff/ahmedshalaby14# Slide 1 Slide 2 Slide 3 Digital Fundamentals CHAPTER
More informationDigital Systems and Binary Numbers
Digital Systems and Binary Numbers Prof. Wangrok Oh Dept. of Information Communications Eng. Chungnam National University Prof. Wangrok Oh(CNU) 1 / 51 Overview 1 Course Summary 2 Binary Numbers 3 Number-Base
More informationCHAPTER 2 Number Systems
CHAPTER 2 Number Systems Objectives After studying this chapter, the student should be able to: Understand the concept of number systems. Distinguish between non-positional and positional number systems.
More information10.1. Unit 10. Signed Representation Systems Binary Arithmetic
0. Unit 0 Signed Representation Systems Binary Arithmetic 0.2 BINARY REPRESENTATION SYSTEMS REVIEW 0.3 Interpreting Binary Strings Given a string of s and 0 s, you need to know the representation system
More information2 Number Systems 2.1. Foundations of Computer Science Cengage Learning
2 Number Systems 2.1 Foundations of Computer Science Cengage Learning 2.2 Objectives After studying this chapter, the student should be able to: Understand the concept of number systems. Distinguish between
More informationCS & IT Conversions. Magnitude 10,000 1,
CS & IT Conversions There are several number systems that you will use when working with computers. These include decimal, binary, octal, and hexadecimal. Knowing how to convert between these number systems
More informationNumeral Systems. -Numeral System -Positional systems -Decimal -Binary -Octal. Subjects:
Numeral Systems -Numeral System -Positional systems -Decimal -Binary -Octal Subjects: Introduction A numeral system (or system of numeration) is a writing system for expressing numbers, that is a mathematical
More informationDIGITAL SYSTEM DESIGN
DIGITAL SYSTEM DESIGN UNIT I: Introduction to Number Systems and Boolean Algebra Digital and Analog Basic Concepts, Some history of Digital Systems-Introduction to number systems, Binary numbers, Number
More informationNumber Systems. TA: Mamun. References: Lecture notes of Introduction to Information Technologies (ITEC 1011) by Dr Scott MacKenzie
Number Systems TA: Mamun References: Lecture notes of Introduction to Information Technologies (ITEC 1011) by Dr Scott MacKenzie Common Number Systems System Base Symbols Decimal 10 0, 1, 9 Binary 2 0,
More information3.1. Unit 3. Binary Representation
3.1 Unit 3 Binary Representation ANALOG VS. DIGITAL 3.2 3.3 Analog vs. Digital The analog world is based on continuous events. Observations can take on (real) any value. The digital world is based on discrete
More informationChapter 1 Preliminaries
Chapter 1 Preliminaries This chapter discusses the major classes of programming languages and the relationship among them. It also discusses the binary and the hexadecimal number systems which are used
More informationNumber Systems CHAPTER Positional Number Systems
CHAPTER 2 Number Systems Inside computers, information is encoded as patterns of bits because it is easy to construct electronic circuits that exhibit the two alternative states, 0 and 1. The meaning of
More informationChapter 3 DATA REPRESENTATION
Page1 Chapter 3 DATA REPRESENTATION Digital Number Systems In digital systems like computers, the quantities are represented by symbols called digits. Many number systems are in use in digital technology
More informationCommon Number Systems
Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, 7 No No Hexadecimal 16 0, 1, 9, A, B, F No No Quantities/Counting
More informationTOPIC: NUMBER SYSTEMS
Ministry of Secondary Education Progressive Comprehensive High School PCHS Mankon Bamenda Department of Computer Studies Republic of Cameroon Peace Work - Fatherland TOPIC: NUMBER SYSTEMS Class: Comp.
More informationCMPS 10 Introduction to Computer Science Lecture Notes
CMPS Introduction to Computer Science Lecture Notes Binary Numbers Until now we have considered the Computing Agent that executes algorithms to be an abstract entity. Now we will be concerned with techniques
More informationFundamentals of Programming (C)
Borrowed from lecturer notes by Omid Jafarinezhad Fundamentals of Programming (C) Group 8 Lecturer: Vahid Khodabakhshi Lecture Number Systems Department of Computer Engineering Outline Numeral Systems
More informationCOE 202: Digital Logic Design Number Systems Part 2. Dr. Ahmad Almulhem ahmadsm AT kfupm Phone: Office:
COE 0: Digital Logic Design Number Systems Part Dr. Ahmad Almulhem Email: ahmadsm AT kfupm Phone: 860-7554 Office: -34 Objectives Arithmetic operations: Binary number system Other number systems Base Conversion
More informationumber Systems bit nibble byte word binary decimal
umber Systems Inside today s computers, data is represented as 1 s and 0 s. These 1 s and 0 s might be stored magnetically on a disk, or as a state in a transistor. To perform useful operations on these
More informationDIGITAL SYSTEM FUNDAMENTALS (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) COURSE / CODE NUMBER SYSTEM
COURSE / CODE DIGITAL SYSTEM FUNDAMENTALS (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) NUMBER SYSTEM A considerable subset of digital systems deals with arithmetic operations. To understand the
More informationCMSC 313 COMPUTER ORGANIZATION & ASSEMBLY LANGUAGE PROGRAMMING LECTURE 01, SPRING 2013
CMSC 313 COMPUTER ORGANIZATION & ASSEMBLY LANGUAGE PROGRAMMING LECTURE 01, SPRING 2013 TOPICS TODAY Course overview Levels of machines Machine models: von Neumann & System Bus Fetch-Execute Cycle Base
More informationLecture 2: Number Systems
Lecture 2: Number Systems Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Prof. Russell Tessier of University of Massachusetts Aby George of Wayne State University Contents
More informationNumber Systems. Decimal numbers. Binary numbers. Chapter 1 <1> 8's column. 1000's column. 2's column. 4's column
1's column 10's column 100's column 1000's column 1's column 2's column 4's column 8's column Number Systems Decimal numbers 5374 10 = Binary numbers 1101 2 = Chapter 1 1's column 10's column 100's
More informationNumber Systems and Binary Arithmetic. Quantitative Analysis II Professor Bob Orr
Number Systems and Binary Arithmetic Quantitative Analysis II Professor Bob Orr Introduction to Numbering Systems We are all familiar with the decimal number system (Base 10). Some other number systems
More informationDLD VIDYA SAGAR P. potharajuvidyasagar.wordpress.com. Vignana Bharathi Institute of Technology UNIT 1 DLD P VIDYA SAGAR
UNIT I Digital Systems: Binary Numbers, Octal, Hexa Decimal and other base numbers, Number base conversions, complements, signed binary numbers, Floating point number representation, binary codes, error
More informationChapter 2 Number Systems and Codes Dr. Xu
Chapter 2 Number Systems and Codes Dr. Xu Chapter 2 Objectives Selected areas covered in this chapter: Converting between number systems. Decimal, binary, hexadecimal. Advantages of the hexadecimal number
More informationCollege of Computer and Information Sciences Department of Computer Science. CSC 220: Computer Organization. Unit1 Number Systems
College of Computer and Information Sciences Department of Computer Science CSC 220: Computer Organization Unit1 Number Systems Common Number Systems System Base Symbols Used by humans? Used in computers?
More informationChap 1. Digital Computers and Information
Chap 1. Digital Computers and Information Spring 004 Overview Digital Systems and Computer Systems Information Representation Number Systems [binary, octal and hexadecimal] Arithmetic Operations Base Conversion
More informationLOGIC DESIGN. Dr. Mahmoud Abo_elfetouh
LOGIC DESIGN Dr. Mahmoud Abo_elfetouh Course objectives This course provides you with a basic understanding of what digital devices are, how they operate, and how they can be designed to perform useful
More informationObjectives. 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
More informationUNIT 2 NUMBER SYSTEM AND PROGRAMMING LANGUAGES
UNIT 2 NUMBER SYSTEM AND PROGRAMMING LANGUAGES Structure 2.0 Introduction 2.1 Unit Objectives 2.2 Number Systems 2.3 Bits and Bytes 2.4 Binary Number System 2.5 Decimal Number System 2.6 Octal Number System
More informationSwitching Circuits and Logic Design Prof. Indranil Sengupta Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur
Switching Circuits and Logic Design Prof. Indranil Sengupta Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture - 02 Octal and Hexadecimal Number Systems Welcome
More informationMs Sandhya Rani Dash UNIT 2: NUMBER SYSTEM AND CODES. 1.1 Introduction
Ms Sandhya Rani Dash UNIT 2: NUMBER SYSTEM AND CODES Structure 2.1 Introduction 2.2 Objectives 2.3 Binary Numbers 2.3.1 Binary-to-Decimal conversion 2.3.2 Decimal-to-Binary Conversion 2.4 Octal Numbers
More informationOutline. What Digit? => Number System. Decimal (base 10) Significant Digits. Lect 03 Number System, Gates, Boolean Algebra. CS221: Digital Design
Lect 3 Number System, Gates, Boolean Algebra CS22: Digital Design Dr. A. Sahu Dept of Comp. Sc. & Engg. Indian Institute of Technology Guwahati Outline Number System Decimal, Binary, Octal, Hex Conversions
More informationChapter 2. Positional number systems. 2.1 Signed number representations Signed magnitude
Chapter 2 Positional number systems A positional number system represents numeric values as sequences of one or more digits. Each digit in the representation is weighted according to its position in the
More informationRui Wang, Assistant professor Dept. of Information and Communication Tongji University.
Data Representation ti and Arithmetic for Computers Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Questions What do you know about
More informationCOMP Overview of Tutorial #2
COMP 1402 Winter 2008 Tutorial #2 Overview of Tutorial #2 Number representation basics Binary conversions Octal conversions Hexadecimal conversions Signed numbers (signed magnitude, one s and two s complement,
More informationNumber System (Different Ways To Say How Many) Fall 2016
Number System (Different Ways To Say How Many) Fall 2016 Introduction to Information and Communication Technologies CSD 102 Email: mehwish.fatima@ciitlahore.edu.pk Website: https://sites.google.com/a/ciitlahore.edu.pk/ict/
More informationNumber Systems and Conversions UNIT 1 NUMBER SYSTEMS & CONVERSIONS. Number Systems (2/2) Number Systems (1/2) Iris Hui-Ru Jiang Spring 2010
Contents Number systems and conversion Binary arithmetic Representation of negative numbers Addition of two s complement numbers Addition of one s complement numbers Binary s Readings Unit.~. UNIT NUMBER
More informationDigital Systems and Binary Numbers
Digital Systems and Binary Numbers ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Spring, 2018 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outline
More informationDigital Fundamentals. CHAPTER 2 Number Systems, Operations, and Codes
Digital Fundamentals CHAPTER 2 Number Systems, Operations, and Codes Decimal Numbers The decimal number system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 The decimal numbering system has a base of
More informationDigital Logic Lecture 4 Binary Codes
Digital Logic Lecture 4 Binary Codes By Ghada Al-Mashaqbeh The Hashemite University Computer Engineering Department Outline Introduction. Character coding. Error detection codes. Gray code. Decimal coding.
More informationEEM 232 Digital System I
EEM 232 Digital System I Instructor : Assist. Prof. Dr. Emin Germen egermen@anadolu.edu.tr Course Book : Logic and Computer Design Fundamentals by Mano & Kime Third Ed/Fourth Ed.. Pearson Grading 1 st
More informationChapter 2. Data Representation in Computer Systems
Chapter 2 Data Representation in Computer Systems Chapter 2 Objectives Understand the fundamentals of numerical data representation and manipulation in digital computers. Master the skill of converting
More informationData Representation COE 301. Computer Organization Prof. Muhamed Mudawar
Data Representation COE 30 Computer Organization Prof. Muhamed Mudawar College of Computer Sciences and Engineering King Fahd University of Petroleum and Minerals Presentation Outline Positional Number
More informationChapter 3: Number Systems and Codes. Textbook: Petruzella, Frank D., Programmable Logic Controllers. McGraw Hill Companies Inc.
Chapter 3: Number Systems and Codes Textbook: Petruzella, Frank D., Programmable Logic Controllers. McGraw Hill Companies Inc., 5 th edition Decimal System The radix or base of a number system determines
More informationDigital Logic. The Binary System is a way of writing numbers using only the digits 0 and 1. This is the method used by the (digital) computer.
Digital Logic 1 Data Representations 1.1 The Binary System The Binary System is a way of writing numbers using only the digits 0 and 1. This is the method used by the (digital) computer. The system we
More informationChapter 1 Emad Felemban
Chapter 1 Emad Felemban Digital Computers and Digital Systems Binary Numbers Number Base Conversion Octal and Hexadecimal Numbers Complements Singed Binary Numbers Binary Codes Binary Storage and Registers
More informationQUIZ: Generations of computer technology. Hardware:
QUIZ: Generations of computer technology Hardware: 1. 2. 3. 4. 5. 1 QUIZ: Generations of computer technology Software: 1. 2. 3. 4. 5. 6. 2 Steampunk! 3 The Telectroscope, 1878-2008 Steampunk Wikipedia
More informationT02 Tutorial Slides for Week 2
T02 Tutorial Slides for Week 2 ENEL 353: Digital Circuits Fall 2017 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 19 September, 2017
More informationIntroduction to Numbering Systems
NUMBER SYSTEM Introduction to Numbering Systems We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are Binary Base 2 Octal Base 8 Hexadecimal
More informationLogic Circuits I ECE 1411 Thursday 4:45pm-7:20pm. Nathan Pihlstrom.
Logic Circuits I ECE 1411 Thursday 4:45pm-7:20pm Nathan Pihlstrom www.uccs.edu/~npihlstr My Background B.S.E.E. from Colorado State University M.S.E.E. from Colorado State University M.B.A. from UCCS Ford
More information1.1. INTRODUCTION 1.2. NUMBER SYSTEMS
Chapter 1. 1.1. INTRODUCTION Digital computers have brought about the information age that we live in today. Computers are important tools because they can locate and process enormous amounts of information
More informationPositional notation Ch Conversions between Decimal and Binary. /continued. Binary to Decimal
Positional notation Ch.. /continued Conversions between Decimal and Binary Binary to Decimal - use the definition of a number in a positional number system with base - evaluate the definition formula using
More informationD I G I T A L C I R C U I T S E E
D I G I T A L C I R C U I T S E E Digital Circuits Basic Scope and Introduction This book covers theory solved examples and previous year gate question for following topics: Number system, Boolean algebra,
More informationBinary. Hexadecimal BINARY CODED DECIMAL
Logical operators Common arithmetic operators, like plus, minus, multiply and divide, works in any number base but the binary number system provides some further operators, called logical operators. Meaning
More informationChapter 2: Number Systems
Chapter 2: Number Systems Logic circuits are used to generate and transmit 1s and 0s to compute and convey information. This two-valued number system is called binary. As presented earlier, there are many
More informationELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-278: Digital Logic Design Fall Notes - Unit 4. hundreds.
ECE-78: Digital Logic Design Fall 6 UNSIGNED INTEGER NUMBERS Notes - Unit 4 DECIMAL NUMBER SYSTEM A decimal digit can take values from to 9: Digit-by-digit representation of a positive integer number (powers
More informationOctal and Hexadecimal Integers
Octal and Hexadecimal Integers CS 350: Computer Organization & Assembler Language Programming A. Why? Octal and hexadecimal numbers are useful for abbreviating long bitstrings. Some operations on octal
More informationELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-2700: Digital Logic Design Winter Notes - Unit 4. hundreds.
UNSIGNED INTEGER NUMBERS Notes - Unit 4 DECIMAL NUMBER SYSTEM A decimal digit can take values from to 9: Digit-by-digit representation of a positive integer number (powers of ): DIGIT 3 4 5 6 7 8 9 Number:
More informationNumber codes nibble byte word double word
Codes Number codes Digital circuits (like processors) represent information with two-valued binary codes. Why? Every signal wire in a digital system can transport one binary digit, or bit. A signal at
More informationDATA REPRESENTATION. By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region. Based on CBSE curriculum Class 11. Neha Tyagi, KV 5 Jaipur II Shift
DATA REPRESENTATION Based on CBSE curriculum Class 11 By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region Neha Tyagi, KV 5 Jaipur II Shift Introduction As we know that computer system stores any
More informationEE 109 Unit 2. Binary Representation Systems
EE 09 Unit 2 Binary Representation Systems ANALOG VS. DIGITAL 2 3 Analog vs. Digital The analog world is based on continuous events. Observations can take on (real) any value. The digital world is based
More informationWhat Is It? Instruction Register Address Register Data Register
What Is It? Consider the following set of 32 binary digits, written in blocks of four so that the example is not impossible to read. 0010 0110 0100 1100 1101 1001 1011 1111 How do we interpret this sequence
More informationFundamentals of Programming
Fundamentals of Programming Lecture 2 Number Systems & Arithmetic Lecturer : Ebrahim Jahandar Some Parts borrowed from slides by IETC1011-Yourk University Common Number Systems System Base Symbols Used
More informationComputer Sc. & IT. Digital Logic. Computer Sciencee & Information Technology. 20 Rank under AIR 100. Postal Correspondence
GATE Postal Correspondence Computer Sc. & IT 1 Digital Logic Computer Sciencee & Information Technology (CS) 20 Rank under AIR 100 Postal Correspondence Examination Oriented Theory, Practice Set Key concepts,
More informationCourse Schedule. CS 221 Computer Architecture. Week 3: Plan. I. Hexadecimals and Character Representations. Hexadecimal Representation
Course Schedule CS 221 Computer Architecture Week 3: Information Representation (2) Fall 2001 W1 Sep 11- Sep 14 Introduction W2 Sep 18- Sep 21 Information Representation (1) (Chapter 3) W3 Sep 25- Sep
More informationMicroprocessors I MICROCOMPUTERS AND MICROPROCESSORS
Microprocessors I Outline of the Lecture Microcomputers and Microprocessors Evolution of Intel 80x86 Family Microprocessors Binary and Hexadecimal Number Systems MICROCOMPUTERS AND MICROPROCESSORS There
More informationLogic and Computer Design Fundamentals. Chapter 1 Digital Computers and Information
Logic and Computer Design Fundamentals Chapter 1 Digital Computers and Information Overview Digital Systems and Computer Systems Information Representation Number Systems [binary, octal and hexadecimal]
More informationUnit 3. Analog vs. Digital. Analog vs. Digital ANALOG VS. DIGITAL. Binary Representation
3.1 3.2 Unit 3 Binary Representation ANALOG VS. DIGITAL 3.3 3.4 Analog vs. Digital The analog world is based on continuous events. Observations can take on (real) any value. The digital world is based
More informationNumber Systems Base r
King Fahd University of Petroleum & Minerals Computer Engineering Dept COE 2 Fundamentals of Computer Engineering Term 22 Dr. Ashraf S. Hasan Mahmoud Rm 22-44 Ext. 724 Email: ashraf@ccse.kfupm.edu.sa 3/7/23
More informationin this web service Cambridge University Press
978-0-51-85748- - Switching and Finite Automata Theory, Third Edition Part 1 Preliminaries 978-0-51-85748- - Switching and Finite Automata Theory, Third Edition CHAPTER 1 Number systems and codes This
More informationDigital Systems and Binary Numbers
Digital Systems and Binary Numbers Mano & Ciletti Chapter 1 By Suleyman TOSUN Ankara University Outline Digital Systems Binary Numbers Number-Base Conversions Octal and Hexadecimal Numbers Complements
More informationIT 1204 Section 2.0. Data Representation and Arithmetic. 2009, University of Colombo School of Computing 1
IT 1204 Section 2.0 Data Representation and Arithmetic 2009, University of Colombo School of Computing 1 What is Analog and Digital The interpretation of an analog signal would correspond to a signal whose
More informationNumbering systems. Dr Abu Arqoub
Numbering systems The decimal numbering system is widely used, because the people Accustomed (معتاد) to use the hand fingers in their counting. But with the development of the computer science another
More informationCS 101: Computer Programming and Utilization
CS 101: Computer Programming and Utilization Jul-Nov 2017 Umesh Bellur (cs101@cse.iitb.ac.in) Lecture 3: Number Representa.ons Representing Numbers Digital Circuits can store and manipulate 0 s and 1 s.
More information6. Binary and Hexadecimal
COMP1917 15s2 6. Binary and Hexadecimal 1 COMP1917: Computing 1 6. Binary and Hexadecimal Reading: Moffat, Section 13.2 Outline Number Systems Binary Computation Converting between Binary and Decimal Octal
More informationWhen using computers, it should have a minimum number of easily identifiable states.
EET 3 Chapter Number Systems (B) /5/4 PAGE Number Systems (B) Number System Characteristics (Tinder) What s important in choosing a number system? Basically, there are four important characteristics desirable
More informationECE 20B, Winter Purpose of Course. Introduction to Electrical Engineering, II. Administration
ECE 20B, Winter 2003 Introduction to Electrical Engineering, II Instructor: Andrew B Kahng (lecture) Email: abk@eceucsdedu Telephone: 858-822-4884 office, 858-353-0550 cell Office: 3802 AP&M Lecture: TuThu
More informationBeyond Base 10: Non-decimal Based Number Systems
Beyond Base : Non-decimal Based Number Systems What is the decimal based number system? How do other number systems work (binary, octal and hex) How to convert to and from nondecimal number systems to
More informationNumber Systems. Both numbers are positive
Number Systems Range of Numbers and Overflow When arithmetic operation such as Addition, Subtraction, Multiplication and Division are performed on numbers the results generated may exceed the range of
More information