Solutions - Homework 1 (Due date: September 25 th ) Presentation and clarity are very important! Show your procedure!
|
|
|
- Corey Allison
- 5 years ago
- Views:
Transcription
1 c 10 =0 c 9 =0 c 8 =0 c 7 =0 c 6 =0 c 5 =0 c 10 =1 c 9 =1 c 8 =1 c 7 =0 c 6 =1 c 5 =1 c 4 =1 c 8 =1 c 7 =1 c 6 =0 c 5 =0 c 8 =0 c 7 =0 c 6 =0 c 5 =0 c 8 =1 c 7 =1 c 6 =1 c 5 =0 c 4 =1 b 7 =0 b 6 =0 b 5 =0 b 4 =0 b 3 =1 b 2 =1 b 1 =1 b 0 =0 c 7 =1 c 6 =1 c 5 =1 c 7 =0 c 6 =0 c 5 =1 ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY Solutions - Homework 1 (Due date: September 25 th ) Presentation and clarity are very important! Show your procedure! PROBLEM 1 (10 PTS) Calculate the result of the additions and subtractions for the following fixed-point numbers UNSIGNED SIGNED UNSIGNED: SIGNED: PROBLEM 2 (10 PTS) Multiply the following signed fixed-point numbers: x x x x x x Instructor: Daniel Llamocca
2 x x x x x x PROBLEM 3 (15 PTS) Get the division result (with x = 4 fractional bits ) for the following signed fixed-point numbers: : To unsigned and then alignment, a = 6: Append x = 4 zeros: Integer Division: = = = Q = , R = Qf = (x = 4) Final result (2C): = : To unsigned and then alignment, a = 6: = = Append x = 4 zeros: Integer Division: Q = 10, R = 1110 Qf = 10.(x = 4) Final result (2C): = : To unsigned (denominator) and then alignment, a = 3: = Append x = 4 zeros: Integer Division: Q = , R = 10 Qf = (x = 4) Final result (2C): = 2C( ) = Instructor: Daniel Llamocca
3 : To unsigned (denominator) and then alignment, a = 6: Append x = 4 zeros: Integer Division: Q = 100, R = Qf = (x = 4) Final result (2C): = 0.01 = PROBLEM 4 (5 PTS) We want to represent numbers between and 179. What is the fixed point format that requires the fewest number of bits for a resolution better or equal than ? 2C representation for integers: 2 n 1 to 2 n 1 1. For 2 n , we have that n 9, so we pick n = 9. For the fractional part, we select the number of fractional bits p that make the resolution better or equal than : Then the Fixed Point format required in [20 11]. 2 p p p = 11 PROBLEM 5 (10 PTS) Complete the table for the following floating point formats (which resemble the IEEE-754 standard) with 16, 24, 48 bits. Only consider ordinary numbers. Exponent Significant bits (E) bits (p) Min Max Range of e Range of significand [ 30,31] [1, ] [ 62,63] [1, ] [ 510,511] [1, ] PROBLEM 6 (20) Calculate the decimal values of the following floating point numbers represented as hexadecimals. Show your procedure. Single (32 bits) Double (64 bits) F DECADE FDEAD378 3DE ABBAA FACEB0E8 7FF32B5A ACCEDE78 FA09D3784D089B7D 80DEADBEE D3787F FA0BEBE80BEEF0A FDB FA09D37809ABC0DE FF ABC0DE DECAFC0FFEE00800 F : e + bias = = 240 e = = 113 Mantissa ([24 23]) = = X = = DECADE: e + bias = = 1 e = = 126 Mantissa = = X = = FDEAD378: e + bias = 1111 = 251 e = = 124 Mantissa = = X = = DE38866: e + bias = 0111 = 123 e = = 4 Mantissa = = X = = Instructor: Daniel Llamocca
4 800ABBAA: e + bias = = 0 Denormal number e = 126 Mantissa = = X = = FACEB0E8: e + bias = = 245 e = = 118 Mantissa = = X = = FF32B5A: e + bias = = 255, f 0 X = NaN ACCEDE78: e + bias = 0001 = 89 e = = 38 Mantissa = = X = = FA09D3784D089B7D: e + bias = = 1952 e = = 929 Mantissa ([53 52]) = = X = = DEADBEE : e + bias = = 13 e = = 1010 Mantissa = = X = = D3787F888800: e + bias = = 0 Denormal number e = 1022 Mantissa = = X = = FA0BEBE80BEEF0A0: e + bias = = 1952 e = = 929 Mantissa = = X = = FDB: e + bias = = 1175 e = = 152 Mantissa = = X = = FA09D37809ABC0DE: e + bias = = 1952 e = = 929 Mantissa = = X = = FF ABC0DE: e + bias = = 2040 e = = 1017 Mantissa = = X = = DECAFC0FFEE00800: e + bias = 100 = 1516 e = = 493 Mantissa = = X = = Instructor: Daniel Llamocca
5 PROBLEM 7 (30) Calculate the result of the following operations with 32-bit floating point numbers. Truncate the results when required. When doing fixed-point division, use 8 fractional bits. Show your procedure. FA FF800FAD CA09D F800FEA + 09ABC0DE 5A09D FDB FC09D F DE32B5A FF DE D959 FA09D378-09ABC0DE FA09D300 4D FE FA09DF00 7F A09D300 0BEEF ABC0DE FF FE0000 FA09D X = FA FF800FAD: FF800FAD: e + bias = = 255, f 0 FF800FAD = NaN X = # + NaN = NaN X = 7F800FEA + 09ABC0DE: 7F800FEA: e + bias = = 255, f 0 7F800FEA = NaN X = NaN + # = NaN X = FC09D F800000: 7F800000: e + bias = = 255, f = 0 7F = + X = # + = + X = 3DE D959: 3DE38866: e + bias = 0111 = 123 e = = 4 Mantissa = DE38866 = D959: e + bias = 01 = 102 e = = 25 Mantissa = D959 = X = X = X = X = = = X = = 3DE3886A X = CA09D : CA98D378: e + bias = = 148 e = = 21 Mantissa = DE38866 = : e + bias = = 0 Denormal number e = = 0 X = = X = = CA09D378 X = 5A09D FDB: 5A09D378: e + bias = 0100 = 180 e = = 53 Mantissa = A09D378 = Instructor: Daniel Llamocca
6 40490FDB: e + bias = = e = = 1 Mantissa = FDB = X = X = The division by 2 52 requires more than p + 1 = 24 bits for proper representation. Thus, we approximate the second operand by 0. X = = X = = 5A09D378 X = 7DE32B5A FF800000: FF800000: e + bias = = 255, f = 0 FF = X = # ( ) = + X = 7F X = FA09D378 09ABC0DE: FA09D378: e + bias = = 244 e = = 117 Mantissa = FA09D378 = ABC0DE: e + bias = = 19 e = = 108 Mantissa = ABC0DE = X = X = The division by requires more than p + 1 = 24 bits for proper representation. Thus, we approximate the second operand by 0. X = = X = = FA09D378 X = FA09D300 4D080000: FA09D300: e + bias = = 244 e = = 117 Mantissa = FA09D300 = D080000: e + bias = 10 = 154 e = = 27 Mantissa = D = X = X = e + bias = = 271 > 254 In this case, there is an overflow. The value X is assigned to. X = = FF X = FE: : e + bias = = 0 Denormal number e = = 0 X = 0 # = 0 X = Instructor: Daniel Llamocca
7 X = FA09DF00 7F800000: 7F800000: e + bias = = 255, f = 0 7F = + X = ( #) = X = = FF X = 7A09D300 0BEEF000: 7A09D300: e + bias = = 244 e = = 117 Mantissa = A09D300 = BEEF000: e + bias = 0001 = 23 e = = 104 Mantissa = BEEF000 = X = X = = = e + bias = = 141 = X = = 4680A35F (four bits were truncated) X = : : e + bias = = 146 e = = 19 Mantissa = = : e + bias = = 128 e = = 1 Mantissa = BEEF000 = X = Alignment: = = Append x = 8 zeros: Integer division Q = 110, R = Qf = Thus: X = = = = e + bias = = 145 = X = = 489B0000 X = ABC0DE: : e + bias = = 0 Denormal number e = = 0 X = 0 # = 0 X = Instructor: Daniel Llamocca
8 X = FF FE0000: FF800000: e + bias = = 255, f = 0 FF = X = # = X = FF X = FA09D FA09D300: e + bias = = 244 e = = 117 Mantissa = FA09D300 = : e + bias = = 144 e = = 17 Mantissa = = X = Alignment: = = Append x = 8 zeros: Integer division Q = , R = Qf = Thus: X = = = = = e + bias = = 226 = X = = F Instructor: Daniel Llamocca
Binary Floating-Point Numbers
Binary Floating-Point Numbers S exponent E significand M F=(-1) s M β E Significand M pure fraction [0, 1-ulp] or [1, 2) for β=2 Normalized form significand has no leading zeros maximum # of significant
Unit 1 - Computer Arithmetic
FIXD-POINT (FX) ARITHMTIC Unit 1 - Comuter Arithmetic INTGR NUMBRS n bit number: b n 1 b n 2 b 0 Decimal Value Range of values UNSIGND n 1 SIGND D = b i 2 i D = 2 n 1 b n 1 + b i 2 i n 2 i=0 i=0 [0, 2
How do computers represent numbers?
How do computers represent numbers? Tips & Tricks Week 1 Topics in Scientific Computing QMUL Semester A 2017/18 1/10 What does digital mean? The term DIGITAL refers to any device that operates on discrete
Chapter 1 Error Analysis
Chapter 1 Error Analysis Several sources of errors are important for numerical data processing: Experimental uncertainty: Input data from an experiment have a limited precision. Instead of the vector of
Chapter 1: Introduction and mathematical preliminaries
Chapter 1: Introduction and mathematical preliminaries Evy Kersalé September 26, 2011 Motivation Most of the mathematical problems you have encountered so far can be solved analytically. However, in real-life,
Conversions between Decimal and Binary
Conversions between Decimal and Binary Binary to Decimal Technique - use the definition of a number in a positional number system with base 2 - evaluate the definition formula ( the formula ) using decimal
9. Datapath Design. Jacob Abraham. Department of Electrical and Computer Engineering The University of Texas at Austin VLSI Design Fall 2017
9. Datapath Design Jacob Abraham Department of Electrical and Computer Engineering The University of Texas at Austin VLSI Design Fall 2017 October 2, 2017 ECE Department, University of Texas at Austin
MATH Dr. Halimah Alshehri Dr. Halimah Alshehri
MATH 1101 haalshehri@ksu.edu.sa 1 Introduction To Number Systems First Section: Binary System Second Section: Octal Number System Third Section: Hexadecimal System 2 Binary System 3 Binary System The binary
Number Representation and Waveform Quantization
1 Number Representation and Waveform Quantization 1 Introduction This lab presents two important concepts for working with digital signals. The first section discusses how numbers are stored in memory.
Numbering Systems. Contents: Binary & Decimal. Converting From: B D, D B. Arithmetic operation on Binary.
Numbering Systems Contents: Binary & Decimal. Converting From: B D, D B. Arithmetic operation on Binary. Addition & Subtraction using Octal & Hexadecimal 2 s Complement, Subtraction Using 2 s Complement.
Divider Implementation
c n = cn-= LCTRICAL AD COTR GIRIG DPARTT, OAKLAD UIVRSITY RCRLA@OU ALGORITH Divider Implementation The division of two unsigned integer numbers A (where A is the dividend and the divisor), results in a
Chapter 4 Number Representations
Chapter 4 Number Representations SKEE2263 Digital Systems Mun im/ismahani/izam {munim@utm.my,e-izam@utm.my,ismahani@fke.utm.my} February 9, 2016 Table of Contents 1 Fundamentals 2 Signed Numbers 3 Fixed-Point
Introduction CSE 541
Introduction CSE 541 1 Numerical methods Solving scientific/engineering problems using computers. Root finding, Chapter 3 Polynomial Interpolation, Chapter 4 Differentiation, Chapter 4 Integration, Chapters
Scientific Notation. Chemistry Honors
Scientific Notation Chemistry Honors Used to easily write very large or very small numbers: 1 mole of a substance consists of 602,000,000,000,000,000,000,000 particles (we ll come back to this in Chapter
Cs302 Quiz for MID TERM Exam Solved
Question # 1 of 10 ( Start time: 01:30:33 PM ) Total Marks: 1 Caveman used a number system that has distinct shapes: 4 5 6 7 Question # 2 of 10 ( Start time: 01:31:25 PM ) Total Marks: 1 TTL based devices
ALU (3) - Division Algorithms
HUMBOLDT-UNIVERSITÄT ZU BERLIN INSTITUT FÜR INFORMATIK Lecture 12 ALU (3) - Division Algorithms Sommersemester 2002 Leitung: Prof. Dr. Miroslaw Malek www.informatik.hu-berlin.de/rok/ca CA - XII - ALU(3)
ECE 372 Microcontroller Design
Data Formats Humor There are 10 types of people in the world: Those who get binary and those who don t. 1 Information vs. Data Information An abstract description of facts, processes or perceptions How
Chapter 4: Radicals and Complex Numbers
Section 4.1: A Review of the Properties of Exponents #1-42: Simplify the expression. 1) x 2 x 3 2) z 4 z 2 3) a 3 a 4) b 2 b 5) 2 3 2 2 6) 3 2 3 7) x 2 x 3 x 8) y 4 y 2 y 9) 10) 11) 12) 13) 14) 15) 16)
Unit 1 Part 1: Significant Figures and Scientific Notation. Objective understand significant figures and their rules. Be able to use scientific
Unit 1 Part 1: Significant Figures and Scientific Notation. Objective understand significant figures and their rules. Be able to use scientific notation in calculations. Significant figures - consist of
ENGIN 112 Intro to Electrical and Computer Engineering
ENGIN 112 Intro to Electrical and Computer Engineering Lecture 3 More Number Systems Overview Hexadecimal numbers Related to binary and octal numbers Conversion between hexadecimal, octal and binary Value
Notes for Chapter 1 of. Scientific Computing with Case Studies
Notes for Chapter 1 of Scientific Computing with Case Studies Dianne P. O Leary SIAM Press, 2008 Mathematical modeling Computer arithmetic Errors 1999-2008 Dianne P. O'Leary 1 Arithmetic and Error What
Arithmetic and Error. How does error arise? How does error arise? Notes for Part 1 of CMSC 460
Notes for Part 1 of CMSC 460 Dianne P. O Leary Preliminaries: Mathematical modeling Computer arithmetic Errors 1999-2006 Dianne P. O'Leary 1 Arithmetic and Error What we need to know about error: -- how
Ex code
Ex. 8.4 7-4-2-1 code Codeconverter 7-4-2-1-code to BCD-code. When encoding the digits 0... 9 sometimes in the past a code having weights 7-4-2-1 instead of the binary code weights 8-4-2-1 was used. In
0,..., r 1 = digits in radix r number system, that is 0 d i r 1 where m i n 1
RADIX r NUMBER SYSTEM Let (N) r be a radix r number in a positional weighting number system, then (N) r = d n 1 r n 1 + + d 0 r 0 d 1 r 1 + + d m r m where: r = radix d i = digit at position i, m i n 1
Complement Arithmetic
Complement Arithmetic Objectives In this lesson, you will learn: How additions and subtractions are performed using the complement representation, What is the Overflow condition, and How to perform arithmetic
Computer Arithmetic. MATH 375 Numerical Analysis. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Computer Arithmetic
Computer Arithmetic MATH 375 Numerical Analysis J. Robert Buchanan Department of Mathematics Fall 2013 Machine Numbers When performing arithmetic on a computer (laptop, desktop, mainframe, cell phone,
DIVIDER IMPLEMENTATION
c n = cn-= DAIL LLAOCCA CLab@OU DIVID IPLTATIO The division of two unsigned integer numbers A (where A is the dividend and the divisor), results in a quotient and a residue. These quantities are related
ECE260: Fundamentals of Computer Engineering
Data Representation & 2 s Complement James Moscola Dept. of Engineering & Computer Science York College of Pennsylvania Based on Computer Organization and Design, 5th Edition by Patterson & Hennessy Data
14:332:231 DIGITAL LOGIC DESIGN. 2 s-complement Representation
4:332:23 DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering Fall 203 Lecture #3: Addition, Subtraction, Multiplication, and Division 2 s-complement Representation RECALL
Chapter 8: Solutions to Exercises
1 DIGITAL ARITHMETIC Miloš D. Ercegovac and Tomás Lang Morgan Kaufmann Publishers, an imprint of Elsevier Science, c 2004 with contributions by Fabrizio Lamberti Exercise 8.1 Fixed-point representation
MATH ASSIGNMENT 03 SOLUTIONS
MATH444.0 ASSIGNMENT 03 SOLUTIONS 4.3 Newton s method can be used to compute reciprocals, without division. To compute /R, let fx) = x R so that fx) = 0 when x = /R. Write down the Newton iteration for
Menu. Review of Number Systems EEL3701 EEL3701. Math. Review of number systems >Binary math >Signed number systems
Menu Review of number systems >Binary math >Signed number systems Look into my... 1 Our decimal (base 10 or radix 10) number system is positional. Ex: 9437 10 = 9x10 3 + 4x10 2 + 3x10 1 + 7x10 0 We have
Name: Chapter 7: Exponents and Polynomials
Name: Chapter 7: Exponents and Polynomials 7-1: Integer Exponents Objectives: Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. You
CSE 241 Digital Systems Spring 2013
CSE 241 Digital Systems Spring 2013 Instructor: Prof. Kui Ren Department of Computer Science and Engineering Lecture slides modified from many online resources and used solely for the educational purpose.
Section 3 Using Scientific Measurements. Look at the specifications for electronic balances. How do the instruments vary in precision?
Lesson Starter Look at the specifications for electronic balances. How do the instruments vary in precision? Discuss using a beaker to measure volume versus using a graduated cylinder. Which is more precise?
Mathematical preliminaries and error analysis
Mathematical preliminaries and error analysis Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan September 12, 2015 Outline 1 Round-off errors and computer arithmetic
Lecture 7. Floating point arithmetic and stability
Lecture 7 Floating point arithmetic and stability 2.5 Machine representation of numbers Scientific notation: 23 }{{} }{{} } 3.14159265 {{} }{{} 10 sign mantissa base exponent (significand) s m β e A floating
What Every Programmer Should Know About Floating-Point Arithmetic DRAFT. Last updated: November 3, Abstract
What Every Programmer Should Know About Floating-Point Arithmetic Last updated: November 3, 2014 Abstract The article provides simple answers to the common recurring questions of novice programmers about
Chapter 1 Mathematical Preliminaries and Error Analysis
Chapter 1 Mathematical Preliminaries and Error Analysis Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 128A Numerical Analysis Limits and Continuity
Multiplication of signed-operands
Multiplication of signed-operands Recall we discussed multiplication of unsigned numbers: Combinatorial array multiplier. Sequential multiplier. Need an approach that works uniformly with unsigned and
Tutorial 2: Expressing Uncertainty (Sig Figs, Scientific Notation and Rounding)
Tutorial 2: Expressing Uncertainty (Sig Figs, Scientific Notation and Rounding) Goals: To be able to convert quantities from one unit to another. To be able to express measurements and answers to the correct
EAD 115. Numerical Solution of Engineering and Scientific Problems. David M. Rocke Department of Applied Science
EAD 115 Numerical Solution of Engineering and Scientific Problems David M. Rocke Department of Applied Science Computer Representation of Numbers Counting numbers (unsigned integers) are the numbers 0,
Alg 1B Chapter 7 Final Exam Review
Name: Class: Date: ID: A Alg B Chapter 7 Final Exam Review Please answer all questions and show your work. Simplify ( 2) 4. 2. Simplify ( 4) 4. 3. Simplify 5 2. 4. Simplify 9x0 y 3 z 8. 5. Simplify 7w0
Optimizing MPC for robust and scalable integer and floating-point arithmetic
Optimizing MPC for robust and scalable integer and floating-point arithmetic Liisi Kerik * Peeter Laud * Jaak Randmets * * Cybernetica AS University of Tartu, Institute of Computer Science January 30,
What Fun! It's Practice with Scientific Notation!
What Fun! It's Practice with Scientific Notation! Review of Scientific Notation Scientific notation provides a place to hold the zeroes that come after a whole number or before a fraction. The number 100,000,000
FLOATING POINT ARITHMETHIC - ERROR ANALYSIS
FLOATING POINT ARITHMETHIC - ERROR ANALYSIS Brief review of floating point arithmetic Model of floating point arithmetic Notation, backward and forward errors 3-1 Roundoff errors and floating-point arithmetic
Lecture 2: Number Representations (2)
Lecture 2: Number Representations (2) ECE 645 Computer Arithmetic 1/29/08 ECE 645 Computer Arithmetic Lecture Roadmap Number systems (cont'd) Floating point number system representations Residue number
Negative Bit Representation Outline
Negative Bit Representation Outline 1. Negative Bit Representation Outline 2. Negative Integers 3. Representing Negativity 4. Which Bit for the Sign? 5. Sign-Value 6. Disadvantages of Sign-Value 7. One
ECE380 Digital Logic. Positional representation
ECE380 Digital Logic Number Representation and Arithmetic Circuits: Number Representation and Unsigned Addition Dr. D. J. Jackson Lecture 16-1 Positional representation First consider integers Begin with
Order of convergence. MA3232 Numerical Analysis Week 3 Jack Carl Kiefer ( ) Question: How fast does x n
Week 3 Jack Carl Kiefer (94-98) Jack Kiefer was an American statistician. Much of his research was on the optimal design of eperiments. However, he also made significant contributions to other areas of
2 ways to write the same number: 6,500: standard form 6.5 x 10 3 : scientific notation
greater than or equal to one, and less than 10 positive exponents: numbers greater than 1 negative exponents: numbers less than 1, (> 0) (fractions) 2 ways to write the same number: 6,500: standard form
Next, we include the several conversion from type to type.
Number Conversions: Binary Decimal; Floating Points In order to communicate with a computer, we need, at some point, to speak the same language. The words of our language are made up of combinations of
NAME DATE PERIOD. A negative exponent is the result of repeated division. Extending the pattern below shows that 4 1 = 1 4 or 1. Example: 6 4 = 1 6 4
Lesson 4.1 Reteach Powers and Exponents A number that is expressed using an exponent is called a power. The base is the number that is multiplied. The exponent tells how many times the base is used as
Numerical Methods - Preliminaries
Numerical Methods - Preliminaries Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Preliminaries 2013 1 / 58 Table of Contents 1 Introduction to Numerical Methods Numerical
1 ERROR ANALYSIS IN COMPUTATION
1 ERROR ANALYSIS IN COMPUTATION 1.2 Round-Off Errors & Computer Arithmetic (a) Computer Representation of Numbers Two types: integer mode (not used in MATLAB) floating-point mode x R ˆx F(β, t, l, u),
Representing Numbers on the Computer
Representing Numbers on the Computer Let s calculate J! Program Maxnumber * * Check what the max numbers are on the computer * Integer N * * N= Do J=,2 N=N*J Print *,J,N Enddo stop end What happened? J
CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 5. Ax = b.
CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 5 GENE H GOLUB Suppose we want to solve We actually have an approximation ξ such that 1 Perturbation Theory Ax = b x = ξ + e The question is, how
NUMBER SYSTEMS. and DATA REPRESENTATION. for COMPUTERS (PROBLEM ANSWERS)
NUMBER SYSTEMS and DATA REPRESENTATION for COMPUTERS (PROBLEM ANSWERS) 05 March 2008 Number Systems and Data Representation 2 Table of Contents Table of Contents... 2 Conversion Between Binary and Hexadecimal
QUADRATIC PROGRAMMING?
QUADRATIC PROGRAMMING? WILLIAM Y. SIT Department of Mathematics, The City College of The City University of New York, New York, NY 10031, USA E-mail: wyscc@cunyvm.cuny.edu This is a talk on how to program
FLOATING POINT ARITHMETHIC - ERROR ANALYSIS
FLOATING POINT ARITHMETHIC - ERROR ANALYSIS Brief review of floating point arithmetic Model of floating point arithmetic Notation, backward and forward errors Roundoff errors and floating-point arithmetic
Chapter 1 Indices & Standard Form
Chapter 1 Indices & Standard Form Section 1.1 Simplifying Only like (same letters go together; same powers and same letter go together) terms can be grouped together. Example: a 2 + 3ab + 4a 2 5ab + 10
UNIT V FINITE WORD LENGTH EFFECTS IN DIGITAL FILTERS PART A 1. Define 1 s complement form? In 1,s complement form the positive number is represented as in the sign magnitude form. To obtain the negative
Number Bases. Ioan Despi. University of New England. August 4, 2013
Number Bases Ioan Despi despi@turing.une.edu.au University of New England August 4, 2013 Outline Ioan Despi AMTH140 2 of 21 1 Frequently Used Number Systems 2 Conversion to Numbers of Different Bases 3
ECS 231 Computer Arithmetic 1 / 27
ECS 231 Computer Arithmetic 1 / 27 Outline 1 Floating-point numbers and representations 2 Floating-point arithmetic 3 Floating-point error analysis 4 Further reading 2 / 27 Outline 1 Floating-point numbers
Introduction to Digital Logic Missouri S&T University CPE 2210 Number Systems
Introduction to Digital Logic Missouri S&T University CPE 2210 Number Systems Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and
Tu: 9/3/13 Math 471, Fall 2013, Section 001 Lecture 1
Tu: 9/3/13 Math 71, Fall 2013, Section 001 Lecture 1 1 Course intro Notes : Take attendance. Instructor introduction. Handout : Course description. Note the exam days (and don t be absent). Bookmark the
Elements of Floating-point Arithmetic
Elements of Floating-point Arithmetic Sanzheng Qiao Department of Computing and Software McMaster University July, 2012 Outline 1 Floating-point Numbers Representations IEEE Floating-point Standards Underflow
Introduction and mathematical preliminaries
Chapter Introduction and mathematical preliminaries Contents. Motivation..................................2 Finite-digit arithmetic.......................... 2.3 Errors in numerical calculations.....................
UNIT 4 NOTES: PROPERTIES & EXPRESSIONS
UNIT 4 NOTES: PROPERTIES & EXPRESSIONS Vocabulary Mathematics: (from Greek mathema, knowledge, study, learning ) Is the study of quantity, structure, space, and change. Algebra: Is the branch of mathematics
8/13/16. Data analysis and modeling: the tools of the trade. Ø Set of numbers. Ø Binary representation of numbers. Ø Floating points.
Data analysis and modeling: the tools of the trade Patrice Koehl Department of Biological Sciences National University of Singapore http://www.cs.ucdavis.edu/~koehl/teaching/bl5229 koehl@cs.ucdavis.edu
Chapter 3 Experimental Error
Chapter 3 Experimental Error Homework Due Friday January 27 Problems: 3-2, 3-5, 3-9, 3-10, 3-11, 3-12, 3-14, 3-19 Chapter 3 Experimental Error Uncertainties They are everywhere!! We need to learn to understand
Sect Exponents: Multiplying and Dividing Common Bases
154 Sect 5.1 - Exponents: Multiplying and Dividing Common Bases Concept #1 Review of Exponential Notation In the exponential expression 4 5, 4 is called the base and 5 is called the exponent. This says
History & Binary Representation
History & Binary Representation C. R. da Cunha 1 1 Instituto de Física, Universidade Federal do Rio Grande do Sul, RS 91501-970, Brazil. August 30, 2017 Abstract In this lesson we will review the history
SECTION Types of Real Numbers The natural numbers, positive integers, or counting numbers, are
SECTION.-.3. Types of Real Numbers The natural numbers, positive integers, or counting numbers, are The negative integers are N = {, 2, 3,...}. {..., 4, 3, 2, } The integers are the positive integers,
Chapter 4: Radicals and Complex Numbers
Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y
8.1 Multiplication Properties of Exponents Objectives 1. Use properties of exponents to multiply exponential expressions.
8.1 Multiplication Properties of Exponents Objectives 1. Use properties of exponents to multiply exponential expressions. 2. Use powers to model real life problems. Multiplication Properties of Exponents
A number that can be written as, where p and q are integers and q Number.
RATIONAL NUMBERS 1.1 Definition of Rational Numbers: What are rational numbers? A number that can be written as, where p and q are integers and q Number. 0, is known as Rational Example:, 12, -18 etc.
Jim Lambers MAT 610 Summer Session Lecture 2 Notes
Jim Lambers MAT 610 Summer Session 2009-10 Lecture 2 Notes These notes correspond to Sections 2.2-2.4 in the text. Vector Norms Given vectors x and y of length one, which are simply scalars x and y, the
Numbers and Arithmetic
Numbers and Arithmetic See: P&H Chapter 2.4 2.6, 3.2, C.5 C.6 Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University Big Picture: Building a Processor memory inst register file alu
Tunable Floating-Point for Energy Efficient Accelerators
Tunable Floating-Point for Energy Efficient Accelerators Alberto Nannarelli DTU Compute, Technical University of Denmark 25 th IEEE Symposium on Computer Arithmetic A. Nannarelli (DTU Compute) Tunable
1. Revision Description Reflect and Review Teasers Answers Recall of Rational Numbers:
1. Revision Description Reflect Review Teasers Answers Recall of Rational Numbers: A rational number is of the form, where p q are integers q 0. Addition or subtraction of rational numbers is possible
Introduction to Digital Logic Missouri S&T University CPE 2210 Number Systems
Introduction to Digital Logic Missouri S&T University CPE 2210 Number Systems Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and
Proposal to Improve Data Format Conversions for a Hybrid Number System Processor
Proposal to Improve Data Format Conversions for a Hybrid Number System Processor LUCIAN JURCA, DANIEL-IOAN CURIAC, AUREL GONTEAN, FLORIN ALEXA Department of Applied Electronics, Department of Automation
14:332:231 DIGITAL LOGIC DESIGN. Why Binary Number System?
:33:3 DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering Fall 3 Lecture #: Binary Number System Complement Number Representation X Y Why Binary Number System? Because
Chapter 1: Preliminaries and Error Analysis
Chapter 1: Error Analysis Peter W. White white@tarleton.edu Department of Tarleton State University Summer 2015 / Numerical Analysis Overview We All Remember Calculus Derivatives: limit definition, sum
6-3 Solving Systems by Elimination
Another method for solving systems of equations is elimination. Like substitution, the goal of elimination is to get one equation that has only one variable. To do this by elimination, you add the two
CHAPTER 2 NUMBER SYSTEMS
CHAPTER 2 NUMBER SYSTEMS The Decimal Number System : We begin our study of the number systems with the familiar decimal number system. The decimal system contains ten unique symbol 0, 1, 2, 3, 4, 5, 6,
Hakim Weatherspoon CS 3410 Computer Science Cornell University
Hakim Weatherspoon CS 3410 Computer Science Cornell University The slides are the product of many rounds of teaching CS 3410 by Professors Weatherspoon, Bala, Bracy, and Sirer. memory inst 32 register
Computer Number Systems
Computer Number Systems All computers are electronic devices and can ultimately do one thing: detect whether an electrical signal is on or off. Therefore, the earliest computer scientists realized that
TI-84+ GC 2: Exponents and Scientific Notation
Rev 6-- Name Date TI-84+ GC : Exponents and Scientific Notation Objectives: Use the caret and square keys to calculate exponents Review scientific notation Input a calculation in scientific notation Recognize
Digital Electronics 1
ELETRONIS ENGINEERING SERIES igital Electronics ombinational Logic ircuits Tertulien Ndjountche igital Electronics Series Editor Robert aptist igital Electronics ombinational Logic ircuits Tertulien
arxiv: v1 [cs.na] 15 Apr 2013
![arxiv: v1 [cs.na] 15 Apr 2013](/thumbs/86/94744477.jpg "arxiv: v1 [cs.na] 15 Apr 2013")
Relative error due to a single bit-flip in floating-point arithmetic Bradley R. Lowery arxiv:1304.4292v1 [cs.na] 15 Apr 2013 February 7, 2014 We consider the error due to a single bit-flip in a floating
Topics Covered in This Chapter:
Introduction to Powers of 10 Topics Covered in This Chapter: I-1: Scientific Notation I-2: Engineering Notation and Metric Prefixes I-3: Converting between Metric Prefixes I-4: Addition and Subtraction
Uncertainties in Measurement
Uncertainties in Measurement Laboratory investigations involve taking measurements of physical quantities. All measurements will involve some degree of experimental uncertainty. QUESTIONS 1. How does one
Elements of Floating-point Arithmetic
Elements of Floating-point Arithmetic Sanzheng Qiao Department of Computing and Software McMaster University July, 2012 Outline 1 Floating-point Numbers Representations IEEE Floating-point Standards Underflow
Floating Point Number Systems. Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le
Floating Point Number Systems Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le 1 Overview Real number system Examples Absolute and relative errors Floating point numbers Roundoff
Section 3-4: Least Common Multiple and Greatest Common Factor
Section -: Fraction Terminology Identify the following as proper fractions, improper fractions, or mixed numbers:, proper fraction;,, improper fractions;, mixed number. Write the following in decimal notation:,,.
Notes on floating point number, numerical computations and pitfalls
Notes on floating point number, numerical computations and pitfalls November 6, 212 1 Floating point numbers An n-digit floating point number in base β has the form x = ±(.d 1 d 2 d n ) β β e where.d 1
Skills Practice Skills Practice for Lesson 4.1
Skills Practice Skills Practice for Lesson.1 Name Date Thinking About Numbers Counting Numbers, Whole Numbers, Integers, Rational and Irrational Numbers Vocabulary Define each term in your own words. 1.