Lecture 2. Binary and Hexadecimal Numbers


 Kristopher Rodger Nichols
 1 years ago
 Views:
Transcription
1 Lecture 2 Binary and Hexadecimal Numbers Purpose: Review binary and hexadecimal number representations Convert directly from one base to another base Review addition and subtraction in binary representations Determine overflow in unsigned and signed binary addition and subtraction Lecture 2 Page 1
2 The Need for Other Bases Humans are used to the decimal number system, also called radix10 or base10. To state the obvious, base10 means that a digit has one of ten possible values, 0 through 9. In computers, numbers are stored in binary, also called radix2 or base2, using arrays of flipflops. Each digit may take one of two values, either 0 or 1. Long strings of these 1 s and 0 s are cumbersome to use, so we will usually represent binary numbers using hexadecimal, also called radix16 or base16. It is important to note that no computer actually stores values in hardware using hexadecimal. This number system is only a convenience for humans. All of these number systems are positional. Lecture 2 Page 2
3 Unsigned Decimal Numbers are represented using the digits 0, 1, 2,, 9. Multidigit numbers are interpreted as in the following example: = Unsigned Binary Numbers are represented using the digits 0 and 1. Multidigit numbers are interpreted as in the following example: = In binary, each digit is called a bit. Since we use binary to represent the values stored in a group of flipflops, we usually specify a binary system by the number of bits (flipflops) being used to store each number. When we write numbers in this system, we will write all bits, including leading 0 s. The number above is expressed in 5bit binary. The number below is in 8bit binary Lecture 2 Page 3
4 Unsigned Hexadecimal Numbers are represented in hexadecimal using the digits 0, 1, 2,, 9, A, B,, F where the letters represent values: A=10, B=11, and so on to F=15. Note that this gives sixteen possible values for each digit. Multidigit numbers are interpreted as in the following example: 76CA 16 = Notes on Bases Since all three number bases will be used, including the correct subscript when a number is written out of context is mandatory. Pronunciation Words like ten, twenty, and onethousand refer to specific numbers of items, regardless of how the numbers are written. To avoid confusion, binary and hexadecimal numbers are spoken by naming the digits followed by binary or hexadecimal. For example, is pronounced one zero zero zero hexadecimal. Onethousand is actually 3E8 16. Lecture 2 Page 4
5 Ranges of Unsigned Number Systems System Lowest Highest Number of Values 4bit binary (1digit hex) 8bit binary (1 byte) (2digit hex) 16bit binary (2 bytes) (4digit hex) nbit binary Lecture 2 Page 5
6 2 s Complement Binary Numbers Most microprocessors today use 2 s Complement numbers to represent systems with positive and negative values. Hardware performs addition and subtraction on binary values the same way whether they represent unsigned systems or 2 s complement systems, and this greatly simplifies the design of the processor The only difference between unsigned binary systems and 2 s comp. binary systems is that the most significant bit in signed systems has a weight of 2 n1. Both systems are also defined by the number of bits being used, and as with unsigned, we must write down all bits. To distinguish between the two, we will use a 2c subscript to indicate a 2 s comp. number. Example: Convert c to decimal. Example: Convert c to decimal. Example: Convert c to decimal. It is very important to note that microprocessors usually view a group of bits as simply a group of bits. It is the human that interprets the group as an unsigned value, signed value, or also as just a group of bits. Lecture 2 Page 6
7 Ranges of Signed Number Systems System Least Greatest Number of Values 4bit binary 8bit binary 16bit binary nbit binary Note that the least representable value has a single 1 in the column with a negative weight and 0 s in the columns with positive weights. The greatest representable number is just the opposite: 1 s in the positively weighted columns and a 0 in the column with a negative weight. Lecture 2 Page 7
8 Sign Bit Since the leftmost column has a negative weight, and the magnitude of that weight is larger than the weights of all the positive columns added together, any number with a 1 in the leftmost column will be negative. If the leftmost bit has a 0, then there is no negative contribution, and the value will be positive. Hence, the sign of the number can be determined by inspection. Negating a 2 s Complement Number The official way to negate a value is to subtract it from zero. This will generate a number with the same magnitude but with the opposite sign. The second method is to perform the 2 s complement, which is the following two steps: 1. Perform the 1 s complement (flip all the bits) 2. Add 1 Example: Negate c (41 10 ) Lecture 2 Page 8
9 Converting Between Number Systems Given the three number systems (binary, hexadecimal, and decimal), there are six possible conversions to allow us to convert directly from one to another. Binary to Decimal: This was covered earlier in this lecture by determining the weights for each column and adding them up, either as unsigned or as signed. Hexadecimal to Decimal: This was also covered earlier in this lecture by determining the weights for each column and adding them up. Decimal to Binary (or Hexadecimal): This conversion is more of a process than the others. 1. Successively divide the decimal number by the new base and keep track of the remainders generated. 2. Stop dividing once the quotient reaches Write the remainders in the opposite order than they were generated. 4. Add leading digits if necessary. Lecture 2 Page 9
10 Example: Convert to 8bit unsigned binary. Example: Convert to 2digit hexadecimal. Lecture 2 Page 10
11 Note: To convert a negative value, first convert the magnitude to the correct number of bits as done above. Then, negate the result to get the final answer. Binary to Hexadecimal: This conversion is the reason that hexadecimal is used in the first place. Since once hexadecimal digit can represent 16 different values, and four bits can also represent 16 (2 4 ) different values, the bits of the binary number will be grouped together in 4 s and replaced by the hexadecimal digit with the same value. Example: Convert the binary numbers below to hexadecimal c Note that a binary number may not always contain an integer multiple of 4 bits. In these cases, always extend the binary number by padding it with zeros, whether the number is unsigned or signed. Example: Convert the binary numbers below to hexadecimal c c Lecture 2 Page 11
12 Hexadecimal to Binary: As with binary to hexadecimal, this conversion is by inspection. Each hexadecimal digit is replaced with the four bits that represents the same value. Example: Convert the following hexadecimal numbers to binary. BEFA 16 73FC 16 Hexadecimal is not usually interpreted as signed or unsigned. It is simply a more convenient method for humans to discuss binary patterns. By default, we will assume that the binary pattern has four bits for every hexadecimal digits used, as done above. However, we can also specify a binary system with any number of bits, as done on the latter half of the previous slide. We must explicitly state the binary system being used, and we will ignore the padding 0 s. Example: Convert the following hexadecimal numbers to the specified binary system. 07B 16 to 9bit signed 1F 16 to 5bit unsigned Lecture 2 Page 12
13 Binary Arithmetic Unsigned and signed addition and subtraction generate the same numerical result. The difference is determining if overflow or underflow (usually grouped together generically as overflow ) occurs. The methods below work for both addition and subtraction. For unsigned: For signed: The above methods are easily implemented in hardware. For humans, there is an alternate approach for signed arithmetic. Addition: If the two numbers being added have the same sign, the answer must have that sign. Otherwise, overflow occurs. Signed addition cannot generate overflow if the two numbers being added have different signs. Subtraction: Lecture 2 Page 13
14 Example: Determine the result, if unsigned overflow occurred, and if signed overflow occurred. (Note that the binary numbers are shown with an unsigned subscript for simplicity.) Lecture 2 Page 14
15 Modular Number Systems Wraparound point for unsigned Wraparound point for signed Here s another way to visualize overflow. Binary systems are modular. For our purposes, modular means that a constant number of digits are used. Above, increasing a value (i.e. add a positive or subtract a negative) moves us clockwise around the number line. Decreasing a value (i.e. subtract a positive or add a negative) moves us counterclockwise. If the wraparound point is crossed during this move, overflow occurs. Lecture 2 Page 15
16 Extending Binary Numbers When performing arithmetic operations, the binary numbers must have the same number of bits. Therefore, it is sometimes necessary to extend the shorter number so that it has the same number of bits as the longer number. This must be done in a manner such that the new, longer number still represents the same value as the shorter number. For unsigned: For signed: Example: Extend the binary numbers below to 16 bits c c Note that a binary number can always be extended correctly. Lecture 2 Page 16
17 Truncating Binary Numbers There are times when a value will be expressed in more bits than needed, and the number of bits being used should be reduced to save space. This is done by discarding the most significant bits of a binary number. Technically, truncation may always be performed. However, we will say that it is not possible if it yields a shorter number that does not represent the same value as the original, longer number. For example, of the possible 16bit numbers, only 512 may be correctly truncated to the 512 possible 9bit numbers. Unsigned: All bits discarded must be 0 s, otherwise the shorter number will not be accurate. Signed: All bits discarded must be the same as the new sign bit of the shorter number, otherwise the shorter number will not be accurate. Example: Determine which of the 16bit values below can be truncated to 8 bits in the specified system c c c c Lecture 2 Page 17
Computer Science 281 Binary and Hexadecimal Review
Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two
More informationEE 3170 Microcontroller Applications
EE 37 Microcontroller Applications Lecture 3 : Digital Computer Fundamentals  Number Representation (.) Based on slides for ECE37 by Profs. Sloan, Davis, Kieckhafer, Tan, and Cischke Number Representation
More informationHere 4 is the least significant digit (LSD) and 2 is the most significant digit (MSD).
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 informationMT1 Number Systems. In general, the number a 3 a 2 a 1 a 0 in a base b number system represents the following number:
MT1 Number Systems MT1.1 Introduction A number system is a well defined structured way of representing or expressing numbers as a combination of the elements of a finite set of mathematical symbols (i.e.,
More informationNumber Representation and Arithmetic in Various Numeral Systems
1 Number Representation and Arithmetic in Various Numeral Systems Computer Organization and Assembly Language Programming 203.8002 Adapted by Yousef Shajrawi, licensed by Huong Nguyen under the Creative
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 informationOct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8
ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: Our standard number system is base, also
More informationDigital Arithmetic. Digital Arithmetic: Operations and Circuits Dr. Farahmand
Digital Arithmetic Digital Arithmetic: Operations and Circuits Dr. Farahmand Binary Arithmetic Digital circuits are frequently used for arithmetic operations Fundamental arithmetic operations on binary
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 twovalued number system is called binary. As presented earlier, there are many
More informationDecimal Numbers: Base 10 Integer Numbers & Arithmetic
Decimal Numbers: Base 10 Integer Numbers & Arithmetic Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 )+(1x10 0 ) Ward 1 Ward 2 Numbers: positional notation Number
More informationNumber Systems Richard E. Haskell
NUMBER SYSTEMS D Number Systems Richard E. Haskell Data inside a computer are represented by binary digits or bits. The logical values of these binary digits are denoted by and, while the corresponding
More informationLecture 1 Introduction, Numbers, and Number System Page 1 of 8
Lecture Introduction, Numbers and Number System Contents.. Number Systems (Appendix B)... 2. Example. Converting to Base 0... 2.2. Number Representation... 2.3. Number Conversion... 3. To convert a number
More informationData Representation Binary Numbers
Data Representation Binary Numbers Integer Conversion Between Decimal and Binary Bases Task accomplished by Repeated division of decimal number by 2 (integer part of decimal number) Repeated multiplication
More information2.1 Binary Numbers. 2.3 Number System Conversion. From Binary to Decimal. From Decimal to Binary. Section 2 Binary Number System Page 1 of 8
Section Binary Number System Page 1 of 8.1 Binary Numbers The number system we use is a positional number system meaning that the position of each digit has an associated weight. The value of a given number
More informationTwo s Complement Arithmetic
Two s Complement Arithmetic We now address the issue of representing integers as binary strings in a computer. There are four formats that have been used in the past; only one is of interest to us. The
More information1 Basic Computing Concepts (4) Data Representations
1 Basic Computing Concepts (4) Data Representations 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
More informationBinary Numbers. Bob Brown Information Technology Department Southern Polytechnic State University
Binary Numbers Bob Brown Information Technology Department Southern Polytechnic State University Positional Number Systems The idea of number is a mathematical abstraction. To use numbers, we must represent
More informationThe largest has a 0 in the sign position and 0's in all other positions:
10.2 Sign Magnitude Representation Sign Magnitude is straightforward method for representing both positive and negative integers. It uses the most significant digit of the digit string to indicate the
More informationCDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012
CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline Data Representation Binary Codes Why 6311 and Excess3? Data Representation (1/2) Each numbering
More information1. Convert the following binary exponential expressions to their 'English'
Answers to Practice Problems Practice Problems  Integer Number System Conversions 1. Convert the decimal integer 427 10 into the following number systems: a. 110101011 2 c. 653 8 b. 120211 3 d. 1AB 16
More informationSolution for Homework 2
Solution for Homework 2 Problem 1 a. What is the minimum number of bits that are required to uniquely represent the characters of English alphabet? (Consider upper case characters alone) The number of
More informationBy the end of the lecture, you should be able to:
Extra Lecture: Number Systems Objectives  To understand: Base of number systems: decimal, binary, octal and hexadecimal Textual information stored as ASCII Binary addition/subtraction, multiplication
More informationCSI 333 Lecture 1 Number Systems
CSI 333 Lecture 1 Number Systems 1 1 / 23 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = 2 10 0 + 9 10 1 + 1 10 2 General: Digit sequence : d n 1 d n 2...
More informationSigned Binary Arithmetic
Signed Binary Arithmetic In the real world of mathematics, computers must represent both positive and negative binary numbers. For example, even when dealing with positive arguments, mathematical operations
More informationUseful Number Systems
Useful Number Systems Decimal Base = 10 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Binary Base = 2 Digit Set = {0, 1} Octal Base = 8 = 2 3 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7} Hexadecimal Base = 16 = 2
More informationCHAPTER THREE. 3.1 Binary Addition. Binary Math and Signed Representations
CHAPTER THREE Binary Math and Signed Representations Representing numbers with bits is one thing. Doing something with them is an entirely different matter. This chapter discusses some of the basic mathematical
More informationNumber Systems and. Data Representation
Number Systems and Data Representation 1 Lecture Outline Number Systems Binary, Octal, Hexadecimal Representation of characters using codes Representation of Numbers Integer, Floating Point, Binary Coded
More informationCOMP2121: Microprocessors and Interfacing
Interfacing Lecture 3: Number Systems (I) http://www.cse.unsw.edu.au/~cs2121 Lecturer: Hui Wu Session 2, 2005 Overview Positional notation Decimal, hexadecimal and binary One complement Two s complement
More informationInteger Numbers. The Number Bases of Integers Textbook Chapter 3
Integer Numbers The Number Bases of Integers Textbook Chapter 3 Number Systems Unary, or marks: /////// = 7 /////// + ////// = ///////////// Grouping lead to Roman Numerals: VII + V = VVII = XII Better:
More informationSection 1.4 Place Value Systems of Numeration in Other Bases
Section.4 Place Value Systems of Numeration in Other Bases Other Bases The HinduArabic system that is used in most of the world today is a positional value system with a base of ten. The simplest reason
More informationToday. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real Time Systems: Binary Arithmetic
Today Binary addition Representing negative numbers 2 Binary Addition Consider the following binary numbers: 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 How do we add these numbers? 3 Binary Addition 0 0 1 0 0 1 1
More informationBinary Representation and Computer Arithmetic
Binary Representation and Computer Arithmetic The decimal system of counting and keeping track of items was first created by Hindu mathematicians in India in A.D. 4. Since it involved the use of fingers
More informationReview of Number Systems Binary, Octal, and Hexadecimal Numbers and Two's Complement
Review of Number Systems Binary, Octal, and Hexadecimal Numbers and Two's Complement Topic 1: Binary, Octal, and Hexadecimal Numbers The number system we generally use in our everyday lives is a decimal
More informationChapter 3: Number Systems
Slide 1/40 Learning Objectives In this chapter you will learn about: Nonpositional number system Positional number system Decimal number system Binary number system Octal number system Hexadecimal number
More informationالدكتور المھندس عادل مانع داخل
الدكتور المھندس عادل مانع داخل / میسان جامعة / كلیة الھندسة قسم الھندسة الكھرباي یة Chapter 1: Digital Systems Discrete Data Examples: 26 letters of the alphabet (A, B etc) 10 decimal digits (0, 1, 2 etc)
More informationPresented By: Ms. Poonam Anand
Presented By: Ms. Poonam Anand Know the different types of numbers Describe positional notation Convert numbers in other bases to base 10 Convert base 10 numbers into numbers of other bases Describe the
More informationChapter I: Digital System and Binary Numbers
Chapter I: Digital System and Binary Numbers 11Digital Systems Digital systems are used in:  Communication  Business transaction  Traffic Control  Medical treatment  Internet The signals in digital
More informationNUMBERING SYSTEMS C HAPTER 1.0 INTRODUCTION 1.1 A REVIEW OF THE DECIMAL SYSTEM 1.2 BINARY NUMBERING SYSTEM
12 Digital Principles Switching Theory C HAPTER 1 NUMBERING SYSTEMS 1.0 INTRODUCTION 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,
More informationArithmetic of Number Systems
2 Arithmetic of Number Systems INTRODUCTION Arithmetic operations in number systems are usually done in binary because designing of logic networks is much easier than decimal. In this chapter we will discuss
More informationChapter 2 Numeric Representation.
Chapter 2 Numeric Representation. Most of the things we encounter in the world around us are analog; they don t just take on one of two values. How then can they be represented digitally? The key is that
More informationGoals. Unary Numbers. Decimal Numbers. 3,148 is. 1000 s 100 s 10 s 1 s. Number Bases 1/12/2009. COMP370 Intro to Computer Architecture 1
Number Bases //9 Goals Numbers Understand binary and hexadecimal numbers Be able to convert between number bases Understand binary fractions COMP37 Introduction to Computer Architecture Unary Numbers Decimal
More informationCPE 323 Data Types and Number Representations
CPE 323 Data Types and Number Representations Aleksandar Milenkovic Numeral Systems: Decimal, binary, hexadecimal, and octal We ordinarily represent numbers using decimal numeral system that has 10 as
More informationComputer is a binary digital system. Data. Unsigned Integers (cont.) Unsigned Integers. Binary (base two) system: Has two states: 0 and 1
Computer Programming Programming Language Is telling the computer how to do something Wikipedia Definition: Applies specific programming languages to solve specific computational problems with solutions
More informationIntroduction Number Systems and Conversion
UNIT 1 Introduction Number Systems and Conversion Objectives 1. Introduction The first part of this unit introduces the material to be studied later. In addition to getting an overview of the material
More informationInteger and Real Numbers Representation in Microprocessor Techniques
Brno University of Technology Integer and Real Numbers Representation in Microprocessor Techniques Microprocessor Techniques and Embedded Systems Lecture 1 Dr. Tomas Fryza 30Sep2011 Contents Numerical
More informationChapter II Binary Data Representation
Chapter II Binary Data Representation The atomic unit of data in computer systems is the bit, which is actually an acronym that stands for BInary digit. It can hold only 2 values or states: 0 or 1, true
More informationBinary Numbers Again. Binary Arithmetic, Subtraction. Binary, Decimal addition
Binary Numbers Again Recall than N binary digits (N bits) can represent unsigned integers from 0 to 2 N 1. 4 bits = 0 to 15 8 bits = 0 to 255 16 bits = 0 to 65535 Besides simply representation, we would
More informationEE 308 Spring Binary, Hex and Decimal Numbers (4bit representation) Binary. Hex. Decimal A B C D E F
EE 8 Spring Binary, Hex and Decimal Numbers (bit representation) Binary Hex 8 9 A B C D E F Decimal 8 9 EE 8 Spring What does a number represent? Binary numbers are a code, and represent what the programmer
More information1 Number System (Lecture 1 and 2 supplement)
1 Number System (Lecture 1 and 2 supplement) By Dr. Taek Kwon Many different number systems perhaps from the prehistoric era have been developed and evolved. Among them, binary number system is one of
More informationNumber Systems and Base Conversions
Number Systems and Base Conversions As you know, the number system that we commonly use is the decimal or base 10 number system. That system has 10 digits, 0 through 9. While it's very convenient for
More information4 Operations On Data
4 Operations On Data 4.1 Source: Foundations of Computer Science Cengage Learning Objectives After studying this chapter, students should be able to: List the three categories of operations performed on
More informationBinary Representation. Number Systems. Positional Notation
Binary Representation The basis of all digital data is binary representation. Binary  means two 1, 0 True, False Hot, Cold On, Off We must be able to handle more than just values for real world problems
More informationBinary Representation. Number Systems. Base 10, Base 2, Base 16. Positional Notation. Conversion of Any Base to Decimal.
Binary Representation The basis of all digital data is binary representation. Binary  means two 1, 0 True, False Hot, Cold On, Off We must be able to handle more than just values for real world problems
More informationComputer Architecture CPIT 210 LAB 1 Manual. Prepared By: Mohammed Ghazi Al Obeidallah.
Computer Architecture CPIT 210 LAB 1 Manual Prepared By: Mohammed Ghazi Al Obeidallah malabaidallah@kau.edu.sa LAB 1 Outline: 1. Students should understand basic concepts of Decimal system, Binary system,
More informationSessions 1, 2 and 3 Number Systems
COMP 1113 Sessions 1, 2 and 3 Number Systems The goal of these three class sessions is to examine ways in which numerical and text information can be both stored in computer memory and how numerical information
More informationRadix Number Systems. Number Systems. Number Systems 4/26/2010. basic idea of a radix number system how do we count:
Number Systems binary, octal, and hexadecimal numbers why used conversions, including to/from decimal negative binary numbers floating point numbers character codes basic idea of a radix number system
More informationNumber Representation
Number Representation Number System :: The Basics We are accustomed to using the socalled decimal number system Ten digits ::,,,3,4,5,6,7,8,9 Every digit position has a weight which is a power of Base
More informationNUMBER SYSTEMS CHAPTER 191
NUMBER SYSTEMS 19.1 The Decimal System 19. The Binary System 19.3 Converting between Binary and Decimal Integers Fractions 19.4 Hexadecimal Notation 19.5 Key Terms and Problems CHAPTER 191 19 CHAPTER
More informationEncoding Systems: Combining Bits to form Bytes
Encoding Systems: Combining Bits to form Bytes Alphanumeric characters are represented in computer storage by combining strings of bits to form unique bit configuration for each character, also called
More informationThe string of digits 101101 in the binary number system represents the quantity
Data Representation Section 3.1 Data Types Registers contain either data or control information Control information is a bit or group of bits used to specify the sequence of command signals needed for
More informationLab 1: Information Representation I  Number Systems
Unit 1: Computer Systems, pages 1 of 7  Department of Computer and Mathematical Sciences CS 1408 Intro to Computer Science with Visual Basic 1 Lab 1: Information Representation I  Number Systems Objectives:
More informationLab 1: Information Representation I  Number Systems
Unit 1: Computer Systems, pages 1 of 7  Department of Computer and Mathematical Sciences CS 1410 Intro to Computer Science with C++ 1 Lab 1: Information Representation I  Number Systems Objectives:
More informationTECH. Arithmetic & Logic Unit. CH09 Computer Arithmetic. Number Systems. ALU Inputs and Outputs. Binary Number System
CH09 Computer Arithmetic CPU combines of ALU and Control Unit, this chapter discusses ALU The Arithmetic and Logic Unit (ALU) Number Systems Integer Representation Integer Arithmetic FloatingPoint Representation
More informationBinary Numbers. Binary Octal Hexadecimal
Binary Numbers Binary Octal Hexadecimal Binary Numbers COUNTING SYSTEMS UNLIMITED... Since you have been using the 10 different digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 all your life, you may wonder how
More informationNUMBER SYSTEMS. William Stallings
NUMBER SYSTEMS William Stallings The Decimal System... The Binary System...3 Converting between Binary and Decimal...3 Integers...4 Fractions...5 Hexadecimal Notation...6 This document available at WilliamStallings.com/StudentSupport.html
More informationSwitching Circuits & Logic Design
Switching Circuits & Logic Design JieHong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 2013 1 1 Number Systems and Conversion Babylonian number system (3100 B.C.)
More informationCHAPTER TWO. 2.1 Unsigned Binary Counting. Numbering Systems
CHAPTER TWO Numbering Systems Chapter one discussed how computers remember numbers using transistors, tiny devices that act like switches with only two positions, on or off. A single transistor, therefore,
More informationChapter 4. Computer Arithmetic
Chapter 4 Computer Arithmetic 4.1 Number Systems A number system uses a specific radix (base). Radices that are power of 2 are widely used in digital systems. These radices include binary (base 2), quaternary
More informationLecture 8: Binary Multiplication & Division
Lecture 8: Binary Multiplication & Division Today s topics: Addition/Subtraction Multiplication Division Reminder: get started early on assignment 3 1 2 s Complement Signed Numbers two = 0 ten 0001 two
More informationDigital System Design Prof. D. Roychoudhury Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur
Digital System Design Prof. D. Roychoudhury Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture  03 Digital Logic  I Today, we will start discussing on Digital
More informationEE 261 Introduction to Logic Circuits. Module #2 Number Systems
EE 261 Introduction to Logic Circuits Module #2 Number Systems Topics A. Number System Formation B. Base Conversions C. Binary Arithmetic D. Signed Numbers E. Signed Arithmetic F. Binary Codes Textbook
More information1. Number Representation
CSEE 3827: Fundamentals of Computer Systems, Spring 2011 1. Number Representation Prof. Martha Kim (martha@cs.columbia.edu) Web: http://www.cs.columbia.edu/~martha/courses/3827/sp11/ Contents (H&H 1.31.4,
More informationCSCC85 Spring 2006: Tutorial 0 Notes
CSCC85 Spring 2006: Tutorial 0 Notes Yani Ioannou January 11 th, 2006 There are 10 types of people in the world, those who understand binary, and those who don t. Contents 1 Number Representations 1 1.1
More informationELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE470/570: MicroprocessorBased System Design Fall 2014.
REVIEW OF NUMBER SYSTEMS Notes Unit 2 BINARY NUMBER SYSTEM In the decimal system, a decimal digit can take values from to 9. For the binary system, the counterpart of the decimal digit is the binary digit,
More informationCHAPTER V NUMBER SYSTEMS AND ARITHMETIC
CHAPTER V1 CHAPTER V CHAPTER V NUMBER SYSTEMS AND ARITHMETIC CHAPTER V2 NUMBER SYSTEMS RADIXR REPRESENTATION Decimal number expansion 73625 10 = ( 7 10 4 ) + ( 3 10 3 ) + ( 6 10 2 ) + ( 2 10 1 ) +(
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Numbers
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics Numbers. Factors 2. Multiples 3. Prime and Composite Numbers 4. Modular Arithmetic 5. Boolean Algebra 6. Modulo 2 Matrix Arithmetic 7. Number Systems
More informationNumeral Systems. The number twentyfive can be represented in many ways: Decimal system (base 10): 25 Roman numerals:
Numeral Systems Which number is larger? 25 8 We need to distinguish between numbers and the symbols that represent them, called numerals. The number 25 is larger than 8, but the numeral 8 above is larger
More informationSystems Architecture
Systems Architecture Lecture 11: Arithmetic for Computers Jeremy R. Johnson Anatole D. Ruslanov William M. Mongan Some or all figures from Computer Organization and Design: The Hardware/Software Approach,
More informationNUMBER SYSTEMS. 1.1 Introduction
NUMBER SYSTEMS 1.1 Introduction There are several number systems which we normally use, such as decimal, binary, octal, hexadecimal, etc. Amongst them we are most familiar with the decimal number system.
More informationDigital Fundamentals
Digital Fundamentals with PLD Programming Floyd Chapter 2 29 Pearson Education Decimal Numbers The position of each digit in a weighted number system is assigned a weight based on the base or radix of
More informationData Representation in Computers
Chapter 3 Data Representation in Computers After studying this chapter the student will be able to: *Learn about binary, octal, decimal and hexadecimal number systems *Learn conversions between two different
More informationFixedpoint Representation of Numbers
Fixedpoint Representation of Numbers Fixed Point Representation of Numbers Signandmagnitude representation Two s complement representation Two s complement binary arithmetic Excess code representation
More informationBINARY NUMBERS A.I FINITEPRECISION NUMBERS
A BINARY NUMBERS The arithmetic used by computers differs in some ways from the arithmetic used by people. The most important difference is that computers perform operations on numbers whose precision
More informationLecture 11: Number Systems
Lecture 11: Number Systems Numeric Data Fixed point Integers (12, 345, 20567 etc) Real fractions (23.45, 23., 0.145 etc.) Floating point such as 23. 45 e 12 Basically an exponent representation Any number
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 informationEEE130 Digital Electronics I Lecture #2
EEE130 Digital Electronics I Lecture #2 Number Systems, Operations and Codes By Dr. Shahrel A. Suandi Topics to be discussed 21 Decimal Numbers 22 Binary Numbers 23 DecimaltoBinary Conversion 24
More informationLecture 2: Number Representation
Lecture 2: Number Representation CSE 30: Computer Organization and Systems Programming Summer Session II 2011 Dr. Ali Irturk Dept. of Computer Science and Engineering University of California, San Diego
More information= Chapter 1. The Binary Number System. 1.1 Why Binary?
Chapter The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base0 system. When you
More informationUNIT 2 : NUMBER SYSTEMS
UNIT 2 : NUMBER SYSTEMS page 2.0 Introduction 1 2.1 Decimal Numbers 2 2.2 The Binary System 3 2.3 The Hexadecimal System 5 2.4 Number Base Conversion 6 2.4.1 Decimal To Binary 6 2.4.2 Decimal to Hex 7
More informationCOMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012
Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about
More informationBinary Numbers. Binary Numbers. Wolfgang Schreiner Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria
Binary Numbers Wolfgang Schreiner Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria Wolfgang.Schreiner@risc.unilinz.ac.at http://www.risc.unilinz.ac.at/people/schreine
More informationBinary Representation
Binary Representation The basis of all digital data is binary representation. Binary  means two 1, 0 True, False Hot, Cold On, Off We must tbe able to handle more than just values for real world problems
More informationNumber Systems & Encoding
Number Systems & Encoding Lecturer: Sri Parameswaran Author: Hui Annie Guo Modified: Sri Parameswaran Week2 1 Lecture overview Basics of computing with digital systems Binary numbers Floating point numbers
More informationLevent EREN levent.eren@ieu.edu.tr A306 Office Phone:4889882 INTRODUCTION TO DIGITAL LOGIC
Levent EREN levent.eren@ieu.edu.tr A306 Office Phone:4889882 1 Number Systems Representation Positive radix, positional number systems A number with radix r is represented by a string of digits: A n
More informationCS101 Lecture 11: Number Systems and Binary Numbers. Aaron Stevens 14 February 2011
CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 14 February 2011 1 2 1 3!!! MATH WARNING!!! TODAY S LECTURE CONTAINS TRACE AMOUNTS OF ARITHMETIC AND ALGEBRA PLEASE BE ADVISED THAT CALCULTORS
More informationSystems I: Computer Organization and Architecture
Systems I: Computer Organization and Architecture Lecture 2: Number Systems and Arithmetic Number Systems  Base The number system that we use is base : 734 = + 7 + 3 + 4 = x + 7x + 3x + 4x = x 3 + 7x
More informationCHAPTER 2 Data Representation in Computer Systems
CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 47 2.2 Positional Numbering Systems 48 2.3 Converting Between Bases 48 2.3.1 Converting Unsigned Whole Numbers 49 2.3.2 Converting Fractions
More informationQuiz for Chapter 3 Arithmetic for Computers 3.10
Date: Quiz for Chapter 3 Arithmetic for Computers 3.10 Not all questions are of equal difficulty. Please review the entire quiz first and then budget your time carefully. Name: Course: Solutions in RED
More informationCSCI 230 Class Notes Binary Number Representations and Arithmetic
CSCI 230 Class otes Binary umber Representations and Arithmetic Mihran Tuceryan with some modifications by Snehasis Mukhopadhyay Jan 22, 1999 1 Decimal otation What does it mean when we write 495? How
More informationCHAPTER 3 Number System and Codes
CHAPTER 3 Number System and Codes 3.1 Introduction On hearing the word number, we immediately think of familiar decimal number system with its 10 digits; 0,1, 2,3,4,5,6, 7, 8 and 9. these numbers are called
More information