Decimal Numbers: Base 10 Integer Numbers & Arithmetic


 Rosalyn Hopkins
 2 years ago
 Views:
Transcription
1 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 Base B => B digits: Base 10 (Decimal): 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Base 2 (Binary): 0, 1 Number representation: d 31 d 30 d 2 d 1 d 0 is a 32 digit number value = d 31 x B 31 + d 30 x B d 2 x B 2 + d 1 x B 1 + d 0 x B 0 Binary: 0, = 1x x x x x x2 + 0x1 = = 90 Notice that 7 digit binary number turns into a 2 digit decimal number A base that converts to binary easily? Hexadecimal Numbers: Base 16 Hexadecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Normal decimal digits + 6 more: picked alphabet Conversion: Binary <> Hex 1 hex digit represents 16 decimal values 4 binary digits represent 16 decimal values => 1 hex digit replaces 4 binary digits Examples: (binary) =? (hex) (binary) = (binary) =? (hex) 3F9(hex) =? (binary) Ward 3 Ward 4
2 Decimal vs Hexadecimal vs Binary Converting Decimal to Binary [1] Examples: (binary) =? (hex) (binary) = (binary) =? (hex) 3F9(hex) =? (binary) A B C D E F 1111 Converting from base 10 to base 2: Take largest binary place in decimal number, put a binary digit there and subtract it from decimal number and continue until decimal number is (64) 2 5 (32) 2 4 (16) 2 3 (8) 2 2 (4) 2 1 (2) 2 0 (1) = = = = Ward 5 Ward 6 Converting Decimal to Binary [2] Converting from base 10 to base 2: Or continually divide decimal number by 2 and keep remainder Number Remainder 85 / / / / / / / What to do with representations of numbers? Just what we do with numbers! Add them Subtract them Multiply them Divide them Compare them Example: = 17 so simple to add in binary that we can build circuits to do it subtraction also just as you would in decimal Ward 7 Ward 8
3 Which base do we use? Decimal: great for humans, especially when doing arithmetic Hex: if human looking at long strings of binary numbers, its much easier to convert to hex and look 4 bits/symbol Terrible for arithmetic Binary: what computers use; you learn how computers do +,,*,/ To a computer, numbers always binary Doesn t matter base in C, just the value: == 0x20 == Use subscripts ten, hex, two in book, slides when might be confusing Limits of Computer Numbers Bits can represent anything! Characters? 26 letter => 5 bits upper/lower case + punctuation => 7 bits (in 8) rest of the world s languages => 16 bits (unicode) Logical values? 0 > False, 1 => True colors? locations / addresses? commands? but N bits => only 2 N things Ward 9 Ward 10 Comparison How to Represent Negative Numbers? How do you tell if X > Y? See if X  Y > 0 So far, unsigned numbers 3 potential systems for handling negative numbers Sign and Magnitude 1 s Complement 2 s Complement Ward 11 Ward 12
4 Sign and Magnitude Obvious solution: define leftmost bit to be sign! 0 => +, 1 =>  Rest of bits can be numerical value of number Representation called sign and magnitude MIPS uses 32bit integers +1 ten would be: And  1 ten in sign and magnitude would be: Shortcomings of sign and magnitude Arithmetic circuit more complicated Special steps depending whether signs are the same or not Also, two zeros 0x = +0 ten 0x = 0 ten What would it mean for programming? Sign and magnitude abandoned Ward 13 Ward 14 Another try: complement the bits Shortcomings of 1 s complement Example: 7 10 = = Called 1 s Complement Note: positive numbers have leading 0s, negative numbers have leadings 1s What is ? How many positive numbers in N bits? How many negative ones? Ward 15 Arithmetic not too hard Still two zeros 0x = +0 ten 0xFFFFFFFF = 0 ten What would it mean for programming? 1 s complement eventually abandoned because another solution was better Ward 16
5 Search for Negative Number Representation Obvious solutions didn t work, find another What is result for unsigned numbers if tried to subtract large number from a small one? Would try to borrow from string of leading 0s, so result would have a string of leading 1s With no obvious better alternative, pick representation that made the hardware simple: leading 0s positive, leading 1s negative xxx is 0, xxx is < 0 This representation called 2 s complement 2 s Complement Number line N1 nonnegatives 2 N1 negatives one zero how many positives? comparison? overflow? Ward 17 Ward 18 2 s Complement two = 0 ten two = 1 ten two = 2 ten two = 2,147,483,645 ten two = 2,147,483,646 ten two = 2,147,483,647 ten two = 2,147,483,648 ten two = 2,147,483,647 ten two = 2,147,483,646 ten two = 3 ten two = 2 ten two = 1 ten One zero 1st bit => 0 or <0, called sign bit 2 s Complement Formula Can represent positive and negative numbers in terms of the bit value times a power of 2: d 31 x d 30 x d 2 x d 1 x d 0 x 2 0 Example two = 1x x x x2 2 +0x2 1 +0x2 0 = = 2,147,483,648 ten + 2,147,483,644 ten = 4 ten Note: need to specify width: we use 32 bits but one negative with no positive 2,147,483,648 ten Ward 19 Ward 20
6 2 s complement shortcut: Negation Invert every 0 to 1 and every 1 to 0, then add 1 to the result Sum of number and its one s complement must be two two = 1 ten Let x be the inverted representation of x Then x + x = 1 x + x + 1 = 0 x + 1 = x Example: 4 to +4 to 4 x : two x : two +1: two () : two +1: two Signed vs Unsigned Numbers C declaration int Declares a signed number Uses two s complement C declaration unsigned int Declares a unsigned number Treats 32bit number as unsigned integer, so most significant bit is part of the number, not a sign bit Ward 21 Ward 22 Example of Values in Unsigned & 2 s Complement Representations Unsigned & 2 s Complement Implementation A computer can use a single piece of hardware to provide unsigned or 2 s complement integer arithmetic; software running on the computer can choose an interpretation for each integer Example: Adding 1 to binary 1001 produces 1010 Unsigned interpretation goes from 9 to 10 2 s complement interpretation goes from 7 to 6 Ward 23 Ward 24
7 Signed vs Unsigned Comparisons Numbers are stored at addresses X = two Y = two Memory is a place to store bits Is X > Y (or X Y > 0)? unsigned: YES signed: NO Converting to decimal to check Unsigned comparison: 4,294,967,292 ten > 1,000,000,000 ten? Signed comparison: 4 ten > 1,000,000,000 ten? = 2 k 1 A word is a fixed number of bits (eg, 32) or bytes (eg, 4) at an address Address is also fixed no of bits Addresses are naturally represented as unsigned numbers Ward 25 Ward 26 Numbering Bits and Bytes Need to choose order for Storage in physical memory system Transmission over serial medium (eg, data network) Bit order Handled by hardware Usually hidden from programmer Byte order Affects multibyte data items such as integers Visible and important to programmers Possible Byte Orders Least significant byte of integer in lowest memory location Known as little endian Most significant byte of integer in lowest memory location Known as big endian Ward 27 Ward 28
8 Illustration of Byte Orders Big Endian & Little Endian Examples Storage of 32bit integer 1 Note: difference is especially important when transferring data between computers for which the byte ordering differs Ward 29 Ward 30 What if integer too big? Binary bit patterns above are simply representatives of numbers Numbers really have an infinite number of digits with almost all being zero except for a few of the rightmost digits Just don t normally show leading zeros If result of add (or ,*/) cannot be represented by these rightmost HW bits, overflow is said to have occurred Sign extension Convert 2 s complement number using n bits to more than n bits (eg, int to long int) Simply replicate the most significant bit (sign bit) of smaller to fill new bits 2 s comp positive number has infinite 0s 2 s comp negative number has infinite 1s Bit representation hides leading bits; sign extension restores some of them 16bit 4 ten to 32bit: two two Ward 31 Ward 32
9 Consequence for Programmers Because 2 s complement hardware performs sign extension, copying an unsigned integer to a larger unsigned integer changes the value; to prevent such errors from occurring, a programmer or a compiler must add code to mask off the extended sign bits And in Conclusion We represent things in computers as particular bit patterns: N bits =>2 N numbers, characters, (data) Decimal for human calculations, binary for computers, hex for convenient way to write binary 2 s complement universal in computing: cannot avoid, so learn Computer operations on the representation correspond to real operations on the real thing Numbers infinite but computers finite, so errors occur (eg, overflow, underflow) Knowing the powers of 2 (eg, 2 0, 2 1, 2 2, 2 3, ) and values of K (2 10 ) and M (2 20 ) will prove invaluable, so learn them Ward 33 Ward 34 Additional Basics: Metric Prefixes Clocks/Seconds T tera G giga 10 9 M mega 10 6 K kilo 10 3 m milli 103 μ micro 106 n nano 109 p pico Bytes/bits Note the difference T tera 2 40 (= 1099 * ) G giga 2 30 (= 1073 * 10 9 ) M mega 2 20 (= 1048 * 10 6 ) K kilo 2 10 (= 1024 * 10 3 ) Ward 35
Lecture 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 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 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 informationComputer 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 informationLecture 2. Binary and Hexadecimal Numbers
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
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationAssembly Language for IntelBased Computers, 4 th Edition. Chapter 1: Basic Concepts
Assembly Language for IntelBased Computers, 4 th Edition Kip R. Irvine Chapter 1: Basic Concepts Slides prepared by Kip R. Irvine Revision date: 07/21/2002 Chapter corrections (Web) Assembly language
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 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 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 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 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! Why Bits (Binary Digits)?!
Number Systems Why Bits (Binary Digits)? Computers are built using digital circuits Inputs and outputs can have only two values True (high voltage) or false (low voltage) Represented as and Can represent
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 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 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 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 informationBits, Data Types, and Operations. University of Texas at Austin CS310H  Computer Organization Spring 2010 Don Fussell
Bits, Data Types, and Operations University of Texas at Austin CS3H  Computer Organization Spring 2 Don Fussell How do we represent data in a computer? At the lowest level, a computer is an electronic
More informationIntroduction to Powers of 10
Introduction to Powers of 10 Topics Covered in This Chapter: I1: Scientific Notation I2: Engineering Notation and Metric Prefixes I3: Converting between Metric Prefixes I4: Addition and Subtraction
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 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 informationCPEN 214  Digital Logic Design Binary Systems
CPEN 4  Digital Logic Design Binary Systems C. Gerousis Digital Design 3 rd Ed., Mano Prentice Hall Digital vs. Analog An analog system has continuous range of values A mercury thermometer Vinyl records
More informationNumbering Systems. InThisAppendix...
G InThisAppendix... Introduction Binary Numbering System Hexadecimal Numbering System Octal Numbering System Binary Coded Decimal (BCD) Numbering System Real (Floating Point) Numbering System BCD/Binary/Decimal/Hex/Octal
More informationBase Conversion written by Cathy Saxton
Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,
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 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 informationActivity 1: Bits and Bytes
ICS3U (Java): Introduction to Computer Science, Grade 11, University Preparation Activity 1: Bits and Bytes The Binary Number System Computers use electrical circuits that include many transistors and
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 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 informationRepresentation of Data
Representation of Data In contrast with higherlevel programming languages, C does not provide strong abstractions for representing data. Indeed, while languages like Racket has a rich notion of data type
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 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 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 informationData Representation. Why Study Data Representation?
Why Study Data Representation? Computers process and store information in binary format For many aspects of programming and networking, the details of data representation must be understood C Programming
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 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 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 informationAssembly Language for IntelBased Computers, 4 th Edition. Chapter 1: Basic Concepts. Chapter Overview. Welcome to Assembly Language
Assembly Language for IntelBased Computers, 4 th Edition Kip R. Irvine Chapter 1: Basic Concepts Slides prepared by Kip R. Irvine Revision date: 10/27/2002 Chapter corrections (Web) Printing a slide show
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 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 informationPart 1 Theory Fundamentals
Part 1 Theory Fundamentals 2 Chapter 1 Information Representation Learning objectives By the end of this chapter you should be able to: show understanding of the basis of different number systems show
More informationNumber Systems and Number Representation
Number Systems and Number Representation 1 For Your Amusement Question: Why do computer programmers confuse Christmas and Halloween? Answer: Because 25 Dec = 31 Oct  http://www.electronicsweekly.com
More informationComputer Number Systems
Computer Number Systems Thorne, Edition 2 : Section 1.3, Appendix I (Irvine, Edition VI : Section 1.3) SYSC3006 1 Starting from What We Already Know Decimal Numbers Based Number Systems : 1. Base defines
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 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 informationCommon Number Systems Number Systems
5/29/204 Common Number Systems Number Systems System Base Symbols Used by humans? Used in computers? Decimal 0 0,, 9 Yes No Binary 2 0, No Yes Octal 8 0,, 7 No No Hexadecimal 6 0,, 9, A, B, F No No Number
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 informationData types. lecture 4
Data types lecture 4 Information in digital computers is represented using binary number system. The base, i.e. radix, of the binary system is 2. Other common number systems: octal (base 8), decimal (base
More information1 Number systems 1.1 DECIMAL SYSTEM
A programmable logical controller uses the binary system rather than the decimal system to process memory cells, inputs, outputs, timers, flags etc.. DECIMAL SYSTEM In order to understand the binary number
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 informationA Short Introduction to Binary Numbers
A Short Introduction to Binary Numbers Brian J. Shelburne Department of Mathematics and Computer Science Wittenberg University 0. Introduction The development of the computer was driven by the need to
More informationBinary Division. Decimal Division. Hardware for Binary Division. Simple 16bit Divider Circuit
Decimal Division Remember 4th grade long division? 43 // quotient 12 521 // divisor dividend 480 4136 5 // remainder Shift divisor left (multiply by 10) until MSB lines up with dividend s Repeat until
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 informationLogic Design. Dr. Yosry A. Azzam
Logic Design Dr. Yosry A. Azzam Binary systems Chapter 1 Agenda Binary Systems : Binary Numbers, Binary Codes, Binary Logic ASCII Code (American Standard Code for Information Interchange) Boolean Algebra
More informationDigital Electronics. 1.0 Introduction to Number Systems. Module
Module 1 www.learnaboutelectronics.org Digital Electronics 1.0 Introduction to What you ll learn in Module 1 Section 1.0. Recognise different number systems and their uses. Section 1.1 in Electronics.
More information1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal:
Exercises 1  number representations Questions 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal: (a) 3012 (b)  435 2. For each of
More information1.3 Data Representation
862828 r4 vs.fm Page 9 Thursday, January 2, 2 2:4 PM.3 Data Representation 9 appears at Level 3, uses short mnemonics such as ADD, SUB, and MOV, which are easily translated to the ISA level. Assembly
More information198:211 Computer Architecture
198:211 Computer Architecture Topics: Lecture 8 (W5) Fall 2012 Data representation 2.1 and 2.2 of the book Floating point 2.4 of the book 1 Computer Architecture What do computers do? Manipulate stored
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 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 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 informationLECTURE 2: Number Systems
LECTURE : Number Systems CIS Fall 7 Instructor: Dr. Song Xing Department of Information Systems California State University, Los Angeles Learning Objectives Review Boolean algebra. Describe numbering systems
More informationCS 16: Assembly Language Programming for the IBM PC and Compatibles
CS 16: Assembly Language Programming for the IBM PC and Compatibles First, a little about you Your name Have you ever worked with/used/played with assembly language? If so, talk about it Why are you taking
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 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 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 informationNUMBER REPRESENTATIONS IN THE COMPUTER for COSC 120
NUMBER REPRESENTATIONS IN THE COMPUTER for COSC 120 First, a reminder of how we represent base ten numbers. Base ten uses ten (decimal) digits: 0, 1, 2,3, 4, 5, 6, 7, 8, 9. In base ten, 10 means ten. Numbers
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 informationPrimitive Data Types Summer 2010 Margaret ReidMiller
Primitive Data Types 15110 Summer 2010 Margaret ReidMiller Data Types Data stored in memory is a string of bits (0 or 1). What does 1000010 mean? 66? 'B'? 9.2E44? How the computer interprets the string
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 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 informationA B C
Data Representation Module 2 CS 272 Sam Houston State University Dr. Tim McGuire Copyright 2001 by Timothy J. McGuire, Ph.D. 1 Positional Number Systems Decimal (base 10) is an example e.g., 435 means
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 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 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 informationChapter 1 Basic Concepts
Chapter 1 Basic Concepts 1.1 Welcome to Assembly Language 1 1.1.1 Good Questions to Ask 2 1.1.2 Assembly language Applications 5 1.1.3 Section Review 6 1.2 Virtual Machine Concept 7 1.2.1 History of PC
More informationCHAPTER 3 Numbers and Numeral Systems
CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,
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 information