History & Binary Representation

Size: px
Start display at page:

Download "History & Binary Representation"

Transcription

1 History & Binary Representation C. R. da Cunha 1 1 Instituto de Física, Universidade Federal do Rio Grande do Sul, RS , Brazil. August 30, 2017 Abstract In this lesson we will review the history of electronics and the development of the first microprocessors. Keywords Microcontrollers; Microprocessors; Electronics; Digital. 1 History Let us begin with the history of semiconductors and contemporary electronics. Although it started in Germany, it flourished at the Bell Labs in the United States. creq@if.ufrgs.br 1

2 1874 Diode effect discovered by Ferdinand Braun at the University of Berlin Diode patented Bell Labs is founded MOS transistor is patented by Julius Lilienfeld Walter Brattain joins Bell Labs Another MOS patent by Oskar Heil Mervin Kelly becomes director of Bell Labs William Shockley joins Bell Labs John Bardeen joins Bell Labs Bardeen and Brattain conceive the Point Contact Transistor Shockley invents the Bipolar Junction Transistor First transistor computer built at the University of Manchester by Dick Grimsdale Transistors are fabricated in Si by Morris Tanenbaum at Bell Labs Nobel Prize for the invention of the transistor Shockley found Shockley Semiconductor Laboratory in Mountain View, CA Robert Noyce, Gordon Moore, Jean Hoerni and the other traitorous eight found Fairchild Semiconductors Jack Kilby at Texas Instruments and Robert Noyce at Fairchild Semiconductors independently develop the fist Integrated Circuit Planar transistor developed by Jean Hoerni at Fairchild Semiconductors Robert Noyce and Gordon Moore found Intel Corporation. Soon after Intel is founded, a new revolution starts by making electronic circuits mimick the behavior of machines such as Charles Babbage s difference engine from 1822, Alan Turing s a-machine from 1936, and John Atanasoff and Clifford Berry s automatic electronic digital computer from Soon after in 1946, John Brainerd at the University of Pennsylvania develop the Electronic Numerical Integrator and Computer (ENIAC). The first transistorized computer appeared in 1955 at the University of Manchester under the leadership of Tom Kilburn. It took another 15 years for the development of integrated processors. 2

3 Figure 1: DIP, PLCC and PGA packagings. Model Year W/Mem. [bits] [MHz] MIPS Applications Intel BCD/ Calculators Intel / Calc./Robots Intel BCD/ Calculators TI TMS / Calculators Intel / Cash Registers G.In. CP1600* / Video Games Zilog Z / Video Games MOS Tech / Video Games Intel MCS / Controllers Intel / Controllers Intel / PC-XT Intel kflops FPU Motorola / Mac/Video Games Intel / Controllers Intel / PC-AT Acorn ARM1/ / Computers Atmel AVR / 8 8 Controllers (*) Microchip s PIC was born from there. 2 Microprocessors vs. Microcontrollers Microprocessors are units responsible for processing the flow of information in an eletronic system, whereas microcontrollers are units that incorporate a processor, a memory and other subunits to perform intelligent operations. Simple microcontrollers can have packagings as simple as a DIP16 such as the Intel Most common microcontrollers are found in a DIP40 package, whereas modern microprocessors are found in packagings such as a 82 PLCC. Some of these packagings are shown in Fig. 1. 3

4 3 Binary Representation Let us begin our study of microcontrollers by reviewing the binary representation and operations. A binary number has only two mnemonics. We will use 1 to represent the high state, and 0 to represent the low state. Thus, we can have a binary number such as This can be converted to decimal by: X ÿ n b n 2 n 1 ˆ 2 0 ` 0 ˆ 2 1 ` 0 ˆ 2 2 ` 1 ˆ 2 3 ` 1 ˆ 2 4 ` 0 ˆ ` 8 ` The reverse operation can be constructed by successively dividing a decimal number by 2 and taking the remainder. For example: 25{2 12 ` rr1s 12{2 6 ` rr0s 6{2 3 ` rr0s 3{2 1 ` rr1s 1{2 0 ` rr1s, where r[x] is the remainder of the operation. Therefore, 25 can be represented as Now, let s take number 9 in binary: 9{2 4 ` rr1s 4{2 2 ` rr0s 2{2 1 ` rr0s 1{2 0 ` rr1s It needs 4 bits to be represented. We could have reached the same result by taking Log 2 p9q «3.17 Ñ 4. Thus, we can represent decimal numbers in groups of four bits. This is called binary coded decimal or BCD. For instance,25 would be represented as It requires more bits to be represented but it has the advantage of simplicity. 3.1 Fixed Point Representation How do we represent real numbers? There are two possibilities, the first is to use fixed point The trick here is to apply a binary point. For example, let us take again number 25 in binary (11001) and apply a point so that we have In this case we have: (1) (2) (3) 4

5 ˆ 2 2 ` 1 ˆ 2 1 ` 0 ˆ 2 0 ` 0 ˆ 2 1 ` 1 ˆ ` 2 ` In our case both 25 and 6.25 have exacly the same representation in binary. The only difference is the point s Complement And negative numbers? One strategy is to use the 1 s complement by just negating the expression. For example, for our 25 we would have and -25 would be in a 6 bit notation. For a 3 bit representation we would have: This has some problems. For example, the 1 s complement of 00 (0) is (-0). Furthermore, arithmetic operations become problematic. Let s take for example 3-1 in a 3 bit representation. This would be , which would produce 001 and a carry-out bit of 1. This has to be added to the result and we obtain 010, which is the expected result s Complement We can improve computations and use a 2 s complement by always adding one. For example, 3 in a 3 bit representation is 011, thus -3 in 1 s complement is 100. In 2 s complement we only have to add 1 and obtain 101. Thus, again in a 3 bit representation we would have: (4) (5) (6) 5

6 This way, not only we avoid the problem of -0 as we also simplify the arithmetics. Let s see that example of 3-1 again now in 2 s complement. This would be =010=2 with a carry bit that can be completely discarded. In 2 s complement, overflow can be detected if summing two numbers of the same sign produces a number with an opposite sign. Let us now see how this works for signed fixed point numbers: (7) Arithmetic For fixed point representation, addition and subtraction are exactly the same operations as we would do for integer numbers. For example: 0.5 ` ` ` Now let s see how it works for multiplication. For simplicity let us drop the radix point and take care of it later. 1.0 ˆ ˆ ` We must now account for the radix point. Since both multiplicands have points after the first bit, the result has to have the point after two bits. Therefore, the result is However, our representation includes only one bit for the fractional part and two for the integer part. We therefore must either truncate or round the result for the appropriate number of bits. It is simple in this case to just truncate it to 00.1, which in decimal is 0.5, the result that we expected. Let us now use two bits for each the integer and the fractional part. Our correspondence table becomes: (8) (9) 6

7 Let s multiply 0.25 ˆ 2.00: (10) 0001 ˆ 1000 ` (11) Our multiplicands have two bits for the fractional part. Therefore, the result should have four bits for its fractional part and we would get Truncating it we get 00.00, which is completely wrong. This happened because we did not account for the sign. One way to compensate for it is by expanding the bit sign: 1000 ˆ ` (12) We must place the point four bits from the right hand side end. Therefore, the truncated result becomes 11.10, which corresponds to -0.5 as expected. Why did we do this 1 filling operation? Because we are multiplying 1ˆ a negative number. We therefore must take its 2 s complement. We obtain it by 1 filling. This is known as sign extension. Let s calculate it now the other way around: 0001 ˆ 1000 ` (13) 7

8 Taking four bits for the radix point and truncating we get 11.10, which is the expected result. Note, however that we took 2 s complement. This happened because we are multiplying one argument by the sign bit. It is like multiplying the argument by -1. Let s now take a look at another example: 0.75 ˆ 0.75: 1101 ˆ ` (14) Placing the point we get If we simply truncate the result we get 1101 which corresponds to The right result would be and we have a quantization error which cannot be accounted for in this representation with a restricted number of bits. For the sake of practice, let s make the same calculation the other way around: Which is exactly the same value ˆ ` (15) 3.2 Floating Point With floating point we have a completely different story. In floating point, a number is represented as: S E 3 E 2 E 1 E 0 M 5 M 4 M 3 M 2 M 1 M 0, (16) where S is a bit for the sign, E is the exponent, and M is the mantissa. This can be represented as p 1q S M ˆ 2 E. Typically, in floating point representation, the mantissa is normalized. For example: ˆ 2 2 (17) ˆ 2 3. Also, the exponent is stored with a 127 bias according to IEEE 754 standard. This means that the exponent is added to 127 ( ) and then stored. For example, 12 (01100) is stored as 139 ( ), and -5 is stored as 122 ( ). 8

9 Let us now put it all together. A number in floating point notation is stored as SIGN EXPONENT MANTISSA. Also, for the mantissa, the 1 at the left hand side of the radix point is dropped. Let s look at some examples: ˆ `1.01 ˆ (18) Arithmetic Addition and subtraction are quite simple. We first must put both operands in the same exponent and them perform a standard sum of the mantissas maintaining the exponent. Multiplication and division are also simple. For the mantissa we must perform a standard multiplication. We sum the exponents and subtract the bias. The sign bit are just added and the carry is dropped. Thus for a floating point representation with 2 bits for the exponent with a bias 2 and 3 bits for the mantissa we have: 2.5 ` ` ˆ 2 1 ` 1.00 ˆ ` ˆ 2 1 ` 1.00 ˆ ˆ ˆ ˆ ˆ ˆ 2 1 ˆ 1.00 ˆ ˆ ˆ (19) (20) 9

Transistor and Integrated Circuits: History

Transistor and Integrated Circuits: History Course Objective Review and practice fundamental chemical engineering concepts (mass, energy, and momentum transport coupled with heterogeneous and homogeneous reactions and thermodynamics). Apply these

More information

ECE 340 Lecture 31 : Narrow Base Diode Class Outline:

ECE 340 Lecture 31 : Narrow Base Diode Class Outline: ECE 340 Lecture 31 : Narrow Base Diode Class Outline: Narrow-Base Diodes Things you should know when you leave Key Questions What is a narrow-base diode? How does current flow in a narrow-base diode? Quick

More information

Chapter 1. Binary Systems 1-1. Outline. ! Introductions. ! Number Base Conversions. ! Binary Arithmetic. ! Binary Codes. ! Binary Elements 1-2

Chapter 1. Binary Systems 1-1. Outline. ! Introductions. ! Number Base Conversions. ! Binary Arithmetic. ! Binary Codes. ! Binary Elements 1-2 Chapter 1 Binary Systems 1-1 Outline! Introductions! Number Base Conversions! Binary Arithmetic! Binary Codes! Binary Elements 1-2 3C Integration 傳輸與介面 IA Connecting 聲音與影像 Consumer Screen Phone Set Top

More information

The Transistor. Thomas J. Bergin Computer History Museum American University

The Transistor. Thomas J. Bergin Computer History Museum American University The Transistor Thomas J. Bergin Computer History Museum American University In the nineteenth century, scientists were rarely inventors: Samuel F.B. Morse, Alexander Graham Bell, Thomas Alva Edison In

More information

Complement Arithmetic

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

More information

How do computers represent numbers?

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

More information

Chapter 4 Number Representations

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

More information

Number Representation and Waveform Quantization

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.

More information

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. 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,

More information

Residue Number Systems Ivor Page 1

Residue Number Systems Ivor Page 1 Residue Number Systems 1 Residue Number Systems Ivor Page 1 7.1 Arithmetic in a modulus system The great speed of arithmetic in Residue Number Systems (RNS) comes from a simple theorem from number theory:

More information

0 volts. 2 volts. 5 volts

0 volts. 2 volts. 5 volts CS101 Binary Storage Devices and Boolean Logic Now that we have discussed number representation, why do computers use the binary representation and not something we are more familiar with, like decimal?

More information

Short column around Transistor. 12/22/2017 JC special topic

Short column around Transistor. 12/22/2017 JC special topic Short column around Transistor 12/22/2017 JC special topic Transistor, FET, CMOS D G S FET Transistor CMOS You can refer, such as, https://www.allaboutcircuits.com/textbook/semiconductors/ Timeline 1925:

More information

Chapter 1 :: From Zero to One

Chapter 1 :: From Zero to One Chapter 1 :: From Zero to One Digital Design and Computer Architecture David Money Harris and Sarah L. Harris Copyright 2007 Elsevier 1- Chapter 1 :: Topics Background The Game Plan The Art of Managing

More information

Field effect = Induction of an electronic charge due to an electric field Example: Planar capacitor

Field effect = Induction of an electronic charge due to an electric field Example: Planar capacitor JFETs AND MESFETs Introduction Field effect = Induction of an electronic charge due to an electric field Example: Planar capacitor Why would an FET made of a planar capacitor with two metal plates, as

More information

Prime Clocks. Michael Stephen Fiske. 10th GI Conference on Autonomous Systems October 23, AEMEA Institute. San Francisco, California

Prime Clocks. Michael Stephen Fiske. 10th GI Conference on Autonomous Systems October 23, AEMEA Institute. San Francisco, California Prime Clocks Michael Stephen Fiske 10th GI Conference on Autonomous Systems October 23, 2017 AEMEA Institute San Francisco, California Motivation for Prime Clocks The mindset of mainstream computer science

More information

Notes for Chapter 1 of. Scientific Computing with Case Studies

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

More information

Arithmetic and Error. How does error arise? How does error arise? Notes for Part 1 of CMSC 460

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

More information

ENG2410 Digital Design Introduction to Digital Systems. Fall 2017 S. Areibi School of Engineering University of Guelph

ENG2410 Digital Design Introduction to Digital Systems. Fall 2017 S. Areibi School of Engineering University of Guelph ENG2410 Digital Design Introduction to Digital Systems Fall 2017 S. Areibi School of Engineering University of Guelph Resources Chapter #1, Mano Sections 1.1 Digital Computers 1.2 Number Systems 1.3 Arithmetic

More information

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. 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.

More information

Hakim Weatherspoon CS 3410 Computer Science Cornell University

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

More information

ENGIN 112 Intro to Electrical and Computer Engineering

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

More information

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

More information

Chapter 1: Introduction and mathematical preliminaries

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,

More information

Answer three questions from Section A and five questions from Section B.

Answer three questions from Section A and five questions from Section B. L.35 PRE-LEAVING CERTIFICATE EXAMINATION, 2017 PHYSICS ORDINARY LEVEL TIME 3 HOURS Answer three questions from Section A and five questions from Section B. Relevant data are listed in the Formulae and

More information

Ex code

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

More information

ALU (3) - Division Algorithms

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)

More information

Quantum Computing 101. ( Everything you wanted to know about quantum computers but were afraid to ask. )

Quantum Computing 101. ( Everything you wanted to know about quantum computers but were afraid to ask. ) Quantum Computing 101 ( Everything you wanted to know about quantum computers but were afraid to ask. ) Copyright Chris Lomont, 2004 2 67 1 = 193707721 761838257287 Took American Mathematician Frank Nelson

More information

ECE260: Fundamentals of Computer Engineering

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

More information

Advanced VLSI Design Prof. A. N. Chandorkar Department of Electrical Engineering Indian Institute of Technology Bombay

Advanced VLSI Design Prof. A. N. Chandorkar Department of Electrical Engineering Indian Institute of Technology Bombay Advanced VLSI Design Prof. A. N. Chandorkar Department of Electrical Engineering Indian Institute of Technology Bombay Lecture - 01 Historical Perspective and Future Trends in CMOS VLSI Circuit and System

More information

The Magic of Negative Numbers in Computers

The Magic of Negative Numbers in Computers IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 12, Issue 4 Ver. I (Jul. - Aug.2016), PP 92-98 www.iosrjournals.org The Magic of Negative Numbers in Computers U. Sridevi

More information

Carry Look Ahead Adders

Carry Look Ahead Adders Carry Look Ahead Adders Lesson Objectives: The objectives of this lesson are to learn about: 1. Carry Look Ahead Adder circuit. 2. Binary Parallel Adder/Subtractor circuit. 3. BCD adder circuit. 4. Binary

More information

! Chris Diorio. ! Gaetano Borrielo. ! Carl Ebeling. ! Computing is in its infancy

! Chris Diorio. ! Gaetano Borrielo. ! Carl Ebeling. ! Computing is in its infancy Welcome to CSE370 Special Thanks!! Instructor: ruce Hemingway " Ts: ryan Nelson and John Hwang " Tool Specialist: Cory Crawford Lecture Materials:! Chris Diorio! Class web " http://www.cs.washington.edu/education/courses/370/currentqtr/

More information

CPE100: Digital Logic Design I

CPE100: Digital Logic Design I Chapter 1 Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu http://www.ee.unlv.edu/~b1morris/cpe100/ CPE100: Digital Logic Design I Section 1004: Dr. Morris From Zero to One Chapter 1 Background:

More information

CHAPTER 2 NUMBER SYSTEMS

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,

More information

Semiconductor Physics and Devices

Semiconductor Physics and Devices Syllabus Advanced Nano Materials Semiconductor Physics and Devices Textbook Donald A. Neamen (McGraw-Hill) Semiconductor Physics and Devices Seong Jun Kang Department of Advanced Materials Engineering

More information

Round-off Errors and Computer Arithmetic - (1.2)

Round-off Errors and Computer Arithmetic - (1.2) Round-off Errors and Comuter Arithmetic - (.). Round-off Errors: Round-off errors is roduced when a calculator or comuter is used to erform real number calculations. That is because the arithmetic erformed

More information

Introduction The Nature of High-Performance Computation

Introduction The Nature of High-Performance Computation 1 Introduction The Nature of High-Performance Computation The need for speed. Since the beginning of the era of the modern digital computer in the early 1940s, computing power has increased at an exponential

More information

Menu. Review of Number Systems EEL3701 EEL3701. Math. Review of number systems >Binary math >Signed number systems

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

More information

Digital Systems Roberto Muscedere Images 2013 Pearson Education Inc. 1

Digital Systems Roberto Muscedere Images 2013 Pearson Education Inc. 1 Digital Systems Digital systems have such a prominent role in everyday life The digital age The technology around us is ubiquitous, that is we don t even notice it anymore Digital systems are used in:

More information

NUMBERS AND CODES CHAPTER Numbers

NUMBERS AND CODES CHAPTER Numbers CHAPTER 2 NUMBERS AND CODES 2.1 Numbers When a number such as 101 is given, it is impossible to determine its numerical value. Some may say it is five. Others may say it is one hundred and one. Could it

More information

Introduction to Computer Systems

Introduction to Computer Systems Introduction to Computer Systems Department of Computer Science and Information Systems Lecturer: Steve Maybank sjmaybank@dcs.bbk.ac.uk Spring 2018 Week 1a: History of Computing 9 January 2018 Birkbeck

More information

Lecture 1: Circuits & Layout

Lecture 1: Circuits & Layout Lecture 1: Circuits & Layout Outline q A Brief History q CMOS Gate esign q Pass Transistors q CMOS Latches & Flip-Flops q Standard Cell Layouts q Stick iagrams 2 A Brief History q 1958: First integrated

More information

MATH Dr. Halimah Alshehri Dr. Halimah Alshehri

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

More information

CPE100: Digital Logic Design I

CPE100: Digital Logic Design I Chapter 1 Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu http://www.ee.unlv.edu/~b1morris/cpe100/ CPE100: Digital Logic Design I Section 1004: Dr. Morris From Zero to One Chapter 1 Background:

More information

14:332:231 DIGITAL LOGIC DESIGN. Why Binary Number System?

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

More information

QUADRATIC PROGRAMMING?

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

More information

Numbers and Arithmetic

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

More information

CMP 334: Seventh Class

CMP 334: Seventh Class CMP 334: Seventh Class Performance HW 5 solution Averages and weighted averages (review) Amdahl's law Ripple-carry adder circuits Binary addition Half-adder circuits Full-adder circuits Subtraction, negative

More information

We are here. Assembly Language. Processors Arithmetic Logic Units. Finite State Machines. Circuits Gates. Transistors

We are here. Assembly Language. Processors Arithmetic Logic Units. Finite State Machines. Circuits Gates. Transistors CSC258 Week 3 1 Logistics If you cannot login to MarkUs, email me your UTORID and name. Check lab marks on MarkUs, if it s recorded wrong, contact Larry within a week after the lab. Quiz 1 average: 86%

More information

0,..., r 1 = digits in radix r number system, that is 0 d i r 1 where m i n 1

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

More information

biologically-inspired computing lecture 14 Informatics luis rocha 2015 biologically Inspired computing INDIANA UNIVERSITY

biologically-inspired computing lecture 14 Informatics luis rocha 2015 biologically Inspired computing INDIANA UNIVERSITY lecture 14 -inspired Sections I485/H400 course outlook Assignments: 35% Students will complete 4/5 assignments based on algorithms presented in class Lab meets in I1 (West) 109 on Lab Wednesdays Lab 0

More information

14:332:231 DIGITAL LOGIC DESIGN. 2 s-complement Representation

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

More information

Numbers and Arithmetic

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

More information

What Every Programmer Should Know About Floating-Point Arithmetic DRAFT. Last updated: November 3, Abstract

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

More information

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 5. Ax = b.

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

More information

CSE 241 Digital Systems Spring 2013

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.

More information

Numbering Systems. Computational Platforms. Scaling and Round-off Noise. Special Purpose. here that is dedicated architecture

Numbering Systems. Computational Platforms. Scaling and Round-off Noise. Special Purpose. here that is dedicated architecture Computational Platforms Numbering Systems Basic Building Blocks Scaling and Round-off Noise Computational Platforms Viktor Öwall viktor.owall@eit.lth.seowall@eit lth Standard Processors or Special Purpose

More information

Graduate Institute of Electronics Engineering, NTU Basic Division Scheme

Graduate Institute of Electronics Engineering, NTU Basic Division Scheme Basic Division Scheme 台灣大學電子所吳安宇博士 2002 ACCESS IC LAB Outline Shift/subtract division algorithm. Programmed division. Restoring hardware dividers. Nonstoring and signed division. Radix-2 SRT divisioin.

More information

Digital Integrated Circuits A Design Perspective. Arithmetic Circuits. Jan M. Rabaey Anantha Chandrakasan Borivoje Nikolic.

Digital Integrated Circuits A Design Perspective. Arithmetic Circuits. Jan M. Rabaey Anantha Chandrakasan Borivoje Nikolic. Digital Integrated Circuits A Design Perspective Jan M. Rabaey Anantha Chandrakasan Borivoje Nikolic Arithmetic Circuits January, 2003 1 A Generic Digital Processor MEMORY INPUT-OUTPUT CONTROL DATAPATH

More information

Chapter 1 Error Analysis

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

More information

Chapter 1: Solutions to Exercises

Chapter 1: Solutions to Exercises 1 DIGITAL ARITHMETIC Miloš D. Ercegovac and Tomás Lang Morgan Kaufmann Publishers, an imprint of Elsevier, c 2004 Exercise 1.1 (a) 1. 9 bits since 2 8 297 2 9 2. 3 radix-8 digits since 8 2 297 8 3 3. 3

More information

EE 434 Lecture 3. Yield Issues and Historical Development

EE 434 Lecture 3. Yield Issues and Historical Development EE 434 Lecture 3 Yield Issues and Historical Development Quiz 1 How many minimum-sized MOS transistors can be placed on a square die that is 500µ on a side in a 90nm process? (Neglect any bonding pads

More information

Digital Systems Overview. Unit 1 Numbering Systems. Why Digital Systems? Levels of Design Abstraction. Dissecting Decimal Numbers

Digital Systems Overview. Unit 1 Numbering Systems. Why Digital Systems? Levels of Design Abstraction. Dissecting Decimal Numbers Unit Numbering Systems Fundamentals of Logic Design EE2369 Prof. Eric MacDonald Fall Semester 2003 Digital Systems Overview Digital Systems are Home PC XBOX or Playstation2 Cell phone Network router Data

More information

Adders, subtractors comparators, multipliers and other ALU elements

Adders, subtractors comparators, multipliers and other ALU elements CSE4: Components and Design Techniques for Digital Systems Adders, subtractors comparators, multipliers and other ALU elements Instructor: Mohsen Imani UC San Diego Slides from: Prof.Tajana Simunic Rosing

More information

FYSE410 DIGITAL ELECTRONICS [1] [2] [3] [4] [5] A number system consists of an ordered set of symbols (digits).

FYSE410 DIGITAL ELECTRONICS [1] [2] [3] [4] [5] A number system consists of an ordered set of symbols (digits). FYSE4 DIGITAL ELECTRONICS Litterature: LECTURE [] [] [4] [5] DIGITAL LOGIC CIRCUIT ANALYSIS & DESIGN Victor P. Nelson, H. Troy Nagle J. David Irwin, ill D. Carroll ISN --4694- DIGITAL DESIGN M. Morris

More information

E&CE 223 Digital Circuits & Systems. Winter Lecture Transparencies (Introduction) M. Sachdev

E&CE 223 Digital Circuits & Systems. Winter Lecture Transparencies (Introduction) M. Sachdev E&CE 223 Digital Circuits & Systems Winter 2004 Lecture Transparencies (Introduction) M. Sachdev 1 of 38 Course Information: People Instructor M. Sachdev, CEIT 4015, ext. 3370, msachdev@uwaterloo.ca Lab

More information

Jim Lambers MAT 610 Summer Session Lecture 2 Notes

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

More information

Gates and Logic: From Transistors to Logic Gates and Logic Circuits

Gates and Logic: From Transistors to Logic Gates and Logic Circuits Gates and Logic: From Transistors to Logic Gates and Logic Circuits Prof. Hakim Weatherspoon CS 3410 Computer Science Cornell University The slides are the product of many rounds of teaching CS 3410 by

More information

ELCT201: DIGITAL LOGIC DESIGN

ELCT201: DIGITAL LOGIC DESIGN ELCT2: DIGITL LOGIC DESIGN Dr. Eng. Haitham Omran, haitham.omran@guc.edu.eg Dr. Eng. Wassim lexan, wassim.joseph@guc.edu.eg Lecture Following the slides of Dr. hmed H. Madian ذو الحجة 438 ه Winter 27 2

More information

CprE 281: Digital Logic

CprE 281: Digital Logic CprE 28: Digital Logic Instructor: Alexander Stoytchev http://www.ece.iastate.edu/~alexs/classes/ Simple Processor CprE 28: Digital Logic Iowa State University, Ames, IA Copyright Alexander Stoytchev Digital

More information

GaN based transistors

GaN based transistors GaN based transistors S FP FP dielectric G SiO 2 Al x Ga 1-x N barrier i-gan Buffer i-sic D Transistors "The Transistor was probably the most important invention of the 20th Century The American Institute

More information

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 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

More information

How fast can we calculate?

How fast can we calculate? November 30, 2013 A touch of History The Colossus Computers developed at Bletchley Park in England during WW2 were probably the first programmable computers. Information about these machines has only been

More information

Binary Floating-Point Numbers

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

More information

Gates and Logic: From switches to Transistors, Logic Gates and Logic Circuits

Gates and Logic: From switches to Transistors, Logic Gates and Logic Circuits Gates and Logic: From switches to Transistors, Logic Gates and Logic Circuits Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University See: P&H ppendix C.2 and C.3 (lso, see C.0 and

More information

Gates and Logic: From Transistors to Logic Gates and Logic Circuits

Gates and Logic: From Transistors to Logic Gates and Logic Circuits Gates and Logic: From Transistors to Logic Gates and Logic Circuits Prof. Hakim Weatherspoon CS 3410 Computer Science Cornell University The slides are the product of many rounds of teaching CS 3410 by

More information

ECE 372 Microcontroller Design

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

More information

ELEC Digital Logic Circuits Fall 2014 Switching Algebra (Chapter 2)

ELEC Digital Logic Circuits Fall 2014 Switching Algebra (Chapter 2) ELEC 2200-002 Digital Logic Circuits Fall 2014 Switching Algebra (Chapter 2) Vishwani D. Agrawal James J. Danaher Professor Department of Electrical and Computer Engineering Auburn University, Auburn,

More information

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 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,

More information

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University

PowerPoints organized by Dr. Michael R. Gustafson II, Duke University Part 1 Chapter 4 Roundoff and Truncation Errors PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction

More information

Chapter 01: Introduction. Lesson 01 Evolution Computers Part 1: Mechanical Systems, Babbage method of Finite Difference Engine and Turing Hypothesis

Chapter 01: Introduction. Lesson 01 Evolution Computers Part 1: Mechanical Systems, Babbage method of Finite Difference Engine and Turing Hypothesis Chapter 01: Introduction Lesson 01 Evolution Computers Part 1: Mechanical Systems, Babbage method of Finite Difference Engine and Turing Hypothesis Objective Understand how mechanical computation systems

More information

Can You Count on Your Computer?

Can You Count on Your Computer? Can You Count on Your Computer? Professor Nick Higham School of Mathematics University of Manchester higham@ma.man.ac.uk http://www.ma.man.ac.uk/~higham/ p. 1/33 p. 2/33 Counting to Six I asked my computer

More information

EE260: Digital Design, Spring n Digital Computers. n Number Systems. n Representations. n Conversions. n Arithmetic Operations.

EE260: Digital Design, Spring n Digital Computers. n Number Systems. n Representations. n Conversions. n Arithmetic Operations. EE 260: Introduction to Digital Design Number Systems Yao Zheng Department of Electrical Engineering University of Hawaiʻi at Mānoa Overview n Digital Computers n Number Systems n Representations n Conversions

More information

Numbers & Arithmetic. Hakim Weatherspoon CS 3410, Spring 2012 Computer Science Cornell University. See: P&H Chapter , 3.2, C.5 C.

Numbers & Arithmetic. Hakim Weatherspoon CS 3410, Spring 2012 Computer Science Cornell University. See: P&H Chapter , 3.2, C.5 C. Numbers & Arithmetic Hakim Weatherspoon CS 3410, Spring 2012 Computer Science Cornell University See: P&H Chapter 2.4-2.6, 3.2, C.5 C.6 Example: Big Picture Computer System Organization and Programming

More information

Proposal to Improve Data Format Conversions for a Hybrid Number System Processor

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

More information

Introduction and mathematical preliminaries

Introduction and mathematical preliminaries Chapter Introduction and mathematical preliminaries Contents. Motivation..................................2 Finite-digit arithmetic.......................... 2.3 Errors in numerical calculations.....................

More information

Lecture 3, Performance

Lecture 3, Performance Repeating some definitions: Lecture 3, Performance CPI MHz MIPS MOPS Clocks Per Instruction megahertz, millions of cycles per second Millions of Instructions Per Second = MHz / CPI Millions of Operations

More information

Mathematical preliminaries and error analysis

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

More information

Lecture 3, Performance

Lecture 3, Performance Lecture 3, Performance Repeating some definitions: CPI Clocks Per Instruction MHz megahertz, millions of cycles per second MIPS Millions of Instructions Per Second = MHz / CPI MOPS Millions of Operations

More information

EECS150 - Digital Design Lecture 27 - misc2

EECS150 - Digital Design Lecture 27 - misc2 EECS150 - Digital Design Lecture 27 - misc2 May 1, 2002 John Wawrzynek Spring 2002 EECS150 - Lec27-misc2 Page 1 Outline Linear Feedback Shift Registers Theory and practice Simple hardware division algorithms

More information

Optimizing MPC for robust and scalable integer and floating-point arithmetic

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,

More information

quantum mechanics is a hugely successful theory... QSIT08.V01 Page 1

quantum mechanics is a hugely successful theory... QSIT08.V01 Page 1 1.0 Introduction to Quantum Systems for Information Technology 1.1 Motivation What is quantum mechanics good for? traditional historical perspective: beginning of 20th century: classical physics fails

More information

1.0 Introduction to Quantum Systems for Information Technology 1.1 Motivation

1.0 Introduction to Quantum Systems for Information Technology 1.1 Motivation QSIT09.V01 Page 1 1.0 Introduction to Quantum Systems for Information Technology 1.1 Motivation What is quantum mechanics good for? traditional historical perspective: beginning of 20th century: classical

More information

DSP Design Lecture 2. Fredrik Edman.

DSP Design Lecture 2. Fredrik Edman. DSP Design Lecture Number representation, scaling, quantization and round-off Noise Fredrik Edman fredrik.edman@eit.lth.se Representation of Numbers Numbers is a way to use symbols to describe and model

More information

1 Floating point arithmetic

1 Floating point arithmetic Introduction to Floating Point Arithmetic Floating point arithmetic Floating point representation (scientific notation) of numbers, for example, takes the following form.346 0 sign fraction base exponent

More information

EE 5211 Analog Integrated Circuit Design. Hua Tang Fall 2012

EE 5211 Analog Integrated Circuit Design. Hua Tang Fall 2012 EE 5211 Analog Integrated Circuit Design Hua Tang Fall 2012 Today s topic: 1. Introduction to Analog IC 2. IC Manufacturing (Chapter 2) Introduction What is Integrated Circuit (IC) vs discrete circuits?

More information

Digital Systems and Information Part II

Digital Systems and Information Part II Digital Systems and Information Part II Overview Arithmetic Operations General Remarks Unsigned and Signed Binary Operations Number representation using Decimal Codes BCD code and Seven-Segment Code Text

More information

Multiplication of signed-operands

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

More information

1.10 (a) Function of AND, OR, NOT, NAND & NOR Logic gates and their input/output.

1.10 (a) Function of AND, OR, NOT, NAND & NOR Logic gates and their input/output. Chapter 1.10 Logic Gates 1.10 (a) Function of AND, OR, NOT, NAND & NOR Logic gates and their input/output. Microprocessors are the central hardware that runs computers. There are several components that

More information