# CS 265. Computer Architecture. Wei Lu, Ph.D., P.Eng.

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CS 265 Computer Architecture Wei Lu, Ph.D., P.Eng. 1

2 Part 1: Data Representation Our goal: revisit and re-establish fundamental of mathematics for the computer architecture course Overview: what are bits & bytes? bit and arithmetic value? bit and hexadecimal? character set in PCs? how to do binary arithmetic operation (+,-,*,/)? integer representation in PCs: signed and unsigned binary and floating point floating point representation? order of byte in computers big endian vs. little endian 2

3 Conversion in Between Float and Binary 3

4 Floating point means a number with an integral part and a fractional part Converting floating point in between decimal and binary Place values: Floating point Integral part Fractional part Example: 4.5 (decimal) = (binary) 4

5 Converting floating point Example: 6.25 = Method: 6 = x 2 = x 2 = 1.0 (Stop when fractional part is 0) Example: (Integral part: convert in the usual way).5 (Fractional part: successive multiplication by 2) 6 = x 2 = x 2 = x 2 = x 2 = x 2 = 0.4 (repeats)

6 One more example to convert floating point Example: 6.4? How the floating binary number stored in computer? 6

7 How Float Represented in Computer 7

8 Floating point representation in computer Computers use a form of scientific notation for floatingpoint representation Numbers written in scientific notation have three basic components: Example: x use base 2 instead of base 10 Sign Mantissa Exponent Exponent specifies the order of magnitude in a base Mantissa specifies most significant part of value Sign specifies negative or positive 8

9 Floating point representation in computer Computer representation of a floating-point number consists of three fixed-size fields: This is the standard arrangement of these fields. 9

10 Floating point representation in computer (1) The one-bit sign field is the sign of the stored value. (2) The size of the exponent field, determines the range of values that can be represented. (3) The size of the significand determines the precision of the representation. 10

11 Floating point representation in computer single-precision (32 bits) floating point representation defined by IEEE standard sign bit "biased" exponent (8 bits) "normalized" mantissa (23 bits) Floating point = (-1) s x (1.m) x 2 (e-bias) Notice that the 1 in 1.m is always assumed. where, bias =

12 Floating point representation in computer double-precision (64 bits) floating point representation defined by IEEE standard sign bit "biased" exponent (11 bits) "normalized" mantissa (52 bits) Floating point = (-1) s x (1.m) x 2 (e-bias) where, bias =

13 Floating point representation: example 1 Given decimal real number 6.25, what is its single precision floating number stored in PC? Step (decimal) = (binary) Step 2. Move the radix point until a single 1 appears on the left, and multiply by the corresponding power of x2 2 so the sign bit is 0 (positive) the biased exponent is = 129 = = e and the normalized mantissa is 1001 (drop the 1, rest zero-fill) C80000 hexadecimal 13

14 Floating point representation: example 2 Given decimal real number 6.2, what is its single precision floating number stored in PC? Step (decimal) = (binary) Step 2. Move the radix point until a single 1 appears on the left, and multiply by the corresponding power of 2 = x2 2 so the sign bit is 0 (positive) the biased exponent is = 129= and the normalized mantissa is C66666 hexadecimal 14

15 Floating point representation: example 3 Double value in memory (in hex): what s its binary: what s its decimal? s (bit 64) = 0 = positive number e (bits 52 to 62) = = = 1026 m (bits 0 to 51) = Value = (-1) 0 x (1.01) b x 2 ( ) = 1x ( )x2 3 =(1+1/4)x8=8+2=10 15

16 Floating point representation: example 4 What decimal floating-point number is represented by 0xC ? Sign Biased exponent Normalized mantissa so the sign is negative the unbiased exponent is = 4 and the unnormalized mantissa is (add the 1 left of the radix point). Move the radix point 4 places to the right: =

17 Byte Order: Big Endian vs. Little Endian 17

18 Byte order Byte ordering, or endianness, is a major computer architectural consideration Byte ordering describes how data stored in memory, which is visible and important to programmer More details later in the course, here we introduce basic concept 18

19 Byte order If we have a two-byte integer, the integer may be stored so that the least significant byte is followed by the most significant byte or vice versa. In little endian machines, the least significant byte is stored in lowest memory location. (DEC, IBM (Intel Pentium)) Big endian machines store the most significant byte in lowest memory. (Motorola, Sun SPARC) 19

20 Big and little endian: an example Given 32 bits (4 bytes) integer (or single-precision floating point numbers) 0x 90AB12CD, the 4 bytes are: 90, AB, 12, CD, How can we store these values into memory? Big Endian In big endian, we store the most significant byte in the smallest address. Here's how it would look: Little Endian In little endian, we store the least significant byte in the smallest address. Here's how it would look: Address Value AB CD Address Value 1000 CD AB

21 Endianness: consequence for programmer For JAVA programmer, we have to know: - Everything in Java binary format files is stored big-endian, i.e. MSB (Most Significant Byte) first. - JAVA is platform independent because and Java hides internal endianness from us no matter what internal byte order used (i.e. Mac is big-endian order and intel PC is little-endian) For C programmer, we have to know: - Most files compiled by C use little-endian in windows PC For both C and JAVA programmer, we have to fix the byte order problem when transferring data files between computers for which the byte ordering differs 21

22 Summary: conceptual issues Bits grouped into sets to represent Integers Characters Floating point values Signed integers can be represented as Sign magnitude One's complement Two's complement Bytes of integer can be numbered in Big-endian order and Little-endian order 22

23 Summary: technical issues Conversion in between Binary, Hexadecimal, Decimal, Single Float and Double Signed Integer represented by Sign magnitude One's complement Two's complement The consequence of byte order to our programmers 23

24 Thank you for your attendance Any questions? 24

### Floating Point Arithmetic

Floating Point Arithmetic CS 365 Floating-Point What can be represented in N bits? Unsigned 0 to 2 N 2s Complement -2 N-1 to 2 N-1-1 But, what about? very large numbers? 9,349,398,989,787,762,244,859,087,678

### IEEE-754 floating-point

IEEE-754 floating-point Real and floating-point numbers Real numbers R form a continuum - Rational numbers are a subset of the reals - Some numbers are irrational, e.g. π Floating-point numbers are an

### Floating-point Arithmetic. where you sum up the integer to the left of the decimal point and the fraction to the right.

Floating-point Arithmetic Reading: pp. 312-328 Floating-Point Representation Non-scientific floating point numbers: A non-integer can be represented as: 2 4 2 3 2 2 2 1 2 0.2-1 2-2 2-3 2-4 where you sum

### Number 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,

### Module 2: Computer Arithmetic

Module 2: Computer Arithmetic 1 B O O K : C O M P U T E R O R G A N I Z A T I O N A N D D E S I G N, 3 E D, D A V I D L. P A T T E R S O N A N D J O H N L. H A N N E S S Y, M O R G A N K A U F M A N N

### Inf2C - 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

### l l l l l l l Base 2; each digit is 0 or 1 l Each bit in place i has value 2 i l Binary representation is used in computers

198:211 Computer Architecture Topics: Lecture 8 (W5) Fall 2012 Data representation 2.1 and 2.2 of the book Floating point 2.4 of the book Computer Architecture What do computers do? Manipulate stored information

### IT 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

### 1. 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

### Floating point. Today! IEEE Floating Point Standard! Rounding! Floating Point Operations! Mathematical properties. Next time. !

Floating point Today! IEEE Floating Point Standard! Rounding! Floating Point Operations! Mathematical properties Next time! The machine model Chris Riesbeck, Fall 2011 Checkpoint IEEE Floating point Floating

### ECE 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

### 15213 Recitation 2: Floating Point

15213 Recitation 2: Floating Point 1 Introduction This handout will introduce and test your knowledge of the floating point representation of real numbers, as defined by the IEEE standard. This information

### 10.1. 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

### 3.5 Floating Point: Overview

3.5 Floating Point: Overview Floating point (FP) numbers Scientific notation Decimal scientific notation Binary scientific notation IEEE 754 FP Standard Floating point representation inside a computer

### Floating-Point Arithmetic

Floating-Point Arithmetic if ((A + A) - A == A) { SelfDestruct() } Reading: Study Chapter 3. L12 Multiplication 1 Approximating Real Numbers on Computers Thus far, we ve entirely ignored one of the most

### Floating Point. The World is Not Just Integers. Programming languages support numbers with fraction

1 Floating Point The World is Not Just Integers Programming languages support numbers with fraction Called floating-point numbers Examples: 3.14159265 (π) 2.71828 (e) 0.000000001 or 1.0 10 9 (seconds in

### Floating Point Puzzles The course that gives CMU its Zip! Floating Point Jan 22, IEEE Floating Point. Fractional Binary Numbers.

class04.ppt 15-213 The course that gives CMU its Zip! Topics Floating Point Jan 22, 2004 IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Floating Point Puzzles For

### CHAPTER 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

### Signed 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,

### Floating Point Numbers. Lecture 9 CAP

Floating Point Numbers Lecture 9 CAP 3103 06-16-2014 Review of Numbers Computers are made to deal with numbers What can we represent in N bits? 2 N things, and no more! They could be Unsigned integers:

### Course 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

### ECE2049: Embedded Computing in Engineering Design C Term Spring Lecture #3: Of Integers and Endians (pt. 2)

ECE2049: Embedded Computing in Engineering Design C Term Spring 2018 Lecture #3: Of Integers and Endians (pt. 2) Reading for Today: Davies Ch 2, MSP430 User's Guide Ch 6.1, 6.3 Reading for Next Class:

### Computer Systems Programming. Practice Midterm. Name:

Computer Systems Programming Practice Midterm Name: 1. (4 pts) (K&R Ch 1-4) What is the output of the following C code? main() { int i = 6; int j = -35; printf( %d %d\n,i++, ++j); i = i >

### Systems I. Floating Point. Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties

Systems I Floating Point Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties IEEE Floating Point IEEE Standard 754 Established in 1985 as uniform standard for

### Chapter 2 Float Point Arithmetic. Real Numbers in Decimal Notation. Real Numbers in Decimal Notation

Chapter 2 Float Point Arithmetic Topics IEEE Floating Point Standard Fractional Binary Numbers Rounding Floating Point Operations Mathematical properties Real Numbers in Decimal Notation Representation

### CHAPTER 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

### CMSC 313 Lecture 03 Multiple-byte data big-endian vs little-endian sign extension Multiplication and division Floating point formats Character Codes

Multiple-byte data CMSC 313 Lecture 03 big-endian vs little-endian sign extension Multiplication and division Floating point formats Character Codes UMBC, CMSC313, Richard Chang 4-5 Chapter

### System Programming CISC 360. Floating Point September 16, 2008

System Programming CISC 360 Floating Point September 16, 2008 Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Powerpoint Lecture Notes for Computer Systems:

### Data Representation Floating Point

Data Representation Floating Point CSCI 2400 / ECE 3217: Computer Architecture Instructor: David Ferry Slides adapted from Bryant & O Hallaron s slides via Jason Fritts Today: Floating Point Background:

### Floating point. Today. IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Next time.

Floating point Today IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Next time The machine model Fabián E. Bustamante, Spring 2010 IEEE Floating point Floating point

### TUTORIAL -1 COMPUTER ORGANISATION TIT-402

TUTORIAL -1 COMPUTER ORGANISATION TIT-402 Q1.(43) 10 = (? ) 2 s (GATE 2000) a)01010101 b)11010101 c)00101011 d)10101011 Q2.The 2 s complement representation of (-539) 10 in hexadecimal is (GATE 2001) a)abe

### The Sign consists of a single bit. If this bit is '1', then the number is negative. If this bit is '0', then the number is positive.

IEEE 754 Standard - Overview Frozen Content Modified by on 13-Sep-2017 Before discussing the actual WB_FPU - Wishbone Floating Point Unit peripheral in detail, it is worth spending some time to look at

### Digital Computers and Machine Representation of Data

Digital Computers and Machine Representation of Data K. Cooper 1 1 Department of Mathematics Washington State University 2013 Computers Machine computation requires a few ingredients: 1 A means of representing

### CS101 Lecture 04: Binary Arithmetic

CS101 Lecture 04: Binary Arithmetic Binary Number Addition Two s complement encoding Briefly: real number representation Aaron Stevens (azs@bu.edu) 25 January 2013 What You ll Learn Today Counting in binary

### Numeric Variable Storage Pattern

Numeric Variable Storage Pattern Sreekanth Middela Srinivas Vanam Rahul Baddula Percept Pharma Services, Bridgewater, NJ ABSTRACT This paper presents the Storage pattern of Numeric Variables within the

### Name: CMSC 313 Fall 2001 Computer Organization & Assembly Language Programming Exam 1. Question Points I. /34 II. /30 III.

CMSC 313 Fall 2001 Computer Organization & Assembly Language Programming Exam 1 Name: Question Points I. /34 II. /30 III. /36 TOTAL: /100 Instructions: 1. This is a closed-book, closed-notes exam. 2. You

### ECE 2020B Fundamentals of Digital Design Spring problems, 6 pages Exam Two 26 February 2014

Instructions: This is a closed book, closed note exam. Calculators are not permitted. If you have a question, raise your hand and I will come to you. Please work the exam in pencil and do not separate

### Floating Point Numbers

Floating Point Numbers Summer 8 Fractional numbers Fractional numbers fixed point Floating point numbers the IEEE 7 floating point standard Floating point operations Rounding modes CMPE Summer 8 Slides

### Programming Studio #1 ECE 190

Programming Studio #1 ECE 190 Programming Studio #1 Announcements In Studio Assignment Introduction to Linux Command-Line Operations Recitation Floating Point Representation Binary & Hexadecimal 2 s Complement

### Characters, Strings, and Floats

Characters, Strings, and Floats CS 350: Computer Organization & Assembler Language Programming 9/6: pp.8,9; 9/28: Activity Q.6 A. Why? We need to represent textual characters in addition to numbers. Floating-point

### CS101 Introduction to computing Floating Point Numbers

CS101 Introduction to computing Floating Point Numbers A. Sahu and S. V.Rao Dept of Comp. Sc. & Engg. Indian Institute of Technology Guwahati 1 Outline Need to floating point number Number representation

### C NUMERIC FORMATS. Overview. IEEE Single-Precision Floating-point Data Format. Figure C-0. Table C-0. Listing C-0.

C NUMERIC FORMATS Figure C-. Table C-. Listing C-. Overview The DSP supports the 32-bit single-precision floating-point data format defined in the IEEE Standard 754/854. In addition, the DSP supports an

### Number Systems and Computer Arithmetic

Number Systems and Computer Arithmetic Counting to four billion two fingers at a time What do all those bits mean now? bits (011011011100010...01) instruction R-format I-format... integer data number text

### CS429: Computer Organization and Architecture

CS429: Computer Organization and Architecture Dr. Bill Young Department of Computer Sciences University of Texas at Austin Last updated: September 18, 2017 at 12:48 CS429 Slideset 4: 1 Topics of this Slideset

### ICS Instructor: Aleksandar Kuzmanovic TA: Ionut Trestian Recitation 2

ICS 2008 Instructor: Aleksandar Kuzmanovic TA: Ionut Trestian Recitation 2 Data Representations Sizes of C Objects (in Bytes) C Data Type Compaq Alpha Typical 32-bit Intel IA32 int 4 4 4 long int 8 4 4

### Number Systems & Encoding

Number Systems & Encoding Lecturer: Sri Parameswaran Author: Hui Annie Guo Modified: Sri Parameswaran Week2 1 Lecture overview Basics of computing with digital systems Binary numbers Floating point numbers

### CS367 Test 1 Review Guide

CS367 Test 1 Review Guide This guide tries to revisit what topics we've covered, and also to briefly suggest/hint at types of questions that might show up on the test. Anything on slides, assigned reading,

### Binary 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

### Binary representation of integer numbers Operations on bits

Outline Binary representation of integer numbers Operations on bits The Bitwise AND Operator The Bitwise Inclusive-OR Operator The Bitwise Exclusive-OR Operator The Ones Complement Operator The Left Shift

### The Perils of Floating Point

The Perils of Floating Point by Bruce M. Bush Copyright (c) 1996 Lahey Computer Systems, Inc. Permission to copy is granted with acknowledgement of the source. Many great engineering and scientific advances

### Byte Ordering. Jin-Soo Kim Computer Systems Laboratory Sungkyunkwan University

Byte Ordering Jin-Soo Kim (jinsookim@skku.edu) Computer Systems Laboratory Sungkyunkwan University http://csl.skku.edu Memory Model Physical memory DRAM chips can read/write 4, 8, 16 bits DRAM modules

### IBM 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

### Computer Systems C S Cynthia Lee

Computer Systems C S 1 0 7 Cynthia Lee 2 Today s Topics LECTURE: Floating point! Real Numbers and Approximation MATH TIME! Some preliminary observations on approximation We know that some non-integer numbers

### Description Hex M E V smallest value > largest denormalized negative infinity number with hex representation 3BB0 ---

CSE2421 HOMEWORK #2 DUE DATE: MONDAY 11/5 11:59pm PROBLEM 2.84 Given a floating-point format with a k-bit exponent and an n-bit fraction, write formulas for the exponent E, significand M, the fraction

### PROGRAMMAZIONE I A.A. 2017/2018

PROGRAMMAZIONE I A.A. 2017/2018 TYPES TYPES Programs have to store and process different kinds of data, such as integers and floating-point numbers, in different ways. To this end, the compiler needs to

### UNIT - I: COMPUTER ARITHMETIC, REGISTER TRANSFER LANGUAGE & MICROOPERATIONS

UNIT - I: COMPUTER ARITHMETIC, REGISTER TRANSFER LANGUAGE & MICROOPERATIONS (09 periods) Computer Arithmetic: Data Representation, Fixed Point Representation, Floating Point Representation, Addition and

### 17. Instruction Sets: Characteristics and Functions

17. Instruction Sets: Characteristics and Functions Chapter 12 Spring 2016 CS430 - Computer Architecture 1 Introduction Section 12.1, 12.2, and 12.3 pp. 406-418 Computer Designer: Machine instruction set

### 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 0 (0 volts) and one for logic

### Math 230 Assembly Programming (AKA Computer Organization) Spring 2008

Math 230 Assembly Programming (AKA Computer Organization) Spring 2008 MIPS Intro II Lect 10 Feb 15, 2008 Adapted from slides developed for: Mary J. Irwin PSU CSE331 Dave Patterson s UCB CS152 M230 L10.1

### Data 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

### 9/23/15. 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

### Computer Arithmetic Ch 8

Computer Arithmetic Ch 8 ALU Integer Representation Integer Arithmetic Floating-Point Representation Floating-Point Arithmetic 1 Arithmetic Logical Unit (ALU) (2) Does all work in CPU (aritmeettis-looginen

### Learning 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 base-2 abstract numerical system, or Binary system. This is a VERY

### COMPUTER HARDWARE. Instruction Set Architecture

COMPUTER HARDWARE Instruction Set Architecture Overview Computer architecture Operand addressing Addressing architecture Addressing modes Elementary instructions Data transfer instructions Data manipulation

### Bits, Bytes, and Integers Part 2

Bits, Bytes, and Integers Part 2 15-213: Introduction to Computer Systems 3 rd Lecture, Jan. 23, 2018 Instructors: Franz Franchetti, Seth Copen Goldstein, Brian Railing 1 First Assignment: Data Lab Due:

### CS321. Introduction to Numerical Methods

CS31 Introduction to Numerical Methods Lecture 1 Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506 0633 August 5, 017 Number

### JAVA Programming Fundamentals

Chapter 4 JAVA Programming Fundamentals By: Deepak Bhinde PGT Comp.Sc. JAVA character set Character set is a set of valid characters that a language can recognize. It may be any letter, digit or any symbol

### FLOATING 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 + 2-2 = 1*2 2 + 1*2

### Co-processor Math Processor. Richa Upadhyay Prabhu. NMIMS s MPSTME February 9, 2016

8087 Math Processor Richa Upadhyay Prabhu NMIMS s MPSTME richa.upadhyay@nmims.edu February 9, 2016 Introduction Need of Math Processor: In application where fast calculation is required Also where there

### Scientific Computing. Error Analysis

ECE257 Numerical Methods and Scientific Computing Error Analysis Today s s class: Introduction to error analysis Approximations Round-Off Errors Introduction Error is the difference between the exact solution

### CSC201, SECTION 002, Fall 2000: Homework Assignment #2

1 of 7 11/8/2003 7:34 PM CSC201, SECTION 002, Fall 2000: Homework Assignment #2 DUE DATE Monday, October 2, at the start of class. INSTRUCTIONS FOR PREPARATION Neat, in order, answers easy to find. Staple

### IEEE Standard 754 Floating Point Numbers

IEEE Standard 754 Floating Point Numbers Steve Hollasch / Last update 2005-Feb-24 IEEE Standard 754 floating point is the most common representation today for real numbers on computers, including Intel-based

### Signed 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

### Chapter 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

### On a 64-bit CPU. Size/Range vary by CPU model and Word size.

On a 64-bit CPU. Size/Range vary by CPU model and Word size. unsigned short x; //range 0 to 65553 signed short x; //range ± 32767 short x; //assumed signed There are (usually) no unsigned floats or doubles.

### 8/30/2016. In Binary, We Have A Binary Point. ECE 120: Introduction to Computing. Fixed-Point Representations Support Fractions

University of Illinois at Urbana-Champaign Dept. of Electrical and Computer Engineering ECE 120: Introduction to Computing Fixed- and Floating-Point Representations In Binary, We Have A Binary Point Let

### Digital 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

### Computer Organization and Levels of Abstraction

Computer Organization and Levels of Abstraction Announcements PS8 Due today PS9 Due July 22 Sound Lab tonight bring machines and headphones! Binary Search Today Review of binary floating point notation

### Objects and Types. COMS W1007 Introduction to Computer Science. Christopher Conway 29 May 2003

Objects and Types COMS W1007 Introduction to Computer Science Christopher Conway 29 May 2003 Java Programs A Java program contains at least one class definition. public class Hello { public static void

### Lecture Notes: Floating-Point Numbers

Lecture Notes: Floating-Point Numbers CS227-Scientific Computing September 8, 2010 What this Lecture is About How computers represent numbers How this affects the accuracy of computation Positional Number

### CS61C L10 MIPS Instruction Representation II, Floating Point I (6)

CS61C L1 MIPS Instruction Representation II, Floating Point I (1) inst.eecs.berkeley.edu/~cs61c CS61C : Machine Structures Lecture #1 Instruction Representation II, Floating Point I 25-1-3 There is one

### 4 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.

### Chapter 2: Number Systems

Chapter 2: Number Systems Logic circuits are used to generate and transmit 1s and 0s to compute and convey information. This two-valued number system is called binary. As presented earlier, there are many

### IEEE Floating Point Numbers Overview

COMP 40: Machine Structure and Assembly Language Programming (Fall 2015) IEEE Floating Point Numbers Overview Noah Mendelsohn Tufts University Email: noah@cs.tufts.edu Web: http://www.cs.tufts.edu/~noah

### 4/8/17. Admin. Assignment 5 BINARY. David Kauchak CS 52 Spring 2017

4/8/17 Admin! Assignment 5 BINARY David Kauchak CS 52 Spring 2017 Diving into your computer Normal computer user 1 After intro CS After 5 weeks of cs52 What now One last note on CS52 memory address binary

### Arithmetic. Chapter 3 Computer Organization and Design

Arithmetic Chapter 3 Computer Organization and Design Addition Addition is similar to decimals 0000 0111 + 0000 0101 = 0000 1100 Subtraction (negate) 0000 0111 + 1111 1011 = 0000 0010 Over(under)flow For

### Classes of Real Numbers 1/2. The Real Line

Classes of Real Numbers All real numbers can be represented by a line: 1/2 π 1 0 1 2 3 4 real numbers The Real Line { integers rational numbers non-integral fractions irrational numbers Rational numbers

### Floating Point. CSC207 Fall 2017

Floating Point CSC207 Fall 2017 Ariane 5 Rocket Launch Ariane 5 rocket explosion In 1996, the European Space Agency s Ariane 5 rocket exploded 40 seconds after launch. During conversion of a 64-bit to

### CS61C : Machine Structures

inst.eecs.berkeley.edu/~cs61c CS61C : Machine Structures Lecture #10 Instruction Representation II, Floating Point I 2005-10-03 Lecturer PSOE, new dad Dan Garcia www.cs.berkeley.edu/~ddgarcia #9 bears

### Floating Point Arithmetic

Floating Point Arithmetic Floating point numbers are frequently used in many applications. Implementation of arithmetic units such as adder, multiplier, etc for Floating point numbers are more complex

### The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Comp 4 Computer Organization Fall 26 Solutions for Problem Set #7 Problem. Bits of Floating-Point Represent the following in single-precision IEEE floating

### Unit 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

### Floating Point. What can be represented in N bits? 0 to 2N-1. 9,349,398,989,787,762,244,859,087, x 1067

MIPS Floating Point Operations Cptr280 Dr Curtis Nelson Floating Point What can be represented in N bits? Unsigned 2 s Complement 0 to 2N-1-2N-1 to 2N-1-1 But, what about- Very large numbers? 9,349,398,989,787,762,244,859,087,678

### 1010 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 1.2-1.4 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

### LAB 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 command-line argument args[0]

### Arithmetic Logic Unit

Arithmetic Logic Unit A.R. Hurson Department of Computer Science Missouri University of Science & Technology A.R. Hurson 1 Arithmetic Logic Unit It is a functional bo designed to perform "basic" arithmetic,

### Appendix. 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)

### CS/EE1012 INTRODUCTION TO COMPUTER ENGINEERING SPRING 2013 HOMEWORK I. Solve all homework and exam problems as shown in class and sample solutions

CS/EE2 INTRODUCTION TO COMPUTER ENGINEERING SPRING 23 DUE : February 22, 23 HOMEWORK I READ : Related portions of the following chapters : È Chapter È Chapter 2 È Appendix E ASSIGNMENT : There are eight