Computer Number Systems Supplement


 Winfred Shields
 1 years ago
 Views:
Transcription
1 Computer Number Systems Supplement Dr. Ken Hoganson, All Rights Reserved. SUPPLEMENT CONTENTS S.1 Decimal System: PowersoftheBase 2 S.2 Converting to Binary: Division/Remainder Algorithm. 3 S.3 Binary Addition. 4 S.4 Bits, Bytes, and Words. 5 S.5 Hexadecimal Number System.. 6 S.6 Negative Numbers 8
2 Computer Number Systems, Dr. Ken Hoganson, 2 S.1 Decimal System: PowersoftheBase The decimal number system is based on powers of the base 10. The place value of each digit is a power of ten. We are so comfortable with this system, that we don t even think about the underlying mechanism. For instance, the number 1259 uses digits in place values that are based on powers of the base (base 10): the 9 is in the 10 0 the 5 is in the 10 1 the 2 is in the 10 2 the 1 is in the 10 3 column  1s column column  10s column column  100s column column s column 1259 is 9 X 1 = X 10 = X 100 = X 1000 = The computer s hardware logic is implemented with transistors, which can work like switches, turning electricity on or off. If we consider on to be a 1, and off to be 0, then internal computer logic can be represented using the Binary number system. The Binary number system uses the same mechanism as the decimal system outlined above, but the base is different  base 2 (binary) rather than base 10 (decimal). The place values for binary are based on powers of the base 2: So, the binary number can be converted to decimal so we can understand it, by using the powersofthebase mechanism: = 1 X 1 = 1 1 X 2 = 2 0 X 4 = 0 0 X 8 = 0 1 X 16 = 16 1 X 32 = 32 0 X 64 = 0 1 X 128 = 128 v SUM = 179 in decimal
3 Computer Number Systems, Dr. Ken Hoganson, 3 S.2 Division/Remainder Algorithm: Converting to Binary Section C.1 explained how the decimal system works, and how the binary system uses the same mechanism. In the process, a way to convert a binary number to a decimal number was discovered, by using the powersofthebase system. To convert in the other direction, from decimal to binary, requires a different method called the division/remainder method. The idea is to repeatedly divide the decimal number to be converted, by the base to be converted into (base 2). The remainders that result are the binary digits. Example: convert 155 to binary: Start from the bottom and work up. Stop 2)1 Q = 0, R = 1 2)2 Q = 1, R = 0 2)4 Q = 2, R = 0 2)9 Q = 4, R = 1 2)19 Q = 9, R = 1 2)38 Q = 19, R = 0 2)77 Q = 38, R = 1 Start: 2 )155 Q = 77, R = 1 Answer is Be careful to read the digits in the correct order. Check the answer with the powersofthebase system: 1 X 1 = 1 1 X 2 = 2 1 X 8 = 8 1 X 16 = 16 1 X 128 =
4 Computer Number Systems, Dr. Ken Hoganson, 4 S.3: Addition in Binary Just as in decimal, binary numbers can be added together. Because the base is different, the carry over to the next column is different. In decimal, when a column adds up to more than 9, a carry is added to the column to the left (the next higher place value). Since the base is 2 in binary with digits of 0 and 1, when a sum evaluates to more than 1, a carry must be added to the column to the left. Examples:
5 Computer Number Systems, Dr. Ken Hoganson, 5 S.4: Bits, Bytes, and Words Bits are organized into groups inside the computer system. The most common grouping is to place eight bits in a byte. A byte just looks like a string of eight zeros and ones: The range of possible binary values that a byte can hold is from to There are 256 possible combinations of zeros and ones arranged in any order in a byte. The number of possible combinations is based on a power of the base: 2 #of bits = the number of combinations Examples: Number of bits Number of combinations = 2 combinations (0 and 1) = 4 combinations, (00, 01, 10, 11) = = 1024 A byte can hold a small number, or a single character. Characters are all the letters of the alphabet in upper and lower case, punctuation symbols, the digits 09, and can include other special symbols. Bytes can be grouped together to form words. A word is simply one or more bytes, but is has a meaning in terms of the computer s power. A computer with a word size of a single byte, can work with and manipulate data eight bits at a time (a rough approximation). A sixteenbit computer (word size of two bytes) is more powerful, because it can access and manipulate 16 bits at a time rather than 8. Typical word sizes for our common personal computers are 32bit (4 bytes) and 64bit (8 bytes). Computer systems include large quantities of bytes billions and trillions or bytes are becoming common. In dealing with these large numbers, a shorthand way to refer to large numbers of bytes has developed. Shorthand Term Roughly Power of 2 Actual K Kilobyte Thousand M or Meg Megabyte Million ,048,576 G or Gig Gigabtye Billion ,073,741,824 T Terabyte Trillion ,099,511,627,776
6 Computer Number Systems, Dr. Ken Hoganson, 6 S.5: Hexadecimal Number System Another number system used in computing is the hexadecimal system. Hexadecimal is base16 number system, that is, just as decimal has a base of 10 (10 digits, 09), and binary is base 2 (2 digits, 01), hexadecimal is base16 (16 digits, 015). But representing the values from is problematic a single numeral is needed to represent those values. The first six letters of the alphabet are used for those integers: A B C D E F Note that just as decimal includes digits for 09, and the 10 is two digits, with the 1 in the tens column, hexadecimal includes digits for 015, and sixteen is represented with a 1 in the sixteen s column. In hexadecimal, 10 is worth sixteen in decimal. The same place value mechanism used in decimal and binary applies to hexadecimal as well, place values are based on powers of the base, in this case, base sixteen. For example, 1B52 in hexadecimal can be converted to our more familiar decimal system: the 2 is in the 16 0 column  1s column the 5 is in the 16 1 column  16s column the B (11) is in the 16 2 column  256s column the 1 is in the 16 3 column s column 1B52 16 is 2 X 1 = X 16 = X 256 = X 4096 = Subscripts are often used to indicate the base of the number, which is not always apparent just from looking at the digits. Hexadecimal turns out to be a useful number system for working with binary digital computers because of the relationship between base16 and base2. Sixteen is a convenient power of the base2: 2 4 = 16. So four binary (base2) digits that span values from (15) cover the same set of value as one hexadecimal digit (015). Thus, a group of four bits can be conveniently represented with a single hexadecimal digit as follows:
7 Computer Number Systems, Dr. Ken Hoganson, 7 Base2, Binary Base16, Hexadecimal A 1011 B 1100 C 1101 D 1110 E 1111 F So if a group of four binary digits can be represented with a single hexadecimal digit, then an 8 bit byte can be represented with two hexadecimal digits: Binary Hexadecimal CB F7 Note that it is far more convenient to talk about digital binary values in hexadecimal than it is in binary. For instance, a sixteen bit binary value: can be easily shared or recorded as C925. Converting from binary to hexadecimal (hex), and hex to binary is easily down without a formal conversion process, simply by grouping bits into groups of four bits, and translating that binary value to its equivalent hex digit. At first the student may need to use decimal as an intermediary: D F Converting multidigit values: Binary Hex , ,1 B1 Hex Binary A8 10, F 3,
8 Computer Number Systems, Dr. Ken Hoganson, 8 Section S.6: Negative Numbers So far we have worked with unsigned binary values, but number systems need to be able to represent both positive and negative numbers. For the purposes of this discussion, we will limit ourselves to values with 8 binary bits. In eight bits, a range of values can be represented. There are 256 possible combinations of 0s and 1s with eight bits, ranging from up to An examination of the range of values follows: Binary Hex Decimal FC FD FE FF 255 There are 256 possible combinations allowing values from 0 to 255. The number of combinations is also based on PowersoftheBase: 2 8 = 256. To represent negative numbers (in eight bits) some of the available values must be dedicated to represent negative numbers, and some to positive value.
9 Computer Number Systems, Dr. Ken Hoganson, 9 SignMagnitude The most obvious way to represent negative numbers is to use one of the digits to represent a sign bit, which indicates whether the number is to be positive or negative. The convention is to use the leftmost bit for the sign bit, with zero meaning a positive number and 1 meaning a negative number. The available combinations of 0s and 1s now have a different meaning: Binary Hex Decimal S E F negative zero? FC FD FE FF 127 Two problems with signmagnitude representation are apparent from the above table of values: 1. There are two representations for zero, both a positive zero and a negative zero. Not only is this incorrect, but the two representations for zero waste a combination that could otherwise be used to represent some other value. 2. Another problem with signmagnitude representation is revealed only when attempting basic mathematics. For instance adding a positive and negative number should work correctly: = 12! The problem with working with positive and negative numbers can be fixed for signmagnitude. Addition circuits can be designed to work correctly for adding numbers of each combination of signs of values: Four different addition circuits can be designed inside the CPU to handle each case, but this requires four times the circuitry and transistors to implement, clearly not efficient. And special cases need to be created for the other operations, not just addition. And each case must also correctly recognize the two representations for zero.
10 Computer Number Systems, Dr. Ken Hoganson, 10 Two s Complement A better approach is a method called Two scomplement. It is more complicated and nonintuitive, and only the unsolvable problems of the signmagnitude representation drive the use of two scomplement. But two scomplement does indeed work correctly and avoids the need for separate circuits to implement math with combinations of positive and negative numbers. In Two s Complement, A single bit is used to represent the sign of the number, and the leftmost bit is still used for the sign. But the meaning of the combinations of bits is different than signmagnitude for the negative numbers. The negative numbers count down from 128 in the progression of bit combinations: Binary Hex Decimal S E F FC FD FE FF 1 It is now difficult to read a negative number, as the meaning of the bits are reversed (complemented). Note that there is now only one representation for zero, and the extra combination allows an extra value to be represented: So the combinations with the zero as the sign bit range from 0 up to 127, and the combinations with the one as the sign bit range from down to 1. Fortunately, there is a simple way to translate or understand the meanings of the negative values, and its how this representation got its name. To convert a positive value to its negative representation in two s complement, a twostep process is used: Start with the binary positive representation: Complement (reverse) all the bits (one s complement) Add one.
11 Computer Number Systems, Dr. Ken Hoganson, 11 Example: find the two s complement representation of 3: A positive 3 in 8bits is: Complementing the bits: Add one = 3 This is the same value for negative 3 shown in the previous table of values. The same two scomplement steps can also be used to translate or convert a negative value: A negative 3 in 8bits is: Complementing the bits: Add one = +3 So a negative two s complement value can be read by finding its positive value equivalent for the magnitude of the number, and remembering that it s the negative of that value. Two scomplement and Math: Two scomplement does indeed solve the problem with working with combinations of signs: = = = +2! Notice that the carry from the addition of ones to the next place value carrys over beyond the eight bits, and inside the computer, this result is noted but the bit is discarded [Somewhat amusingly described as thrown into the bit bucket, though there is no actual bitbucket inside the machine]. Another example: = = = 12!
12 Computer Number Systems, Dr. Ken Hoganson, 12 Supplement Exercises: Work out the following problems on paper (show your work). Convert from binary to decimal: Work the following problems, converting decimal to binary (show all work) Work the following programs, representing the following decimal numbers in two s complement binary in eight bits
Number Systems Using and Converting Between Decimal, Binary, Octal and Hexadecimal Number Systems
Number Systems Using and Converting Between Decimal, Binary, Octal and Hexadecimal Number Systems In everyday life, we humans most often count using decimal or base10 numbers. In computer science, it
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 informationNumbers and Representations
Çetin Kaya Koç http://koclab.cs.ucsb.edu/teaching/cs192 koc@cs.ucsb.edu Çetin Kaya Koç http://koclab.cs.ucsb.edu Fall 2016 1 / 38 Outline Computational Thinking Representations of integers Binary and decimal
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 informationSigned umbers. Sign/Magnitude otation
Signed umbers So far we have discussed unsigned number representations. In particular, we have looked at the binary number system and shorthand methods in representing binary codes. With m binary digits,
More informationA complement number system is used to represent positive and negative integers. A complement number system is based on a fixed length representation
Complement Number Systems A complement number system is used to represent positive and negative integers A complement number system is based on a fixed length representation of numbers Pretend that integers
More informationNUMBER SYSTEMS AND CODES
C H A P T E R 69 Learning Objectives Number Systems The Decimal Number System Binary Number System Binary to Decimal Conversion Binary Fractions DoubleDadd Method Decimal to Binary Conversion Shifting
More informationReview of Data Representation & Binary Operations Dhananjai M. Rao CSA Department Miami University
Review of Data Representation & Binary Operations Dhananjai M. Rao () CSA Department Miami University 1. Introduction In digital computers all data including numbers, characters, and strings are ultimately
More informationBinary, Hexadecimal and Octal number system
Binary, Hexadecimal and Octal number system Binary, hexadecimal, and octal refer to different number systems. The one that we typically use is called decimal. These number systems refer to the number of
More information1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM
1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM 1.1 Introduction Given that digital logic and memory devices are based on two electrical states (on and off), it is natural to use a number
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 informationAppendix. Numbering Systems. In This Appendix...
Numbering Systems ppendix In This ppendix... Introduction... inary Numbering System... exadecimal Numbering System... Octal Numbering System... inary oded ecimal () Numbering System... 5 Real (Floating
More informationAppendix. Numbering Systems. In this Appendix
Numbering Systems ppendix n this ppendix ntroduction... inary Numbering System... exadecimal Numbering System... Octal Numbering System... inary oded ecimal () Numbering System... 5 Real (Floating Point)
More informationChapter 2 Exercises and Answers
Chapter 2 Exercises and nswers nswers are in blue. For Exercises 5, match the following numbers with their definition.. Number. Natural number C. Integer number D. Negative number E. Rational number unit
More informationThe type of all data used in a C (or C++) program must be specified
The type of all data used in a C (or C++) program must be specified A data type is a description of the data being represented That is, a set of possible values and a set of operations on those values
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 informationREPRESENTING INFORMATION:
REPRESENTING INFORMATION: BINARY, HEX, ASCII CORRESPONDING READING: WELL, NONE IN YOUR TEXT. SO LISTEN CAREFULLY IN LECTURE (BECAUSE IT WILL BE ON THE EXAM(S))! CMSC 150: Fall 2015 Controlling Information
More informationData Representation. DRAM uses a single capacitor to store and a transistor to select. SRAM typically uses 6 transistors.
Data Representation Data Representation Goal: Store numbers, characters, sets, database records in the computer. What we got: Circuit that stores 2 voltages, one for logic ( volts) and one for logic (3.3
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 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 informationLogic, Words, and Integers
Computer Science 52 Logic, Words, and Integers 1 Words and Data The basic unit of information in a computer is the bit; it is simply a quantity that takes one of two values, 0 or 1. A sequence of k bits
More informationDRAM uses a single capacitor to store and a transistor to select. SRAM typically uses 6 transistors.
Data Representation Data Representation Goal: Store numbers, characters, sets, database records in the computer. What we got: Circuit that stores 2 voltages, one for logic 0 (0 volts) and one for logic
More informationDigital Logic Lecture 2 Number Systems
Digital Logic Lecture 2 Number Systems By Ghada AlMashaqbeh The Hashemite University Computer Engineering Department Outline Introduction. Basic definitions. Number systems types. Conversion between different
More information1010 2?= ?= CS 64 Lecture 2 Data Representation. Decimal Numbers: Base 10. Reading: FLD Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
CS 64 Lecture 2 Data Representation Reading: FLD 1.21.4 Decimal Numbers: Base 10 Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 ) + (1x10 0 ) 1010 10?= 1010 2?= 1
More informationInf2C  Computer Systems Lecture 2 Data Representation
Inf2C  Computer Systems Lecture 2 Data Representation Boris Grot School of Informatics University of Edinburgh Last lecture Moore s law Types of computer systems Computer components Computer system stack
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 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 informationECE 550D Fundamentals of Computer Systems and Engineering. Fall 2017
ECE 550D Fundamentals of Computer Systems and Engineering Fall 2017 Combinational Logic Prof. John Board Duke University Slides are derived from work by Profs. Tyler Bletsch and Andrew Hilton (Duke) Last
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 informationExcerpt from "Art of Problem Solving Volume 1: the Basics" 2014 AoPS Inc.
Chapter 5 Using the Integers In spite of their being a rather restricted class of numbers, the integers have a lot of interesting properties and uses. Math which involves the properties of integers is
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 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 information9/23/15. Agenda. Goals of this Lecture. For Your Amusement. Number Systems and Number Representation. The Binary Number System
For Your Amusement Number Systems and Number Representation Jennifer Rexford Question: Why do computer programmers confuse Christmas and Halloween? Answer: Because 25 Dec = 31 Oct  http://www.electronicsweekly.com
More informationNumber Bases LESSON TWO. Computer Science. By John Owen
Number Bases LESSON TWO By John Owen Computer Science Objective In the last lesson you learned about different Number Bases used by the computer, which were Base Two binary Base Eight octal Base Sixteen
More informationEE 109 Unit 6 Binary Arithmetic
EE 109 Unit 6 Binary Arithmetic 1 2 Semester Transition Point At this point we are going to start to transition in our class to look more at the hardware organization and the lowlevel software that is
More informationFLOATING POINT NUMBERS
FLOATING POINT NUMBERS Robert P. Webber, Longwood University We have seen how decimal fractions can be converted to binary. For instance, we can write 6.25 10 as 4 + 2 + ¼ = 2 2 + 2 1 + 22 = 1*2 2 + 1*2
More informationIBM 370 Basic Data Types
IBM 370 Basic Data Types This lecture discusses the basic data types used on the IBM 370, 1. Two s complement binary numbers 2. EBCDIC (Extended Binary Coded Decimal Interchange Code) 3. Zoned Decimal
More informationChapter 2 Bits, Data Types, and Operations
Chapter 2 Bits, Data Types, and Operations How do we represent data in a computer? At the lowest level, a computer is an electronic machine. works by controlling the flow of electrons Easy to recognize
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 informationLesson Plan. Preparation
Math Practicum in Information Technology Lesson Plan Performance Objective Upon completion of this lesson, each student will be able to convert between different numbering systems and correctly write mathematical
More informationCS2630: Computer Organization Homework 1 Bits, bytes, and memory organization Due January 25, 2017, 11:59pm
CS2630: Computer Organization Homework 1 Bits, bytes, and memory organization Due January 25, 2017, 11:59pm Instructions: Show your work. Correct answers with no work will not receive full credit. Whether
More informationLearning the Binary System
Learning the Binary System www.brainlubeonline.com/counting_on_binary/ Formated to L A TEX: /25/22 Abstract This is a document on the base2 abstract numerical system, or Binary system. This is a VERY
More informationLAB A Translating Data to Binary
LAB A Translating Data to Binary Create a directory for this lab and perform in it the following groups of tasks: LabA1.java 1. Write the Java app LabA1 that takes an int via a commandline argument args[0]
More informationChapter 1. Digital Systems and Binary Numbers
Chapter 1. Digital Systems and Binary Numbers Tong In Oh 1 1.1 Digital Systems Digital age Characteristic of digital system Generality and flexibility Represent and manipulate discrete elements of information
More informationBinary Addition & Subtraction. Unsigned and Sign & Magnitude numbers
Binary Addition & Subtraction Unsigned and Sign & Magnitude numbers Addition and subtraction of unsigned or sign & magnitude binary numbers by hand proceeds exactly as with decimal numbers. (In fact this
More informationunused unused unused unused unused unused
BCD numbers. In some applications, such as in the financial industry, the errors that can creep in due to converting numbers back and forth between decimal and binary is unacceptable. For these applications
More informationDivisibility Rules and Their Explanations
Divisibility Rules and Their Explanations Increase Your Number Sense These divisibility rules apply to determining the divisibility of a positive integer (1, 2, 3, ) by another positive integer or 0 (although
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 information±M R ±E, S M CHARACTERISTIC MANTISSA 1 k j
ENEE 350 c C. B. Silio, Jan., 2010 FLOATING POINT REPRESENTATIONS It is assumed that the student is familiar with the discussion in Appendix B of the text by A. Tanenbaum, Structured Computer Organization,
More informationMATH 104B OCTAL, BINARY, AND HEXADECIMALS NUMBERS
MATH 104B OCTAL, BINARY, AND HEXADECIMALS NUMBERS A: Review: Decimal or Base Ten Numbers When we see a number like 2,578 we know the 2 counts for more than the 7, even though 7 is a larger number than
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 informationChapter 4: Data Representations
Chapter 4: Data Representations Integer Representations o unsigned o signmagnitude o one's complement o two's complement o bias o comparison o sign extension o overflow Character Representations Floating
More informationMicrocomputers. Outline. Number Systems and Digital Logic Review
Microcomputers Number Systems and Digital Logic Review Lecture 11 Outline Number systems and formats Common number systems Base Conversion Integer representation Signed integer representation Binary coded
More informationBasic Arithmetic Operations
Basic Arithmetic Operations Learning Outcome When you complete this module you will be able to: Perform basic arithmetic operations without the use of a calculator. Learning Objectives Here is what you
More informationSigned Binary Numbers
Signed Binary Numbers Unsigned Binary Numbers We write numbers with as many digits as we need: 0, 99, 65536, 15000, 1979, However, memory locations and CPU registers always hold a constant, fixed number
More informationEE 109 Unit 3. Analog vs. Digital. Analog vs. Digital. Binary Representation Systems ANALOG VS. DIGITAL
3. 3. EE 9 Unit 3 Binary Representation Systems ANALOG VS. DIGITAL 3.3 3. Analog vs. Digital The analog world is based on continuous events. Observations can take on any (real) value. The digital world
More informationDigital Fundamentals
Digital Fundamentals Tenth Edition Floyd Chapter 1 Modified by Yuttapong Jiraraksopakun Floyd, Digital Fundamentals, 10 th 2008 Pearson Education ENE, KMUTT ed 2009 Analog Quantities Most natural quantities
More informationColour and Number Representation. From Hex to Binary and Back. Colour and Number Representation. Colour and Number Representation
Colour and Number Representation From Hex to Binary and Back summary: colour representation easy: replace each hexadecimal "digit" with the corresponding four binary digits using the conversion table examples:
More informationCHAPTER 2 Data Representation in Computer Systems
CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 37 2.2 Positional Numbering Systems 38 2.3 Decimal to Binary Conversions 38 2.3.1 Converting Unsigned Whole Numbers 39 2.3.2 Converting
More informationLecture (03) Binary Codes Registers and Logic Gates
Lecture (03) Binary Codes Registers and Logic Gates By: Dr. Ahmed ElShafee Binary Codes Digital systems use signals that have two distinct values and circuit elements that have two stable states. binary
More informationNote: This case study utilizes Packet Tracer. Please see the Chapter 4 Packet Tracer file located in Supplemental Materials.
Part 1 Variable Length Subnet Mask (VLSM) Note: This case study utilizes Packet Tracer Please see the Chapter 4 Packet Tracer file located in Supplemental Materials An organization has been assigned the
More informationNumber Systems. Binary Numbers. Appendix. Decimal notation represents numbers as powers of 10, for example
Appendix F Number Systems Binary Numbers Decimal notation represents numbers as powers of 10, for example 1729 1 103 7 102 2 101 9 100 decimal = + + + There is no particular reason for the choice of 10,
More information,,,, Number Place Names. tens hundreds. ten thousands hundred thousands. ten trillions. hundred millions. ten billions hundred billions.
Number Place Names NSPV 1 Memorizing the most common number place names will be much easier once you recognize how the pattern of tens and hundreds names are repeated as prefixes for each group of three
More informationLecture 1: Digital Systems and Number Systems
Lecture 1: Digital Systems and Number Systems Matthew Shuman September 26th, 2012 The Digital Abstraction 1.3 in Text Analog Systems Analog systems are continuous. Look at the analog clock in figure 1.
More informationFundamentals of Programming Session 2
Fundamentals of Programming Session 2 Instructor: Reza EntezariMaleki Email: entezari@ce.sharif.edu 1 Fall 2013 Sharif University of Technology Outlines Programming Language Binary numbers Addition Subtraction
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 information4 Operations On Data 4.1. Foundations of Computer Science Cengage Learning
4 Operations On Data 4.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List the three categories of operations performed on data.
More informationIn this lesson, we will use the order of operations to evaluate and simplify expressions that contain numbers and variables.
Show Me: Expressions M8081 Could we sit in a classroom on the other side of the world and still make sense of the mathematics? The answer is yes! Of course, we might not understand exactly what the teacher
More information4 Operations On Data 4.1. Foundations of Computer Science Cengage Learning
4 Operations On Data 4.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List the three categories of operations performed on data.
More informationBits and Bytes. Here is a sort of glossary of computer buzzwords you will encounter in computer use:
Bits and Bytes Here is a sort of glossary of computer buzzwords you will encounter in computer use: Bit Computer processors can only tell if a wire is on or off. Luckily, they can look at lots of wires
More informationDigital Electronics A Practical Approach with VHDL William Kleitz Ninth Edition
Digital Electronics A Practical Approach with VHDL William Kleitz Ninth Edition Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world Visit
More informationCHAPTER 2 Data Representation in Computer Systems
CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 37 2.2 Positional Numbering Systems 38 2.3 Decimal to Binary Conversions 38 2.3.1 Converting Unsigned Whole Numbers 39 2.3.2 Converting
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 informationCS 265. Computer Architecture. Wei Lu, Ph.D., P.Eng.
CS 265 Computer Architecture Wei Lu, Ph.D., P.Eng. 1 Part 1: Data Representation Our goal: revisit and reestablish fundamental of mathematics for the computer architecture course Overview: what are bits
More informationComputer Programming C++ (wg) CCOs
Computer Programming C++ (wg) CCOs I. The student will analyze the different systems, and languages of the computer. (SM 1.4, 3.1, 3.4, 3.6) II. The student will write, compile, link and run a simple C++
More informationHere is a C function that will print a selected block of bytes from such a memory block, using an arraybased view of the necessary logic:
Pointer Manipulations Pointer Casts and Data Accesses Viewing Memory The contents of a block of memory may be viewed as a collection of hex nybbles indicating the contents of the byte in the memory region;
More informationElectronics Engineering ECE / E & T
STUDENT COPY DIGITAL ELECTRONICS 1 SAMPLE STUDY MATERIAL Electronics Engineering ECE / E & T Postal Correspondence Course GATE, IES & PSUs Digital Electronics 2015 ENGINEERS INSTITUTE OF INDIA. All Rights
More informationCS 101: Computer Programming and Utilization
CS 101: Computer Programming and Utilization JulNov 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 informationBawar Abid Abdalla. Assistant Lecturer Software Engineering Department Koya University
Logic Design First Stage Lecture No.5 Boolean Algebra Bawar Abid Abdalla Assistant Lecturer Software Engineering Department Koya University Boolean Operations Laws of Boolean Algebra Rules of Boolean Algebra
More informationECE 372 Microcontroller Design Assembly Programming Arrays. ECE 372 Microcontroller Design Assembly Programming Arrays
Assembly Programming Arrays Assembly Programming Arrays Array For Loop Example: unsigned short a[]; for(j=; j
More informationProgramming Studio #1 ECE 190
Programming Studio #1 ECE 190 Programming Studio #1 Announcements In Studio Assignment Introduction to Linux CommandLine Operations Recitation Floating Point Representation Binary & Hexadecimal 2 s Complement
More informationChapter 4: Computer Codes. In this chapter you will learn about:
Ref. Page Slide 1/30 Learning Objectives In this chapter you will learn about: Computer data Computer codes: representation of data in binary Most commonly used computer codes Collating sequence Ref. Page
More informationBits, Bytes, and Integers
Bits, Bytes, and Integers with contributions from Dr. Bin Ren, College of William & Mary 1 Bits, Bytes, and Integers Representing information as bits Bitlevel manipulations Integers Representation: unsigned
More informationThe Binary Number System
The Binary Number System Robert B. Heckendorn University of Idaho August 24, 2017 Numbers are said to be represented by a placevalue system, where the value of a symbol depends on where it is... its place.
More informationIntroduction to Assembly language
Introduction to Assembly language 1 USING THE AVR MICROPROCESSOR Outline Introduction to Assembly Code The AVR Microprocessor Binary/Hex Numbers Breaking down an example microprocessor program AVR instructions
More informationChapter 3: part 3 Binary Subtraction
Chapter 3: part 3 Binary Subtraction Iterative combinational circuits Binary adders Half and full adders Ripple carry and carry lookahead adders Binary subtraction Binary addersubtractors Signed binary
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 informationMath 340 Fall 2014, Victor Matveev. Binary system, roundoff errors, loss of significance, and double precision accuracy.
Math 340 Fall 2014, Victor Matveev Binary system, roundoff errors, loss of significance, and double precision accuracy. 1. Bits and the binary number system A bit is one digit in a binary representation
More informationChapter 1. Data Storage Pearson AddisonWesley. All rights reserved
Chapter 1 Data Storage 2007 Pearson AddisonWesley. All rights reserved Chapter 1: Data Storage 1.1 Bits and Their Storage 1.2 Main Memory 1.3 Mass Storage 1.4 Representing Information as Bit Patterns
More informationVARIABLES. Aim Understanding how computer programs store values, and how they are accessed and used in computer programs.
Lesson 2 VARIABLES Aim Understanding how computer programs store values, and how they are accessed and used in computer programs. WHAT ARE VARIABLES? When you input data (i.e. information) into a computer
More informationLecture 5: Arithmetic and Algebra Steven Skiena. skiena
Lecture 5: Arithmetic and Algebra Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena How Long is Long? Today s PCs are
More informationNumeric Precision 101
www.sas.com > Service and Support > Technical Support TS Home Intro to Services News and Info Contact TS Site Map FAQ Feedback TS654 Numeric Precision 101 This paper is intended as a basic introduction
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 FetchExecute Cycle Base
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 informationHow & Why We Subnet Lab Workbook
i How & Why We Subnet Lab Workbook ii CertificationKits.com How & Why We Subnet Workbook Copyright 2013 CertificationKits LLC All rights reserved. No part of this book maybe be reproduced or transmitted
More informationCommon Core State Standards Mathematics (Subset K5 Counting and Cardinality, Operations and Algebraic Thinking, Number and Operations in Base 10)
Kindergarten 1 Common Core State Standards Mathematics (Subset K5 Counting and Cardinality,, Number and Operations in Base 10) Kindergarten Counting and Cardinality Know number names and the count sequence.
More informationChapter 2 Data Representations
Computer Engineering Chapter 2 Data Representations Hiroaki Kobayashi 4/21/2008 4/21/2008 1 Agenda in Chapter 2 Translation between binary numbers and decimal numbers Data Representations for Integers
More informationCS/ECE 252: INTRODUCTION TO COMPUTER ENGINEERING UNIVERSITY OF WISCONSIN MADISON
CS/ECE 252: INTRODUCTION TO COMPUTER ENGINEERING UNIVERSITY OF WISCONSIN MADISON Prof. Gurindar Sohi, Kai Zhao TAs: Annie Lin, Mohit Verma, Neha Mittal, Daniel Griffin, Yuzhe Ma Examination 1 In Class
More informationBut first, encode deck of cards. Integer Representation. Two possible representations. Two better representations WELLESLEY CS 240 9/8/15
Integer Representation Representation of integers: unsigned and signed Sign extension Arithmetic and shifting Casting But first, encode deck of cards. cards in suits How do we encode suits, face cards?
More informationNet Masks and IP Addresses
Net Masks and IP Addresses Professor Don Colton Brigham Young University Hawaii 1 Introduction IPv4 is currently the main addressing method on the Internet. Students who plan to use networking skills in
More information