EE 3170 Microcontroller Applications

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "EE 3170 Microcontroller Applications"

Transcription

1 EE 37 Microcontroller Applications Lecture 3 : Digital Computer Fundamentals - Number Representation (.) Based on slides for ECE37 by Profs. Sloan, Davis, Kieckhafer, Tan, and Cischke Number Representation Integers in different radices (bases) decimal, binary, octal, hexadecimal Unsigned Binary Numbers Twos Complement Signed Binary Numbers Definition, Properties, Arithmetic The ASCII Character Code EE37/CC/Lecture#3 EE37/CC/Lecture#3 2 Preliminaries Three Systems of Number Representation Radix = the base of a number system Given radix k : one digit has one of k values need k symbols for values [ k-] Common Radices: decimal: k =, symbols = [ 9] binary: k = 2, symbols = [,] octal: k = 8, symbols = [...7] hexadecimal: k = 6, symbols = [ 9, a, b, c, d, e, f ] Three systems of representing unsigned numbers, all of them positional, but each based on a different radix, or base: Decimal (radix-): This is the standard representation that you re used to. Binary (radix-2): This is the actual representation used in all digital systems at the hardware level. Hexadecimal, or hex, (radix-6): This is closely related to binary, often used to represent values in digital systems using fewer digits than required by binary. The octal (radix-8) representation is also common, but we won t use it in this course. EE37/CC/Lecture#3 3 EE37/CC/Lecture#3 4

2 Binary Numbers Binary to Decimal Exercise Binary (base 2) numbers are arranged like decimal numbers in positions. Each position stands for a power of 2. Example: 2 is x2 5 + x2 4 + x2 3 + x2 2 + x2 + x2 = = 53 Changing from binary to decimal just means adding the powers of 2 for which the position has a. 2 A. 239 B. 247 C. 25 D. 253 E. 255 is EE37/CC/Lecture#3 5 EE37/CC/Lecture#3 6 How Do We Change from Decimal to Binary? Divide the decimal number repeatedly by 2. Put the remainders from right to left. 53/2 = 26 R 26/2 = 3 R 3/2 = 6 R 6/2 = 3 R 3/2 = R /2 = R Decimal to Binary Exercise 99 is A. B. C. D. E. 53 = 2 EE37/CC/Lecture#3 7 EE37/CC/Lecture#3 8 2

3 How Many Bits Do We Need? The range of unsigned eight-bit binary numbers is: The range of unsigned 6-bit binary numbers is: The range of unsigned n-bit binary numbers is: range = [, 2 n -] How Many Bits Do We Need? The number of binary digits (bits) we need depends on the size of the number we want to represent. 8 bits gives 256 numbers 6 bits gives 65.5k numbers 24 bits gives 6.7M numbers 32 bits gives 4.3 G numbers EE37/CC/Lecture#3 9 EE37/CC/Lecture#3 Observations Computer Prefixes Modern computers use multiples of 8 bits. Sometimes these have special names. 8 bits 6 bits 32 bits byte word Standard computer prefixes are based on powers of 2 rather than powers of and some are pronounced differently. Symbol K M G T Pronounced kay meg gig tera Power of bits long word P peta 5 We will mainly use 8 and 6 bits in EE37. E exa 6 Know at least the first four of these. EE37/CC/Lecture#3 EE37/CC/Lecture#3 2 3

4 Grouping Bits into Hex Long bit strings are hard to remember recognize work with Hence we group four bits together in a hexadecimal (base 6) digit or nibble. Hex Decimal Binary A B C 2 D 3 E 4 F 5 EE37/CC/Lecture#3 3 How Do We Change Binary to Hex? Starting at right, group bits by four. Change each group into its hex number. Put $ (Motorola s hex designator) in front. % is Motorola s binary designator. Decimal is default (no designator.) Note: Just pad on the left with enough s to get a multiple of four bits. Example: % D E 9 B $DE9B EE37/CC/Lecture#3 4 Binary to Hex Exercise % is A. $B586 B. $BD8E C. $E8DB How Do We Change Hex to Binary? Replace each hex digit by its 4-bit binary equivalent. Put a % (Motorola s binary designator) in front. Example: $A476 A % EE37/CC/Lecture#3 5 EE37/CC/Lecture#3 6 4

5 Hex to Binary Exercise Summary: Conversion Between Radices $28A7 is A. % B. % C. % From: binary or hexadecimal To: decimal use the radix expansions Binary to or from Hex Each hex digit = exactly 4 binary bits. So, one (8-bit) byte can hold 2 hex digits Simply collect bits into groups of 4 (starting at right): EE37/CC/Lecture#3 7 EE37/CC/Lecture#3 8 Boolean Values Unsigned Numbers Two states ON OFF +5 V V open closed clockwise counterclockwise all s any nonzero value EE37/CC/Lecture#3 9 Unsigned numbers Use all bits for magnitude Can be thought of as positive. b7 b6 b5 b4 b3 b2 b b Each bit represents 2 n Range is to 255 (2 8 -) Observations If the right bit is, the number is even. If the right n bits are, the number is divisible by 2 n If the left bit of an 8-bit number is, the number is at least 28. EE37/CC/Lecture#3 2 5

6 How Can We Represent Signed Numbers? We need one bit for the sign. Use the leftmost bit. = +, = - Other bits are magnitude. Three possibilities shown with 4 bits. +5 =, -5 = sign-magnitude +5 =, -5 = s complement +5 =, -5 = 2 s complement Observations? Twos Complement Signed Numbers Definition: negation = complement all bits then add. The most-significant bit is the sign-bit = for positive numbers = for negative numbers Range = [-2 n-, 2 n- -] Uses the same arithmetic hardware as unsigned overflow bits are discarded Subtracting a number is equivalent to adding its twos complement negative number (pp.8) EE37/CC/Lecture#3 2 EE37/CC/Lecture# bit 2 s complement How Can We Find 2 s Complement Representations? 2 s complement is used in modern computers. Consider 3-bit 2 s complement. Range is -4 to +3. Number b3 b2 b For positive numbers Write binary equivalent Add s at left as needed for total of n bits. Example: 8-bit 2 s complement representation of = = in 2 s complement -4 EE37/CC/Lecture#3 23 EE37/CC/Lecture#3 24 6

7 How Can We Find 2 s Complement Representations? For negative numbers Write binary equivalent of positive number Add s at left as needed for n bits total. Complement each bit. ( ; ) Add. Example: 8-bit 2 s complement rep. of = = in 2 s complement -7 = + = EE37/CC/Lecture# s Complement Exercise -23 in 6-bit 2 s complement is A. B. C. D. EE37/CC/Lecture#3 26 Observations To find the magnitude of a negative 2 s complement number left bit = Complement each bit Add Example: + = or 7 2 s complement represents both positive and negative numbers. Observations We can negate a positive 2 s complement number with exactly the same procedure. Complement each bit. Add. Example: + = or -7 decimal EE37/CC/Lecture#3 27 EE37/CC/Lecture#3 28 7

8 Magnitude of 2 s Complement Number The magnitude of the 6-bit 2 s complement number is A. B. 2 C. 3 D. 4 E. 22 Range of 2 s Complement Numbers Range of n-bit 2 s complement numbers is -2 n- to 2 n- -. Examples: 4-bit to to +7 8-bit to to +27 EE37/CC/Lecture#3 29 EE37/CC/Lecture#3 3 Range of 6-bit 2 s Complement A. -64 to +63 B. -64 to +64 C. -32 to +32 D. -32 to +3 E. -3 to +32 How Do We Change 8-bit 2 s Complement to More Bits? Sign Extension Copy the sign bit to the left. Examples: 8-bit to 6-bit 8-bit to 32-bit EE37/CC/Lecture#3 3 EE37/CC/Lecture#3 32 8

9 How Do We Encode Decimal Numbers? Binary Coded Decimal (BCD) is handy for decimal numbers. Each digit is encoded in 4-bit binary. Why BCD? Convenient for certain I/O devices that work w/ decimal numbers Examples: 97 = BCD How Do We Know if Numbers Are Signed or Unsigned? You, the programmer, decide whether numbers are signed or unsigned. The computer doesn t know what a bit pattern means until you tell it. The bit pattern could be a character. 2 = BCD EE37/CC/Lecture#3 33 EE37/CC/Lecture#3 34 Character Representations Characters must be represented as binary numbers upper-case, lower-case, Numerals, Punctuation, control characters (non-printing) How many Bits? 7 bits : 28 Characters = EBCDIC & ASCII 8 bits : 256 Characters = Extended ASCII 6 bits : 26 Characters = Unicode How Do We Represent Characters? ASCII (American Standard Code for Information Interchange) Standard (7-bit) 27 characters Extended (8-bit) 255 characters Example: M = $4D = % from table EE37/CC/Lecture#3 35 EE37/CC/Lecture#3 36 9

10 ASCII Character Code Control Characters: First 32 values [ 2 6 ] Numerals: number X = 3X 6 i.e. [ ] very easy to recognize and decode Upper Case = [4 6 5A 6 ] = [,,] Lower Case = [6 6 7A 6 ] = [,,] Punctuation is scattered through remaining patterns ASCII EE37/CC/Lecture#3 37 EE37/CC/Lecture#3 38 Summary ASCII Extended Converting numbers among decimal, binary and hex (EE27) Two s complement numbers (EE27) Ranges of signed and unsigned numbers How to represent characters EE37/CC/Lecture#3 39 EE37/CC/Lecture#3 4

By the end of the lecture, you should be able to:

By the end of the lecture, you should be able to: Extra Lecture: Number Systems Objectives - To understand: Base of number systems: decimal, binary, octal and hexadecimal Textual information stored as ASCII Binary addition/subtraction, multiplication

More information

Lecture 2. Binary and Hexadecimal Numbers

Lecture 2. Binary and Hexadecimal Numbers Lecture 2 Binary and Hexadecimal Numbers Purpose: Review binary and hexadecimal number representations Convert directly from one base to another base Review addition and subtraction in binary representations

More information

Signed Binary Arithmetic

Signed Binary Arithmetic Signed Binary Arithmetic In the real world of mathematics, computers must represent both positive and negative binary numbers. For example, even when dealing with positive arguments, mathematical operations

More information

Encoding Systems: Combining Bits to form Bytes

Encoding Systems: Combining Bits to form Bytes Encoding Systems: Combining Bits to form Bytes Alphanumeric characters are represented in computer storage by combining strings of bits to form unique bit configuration for each character, also called

More information

2.1 Binary Numbers. 2.3 Number System Conversion. From Binary to Decimal. From Decimal to Binary. Section 2 Binary Number System Page 1 of 8

2.1 Binary Numbers. 2.3 Number System Conversion. From Binary to Decimal. From Decimal to Binary. Section 2 Binary Number System Page 1 of 8 Section Binary Number System Page 1 of 8.1 Binary Numbers The number system we use is a positional number system meaning that the position of each digit has an associated weight. The value of a given number

More information

Chapter 4: Computer Codes

Chapter 4: Computer Codes Slide 1/30 Learning Objectives In this chapter you will learn about: Computer data Computer codes: representation of data in binary Most commonly used computer codes Collating sequence 36 Slide 2/30 Data

More information

Number Representation

Number Representation Number Representation Number System :: The Basics We are accustomed to using the so-called decimal number system Ten digits ::,,,3,4,5,6,7,8,9 Every digit position has a weight which is a power of Base

More information

COMP2121: Microprocessors and Interfacing

COMP2121: Microprocessors and Interfacing Interfacing Lecture 3: Number Systems (I) http://www.cse.unsw.edu.au/~cs2121 Lecturer: Hui Wu Session 2, 2005 Overview Positional notation Decimal, hexadecimal and binary One complement Two s complement

More information

Oct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8

Oct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8 ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: -Our standard number system is base, also

More information

Chap 3 Data Representation

Chap 3 Data Representation Chap 3 Data Representation 3-11 Data Types How to representation and conversion between these data types? 3-11 Data Types : Number System Radix : Decimal : radix 10 Binary : radix 2 3-11 Data Types : Number

More information

Here 4 is the least significant digit (LSD) and 2 is the most significant digit (MSD).

Here 4 is the least significant digit (LSD) and 2 is the most significant digit (MSD). Number System Introduction Number systems provide the basis for all operations in information processing systems. In a number system the information is divided into a group of symbols; for example, 26

More information

1 Basic Computing Concepts (4) Data Representations

1 Basic Computing Concepts (4) Data Representations 1 Basic Computing Concepts (4) Data Representations The Binary System The Binary System is a way of writing numbers using only the digits 0 and 1. This is the method used by the (digital) computer. The

More information

الدكتور المھندس عادل مانع داخل

الدكتور المھندس عادل مانع داخل الدكتور المھندس عادل مانع داخل / میسان جامعة / كلیة الھندسة قسم الھندسة الكھرباي یة Chapter 1: Digital Systems Discrete Data Examples: 26 letters of the alphabet (A, B etc) 10 decimal digits (0, 1, 2 etc)

More information

Chapter II Binary Data Representation

Chapter II Binary Data Representation Chapter II Binary Data Representation The atomic unit of data in computer systems is the bit, which is actually an acronym that stands for BInary digit. It can hold only 2 values or states: 0 or 1, true

More information

Number Representation and Arithmetic in Various Numeral Systems

Number Representation and Arithmetic in Various Numeral Systems 1 Number Representation and Arithmetic in Various Numeral Systems Computer Organization and Assembly Language Programming 203.8002 Adapted by Yousef Shajrawi, licensed by Huong Nguyen under the Creative

More information

Number Representation

Number Representation Number Representation CS10001: Programming & Data Structures Pallab Dasgupta Professor, Dept. of Computer Sc. & Engg., Indian Institute of Technology Kharagpur Topics to be Discussed How are numeric data

More information

Data Representation in Computers

Data Representation in Computers Chapter 3 Data Representation in Computers After studying this chapter the student will be able to: *Learn about binary, octal, decimal and hexadecimal number systems *Learn conversions between two different

More information

Review of Number Systems The study of number systems is important from the viewpoint of understanding how data are represented before they can be processed by any digital system including a computer. Different

More information

Logic Design. Dr. Yosry A. Azzam

Logic Design. Dr. Yosry A. Azzam Logic Design Dr. Yosry A. Azzam Binary systems Chapter 1 Agenda Binary Systems : Binary Numbers, Binary Codes, Binary Logic ASCII Code (American Standard Code for Information Interchange) Boolean Algebra

More information

Number Systems & Encoding

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

More information

The largest has a 0 in the sign position and 0's in all other positions:

The largest has a 0 in the sign position and 0's in all other positions: 10.2 Sign Magnitude Representation Sign Magnitude is straight-forward method for representing both positive and negative integers. It uses the most significant digit of the digit string to indicate the

More information

Lecture 2: Number System

Lecture 2: Number System Lecture 2: Number System Today s Topics Review binary and hexadecimal number representation Convert directly from one base to another base Review addition and subtraction in binary representation Determine

More information

CPE 323 Data Types and Number Representations

CPE 323 Data Types and Number Representations CPE 323 Data Types and Number Representations Aleksandar Milenkovic Numeral Systems: Decimal, binary, hexadecimal, and octal We ordinarily represent numbers using decimal numeral system that has 10 as

More information

Digital Arithmetic. Digital Arithmetic: Operations and Circuits Dr. Farahmand

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

More information

P A R T DIGITAL TECHNOLOGY

P A R T DIGITAL TECHNOLOGY P A R T A DIGITAL TECHNOLOGY 1 CHAPTER NUMBERING SYSTEMS 1.0 INTRODUCTION This chapter discusses several important concepts including the binary, octal and hexadecimal numbering systems, binary data organization

More information

CHAPTER 2 Data Representation in Computer Systems

CHAPTER 2 Data Representation in Computer Systems CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 47 2.2 Positional Numbering Systems 48 2.3 Converting Between Bases 48 2.3.1 Converting Unsigned Whole Numbers 49 2.3.2 Converting Fractions

More information

Decimal Numbers: Base 10 Integer Numbers & Arithmetic

Decimal Numbers: Base 10 Integer Numbers & Arithmetic Decimal Numbers: Base 10 Integer Numbers & Arithmetic Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 )+(1x10 0 ) Ward 1 Ward 2 Numbers: positional notation Number

More information

NUMBERING SYSTEMS C HAPTER 1.0 INTRODUCTION 1.1 A REVIEW OF THE DECIMAL SYSTEM 1.2 BINARY NUMBERING SYSTEM

NUMBERING SYSTEMS C HAPTER 1.0 INTRODUCTION 1.1 A REVIEW OF THE DECIMAL SYSTEM 1.2 BINARY NUMBERING SYSTEM 12 Digital Principles Switching Theory C HAPTER 1 NUMBERING SYSTEMS 1.0 INTRODUCTION Inside today s computers, data is represented as 1 s and 0 s. These 1 s and 0 s might be stored magnetically on a disk,

More information

Solution for Homework 2

Solution for Homework 2 Solution for Homework 2 Problem 1 a. What is the minimum number of bits that are required to uniquely represent the characters of English alphabet? (Consider upper case characters alone) The number of

More information

Number Systems and. Data Representation

Number Systems and. Data Representation Number Systems and Data Representation 1 Lecture Outline Number Systems Binary, Octal, Hexadecimal Representation of characters using codes Representation of Numbers Integer, Floating Point, Binary Coded

More information

Data Representation. Representing Data

Data Representation. Representing Data Data Representation COMP 1002/1402 Representing Data A computer s basic unit of information is: a bit (Binary digit) An addressable memory cell is a byte (8 bits) Capable of storing one character 10101010

More information

Chapter No.5 DATA REPRESENTATION

Chapter No.5 DATA REPRESENTATION Chapter No.5 DATA REPRESENTATION Q.5.01 Complete the following statements. i) Data is a collection of ii) Data becomes information when properly. iii) Octal equivalent of binary number 1100010 is iv) 2

More information

CDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012

CDA 3200 Digital Systems. Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 CDA 3200 Digital Systems Instructor: Dr. Janusz Zalewski Developed by: Dr. Dahai Guo Spring 2012 Outline Data Representation Binary Codes Why 6-3-1-1 and Excess-3? Data Representation (1/2) Each numbering

More information

Data types. lecture 4

Data types. lecture 4 Data types lecture 4 Information in digital computers is represented using binary number system. The base, i.e. radix, of the binary system is 2. Other common number systems: octal (base 8), decimal (base

More information

EE 261 Introduction to Logic Circuits. Module #2 Number Systems

EE 261 Introduction to Logic Circuits. Module #2 Number Systems EE 261 Introduction to Logic Circuits Module #2 Number Systems Topics A. Number System Formation B. Base Conversions C. Binary Arithmetic D. Signed Numbers E. Signed Arithmetic F. Binary Codes Textbook

More information

EEE130 Digital Electronics I Lecture #2

EEE130 Digital Electronics I Lecture #2 EEE130 Digital Electronics I Lecture #2 -Number Systems, Operations and Codes- By Dr. Shahrel A. Suandi Topics to be discussed 2-1 Decimal Numbers 2-2 Binary Numbers 2-3 Decimal-to-Binary Conversion 2-4

More information

Lecture 1 Introduction, Numbers, and Number System Page 1 of 8

Lecture 1 Introduction, Numbers, and Number System Page 1 of 8 Lecture Introduction, Numbers and Number System Contents.. Number Systems (Appendix B)... 2. Example. Converting to Base 0... 2.2. Number Representation... 2.3. Number Conversion... 3. To convert a number

More information

CHAPTER 1 BINARY SYSTEM

CHAPTER 1 BINARY SYSTEM STUDENT COPY DIGITAL & MICROPROCESSORS 3 CHAPTER 1 BINARY SYSTEM Base Conversion: A number a n, a n 1 a 2, a 1 a 0 a 1 a 2 a 3 expressed in a base r system has coefficient multiplied by powers of r. n

More information

CSC 1103: Digital Logic. Lecture Six: Data Representation

CSC 1103: Digital Logic. Lecture Six: Data Representation CSC 1103: Digital Logic Lecture Six: Data Representation Martin Ngobye mngobye@must.ac.ug Mbarara University of Science and Technology MAN (MUST) CSC 1103 1 / 32 Outline 1 Digital Computers 2 Number Systems

More information

1 / 40. Data Representation. January 9 14, 2013

1 / 40. Data Representation. January 9 14, 2013 1 / 40 Data Representation January 9 14, 2013 Quick logistical notes In class exercises Bring paper and pencil (or laptop) to each lecture! Goals: break up lectures, keep you engaged chance to work through

More information

Data Representation. Data Representation, Storage, and Retrieval. Data Representation. Data Representation. Data Representation. Data Representation

Data Representation. Data Representation, Storage, and Retrieval. Data Representation. Data Representation. Data Representation. Data Representation , Storage, and Retrieval ULM/HHIM Summer Program Project 3, Day 3, Part 3 Digital computers convert the data they process into a digital value. Text Audio Images/Graphics Video Digitizing 00000000... 6/8/20

More information

Digital Fundamentals

Digital Fundamentals Digital Fundamentals with PLD Programming Floyd Chapter 2 29 Pearson Education Decimal Numbers The position of each digit in a weighted number system is assigned a weight based on the base or radix of

More information

Systems I: Computer Organization and Architecture

Systems I: Computer Organization and Architecture Systems I: Computer Organization and Architecture Lecture 2: Number Systems and Arithmetic Number Systems - Base The number system that we use is base : 734 = + 7 + 3 + 4 = x + 7x + 3x + 4x = x 3 + 7x

More information

Computer Number Systems

Computer Number Systems Computer Number Systems Thorne, Edition 2 : Section 1.3, Appendix I (Irvine, Edition VI : Section 1.3) SYSC3006 1 Starting from What We Already Know Decimal Numbers Based Number Systems : 1. Base defines

More information

Binary Representation. Number Systems. Positional Notation

Binary Representation. Number Systems. Positional Notation Binary Representation The basis of all digital data is binary representation. Binary - means two 1, 0 True, False Hot, Cold On, Off We must be able to handle more than just values for real world problems

More information

Binary Representation. Number Systems. Base 10, Base 2, Base 16. Positional Notation. Conversion of Any Base to Decimal.

Binary Representation. Number Systems. Base 10, Base 2, Base 16. Positional Notation. Conversion of Any Base to Decimal. Binary Representation The basis of all digital data is binary representation. Binary - means two 1, 0 True, False Hot, Cold On, Off We must be able to handle more than just values for real world problems

More information

Radix Number Systems. Number Systems. Number Systems 4/26/2010. basic idea of a radix number system how do we count:

Radix Number Systems. Number Systems. Number Systems 4/26/2010. basic idea of a radix number system how do we count: Number Systems binary, octal, and hexadecimal numbers why used conversions, including to/from decimal negative binary numbers floating point numbers character codes basic idea of a radix number system

More information

D r = d p-1 d p-2.. d 1 d 0.d -1 d -2. D -n. EECC341 - Shaaban

D r = d p-1 d p-2.. d 1 d 0.d -1 d -2. D -n. EECC341 - Shaaban Positional Number Systems A number system consists of an order set of symbols (digits) with relations defined for +,-,*, / The radix (or base) of the number system is the total number of digits allowed

More information

MT1 Number Systems. In general, the number a 3 a 2 a 1 a 0 in a base b number system represents the following number:

MT1 Number Systems. In general, the number a 3 a 2 a 1 a 0 in a base b number system represents the following number: MT1 Number Systems MT1.1 Introduction A number system is a well defined structured way of representing or expressing numbers as a combination of the elements of a finite set of mathematical symbols (i.e.,

More information

Codes and number systems

Codes and number systems Coding Codes and number systems Assume that you want to communicate with your friend with a flashlight in a night, what will you do? Introduction to Computer Yung-Yu Chuang light painting? What s the problem?

More information

Number Representation

Number Representation Number Representation COMP375 Computer Organization and darchitecture t How do we represent data in a computer? At the lowest level, a computer is an electronic machine. works by controlling the flow of

More information

Computer Science 281 Binary and Hexadecimal Review

Computer Science 281 Binary and Hexadecimal Review Computer Science 281 Binary and Hexadecimal Review 1 The Binary Number System Computers store everything, both instructions and data, by using many, many transistors, each of which can be in one of two

More information

2011, The McGraw-Hill Companies, Inc. Chapter 3

2011, The McGraw-Hill Companies, Inc. Chapter 3 Chapter 3 3.1 Decimal System The radix or base of a number system determines the total number of different symbols or digits used by that system. The decimal system has a base of 10 with the digits 0 through

More information

Binary Numbers. Bob Brown Information Technology Department Southern Polytechnic State University

Binary Numbers. Bob Brown Information Technology Department Southern Polytechnic State University Binary Numbers Bob Brown Information Technology Department Southern Polytechnic State University Positional Number Systems The idea of number is a mathematical abstraction. To use numbers, we must represent

More information

Switching Circuits & Logic Design

Switching Circuits & Logic Design Switching Circuits & Logic Design Jie-Hong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 2013 1 1 Number Systems and Conversion Babylonian number system (3100 B.C.)

More information

Review of Number Systems Binary, Octal, and Hexadecimal Numbers and Two's Complement

Review of Number Systems Binary, Octal, and Hexadecimal Numbers and Two's Complement Review of Number Systems Binary, Octal, and Hexadecimal Numbers and Two's Complement Topic 1: Binary, Octal, and Hexadecimal Numbers The number system we generally use in our everyday lives is a decimal

More information

Integer Numbers. The Number Bases of Integers Textbook Chapter 3

Integer Numbers. The Number Bases of Integers Textbook Chapter 3 Integer Numbers The Number Bases of Integers Textbook Chapter 3 Number Systems Unary, or marks: /////// = 7 /////// + ////// = ///////////// Grouping lead to Roman Numerals: VII + V = VVII = XII Better:

More information

Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 INTRODUCTION TO DIGITAL LOGIC

Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 INTRODUCTION TO DIGITAL LOGIC Levent EREN levent.eren@ieu.edu.tr A-306 Office Phone:488-9882 1 Number Systems Representation Positive radix, positional number systems A number with radix r is represented by a string of digits: A n

More information

Lecture 2: Number Representation

Lecture 2: Number Representation Lecture 2: Number Representation CSE 30: Computer Organization and Systems Programming Summer Session II 2011 Dr. Ali Irturk Dept. of Computer Science and Engineering University of California, San Diego

More information

2. Number Systems - Positional Number Systems (1) - 2. Number Systems - Binary Numbers - 2. Number Systems - Positional Number Systems (2) -

2. Number Systems - Positional Number Systems (1) - 2. Number Systems - Binary Numbers - 2. Number Systems - Positional Number Systems (2) - Sistemas Digitais I LESI - 2º ano Lesson 2 - Number Systems Prof. João Miguel Fernandes (miguel@di.uminho.pt) Dept. Informática - Positional Number Systems (1) - We use daily a positional number system.

More information

Number Systems Richard E. Haskell

Number Systems Richard E. Haskell NUMBER SYSTEMS D Number Systems Richard E. Haskell Data inside a computer are represented by binary digits or bits. The logical values of these binary digits are denoted by and, while the corresponding

More information

CSCC85 Spring 2006: Tutorial 0 Notes

CSCC85 Spring 2006: Tutorial 0 Notes CSCC85 Spring 2006: Tutorial 0 Notes Yani Ioannou January 11 th, 2006 There are 10 types of people in the world, those who understand binary, and those who don t. Contents 1 Number Representations 1 1.1

More information

Introduction Number Systems and Conversion

Introduction Number Systems and Conversion UNIT 1 Introduction Number Systems and Conversion Objectives 1. Introduction The first part of this unit introduces the material to be studied later. In addition to getting an overview of the material

More information

The string of digits 101101 in the binary number system represents the quantity

The string of digits 101101 in the binary number system represents the quantity Data Representation Section 3.1 Data Types Registers contain either data or control information Control information is a bit or group of bits used to specify the sequence of command signals needed for

More information

1. Convert the following binary exponential expressions to their 'English'

1. Convert the following binary exponential expressions to their 'English' Answers to Practice Problems Practice Problems - Integer Number System Conversions 1. Convert the decimal integer 427 10 into the following number systems: a. 110101011 2 c. 653 8 b. 120211 3 d. 1AB 16

More information

A B C

A B C Data Representation Module 2 CS 272 Sam Houston State University Dr. Tim McGuire Copyright 2001 by Timothy J. McGuire, Ph.D. 1 Positional Number Systems Decimal (base 10) is an example e.g., 435 means

More information

Using 0 s and 1 s to Represent Stuff

Using 0 s and 1 s to Represent Stuff Using 0 s and 1 s to Represent Stuff First, how many things can we represent? In one bit, two things. In two bits, four things. In three bits,? How many things do we want to represent? For numbers, at

More information

Part 1 Theory Fundamentals

Part 1 Theory Fundamentals Part 1 Theory Fundamentals 2 Chapter 1 Information Representation Learning objectives By the end of this chapter you should be able to: show understanding of the basis of different number systems show

More information

EE 308 Spring Binary, Hex and Decimal Numbers (4-bit representation) Binary. Hex. Decimal A B C D E F

EE 308 Spring Binary, Hex and Decimal Numbers (4-bit representation) Binary. Hex. Decimal A B C D E F EE 8 Spring Binary, Hex and Decimal Numbers (-bit representation) Binary Hex 8 9 A B C D E F Decimal 8 9 EE 8 Spring What does a number represent? Binary numbers are a code, and represent what the programmer

More information

Chapter 2 Numeric Representation.

Chapter 2 Numeric Representation. Chapter 2 Numeric Representation. Most of the things we encounter in the world around us are analog; they don t just take on one of two values. How then can they be represented digitally? The key is that

More information

Useful Number Systems

Useful Number Systems Useful Number Systems Decimal Base = 10 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Binary Base = 2 Digit Set = {0, 1} Octal Base = 8 = 2 3 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7} Hexadecimal Base = 16 = 2

More information

CHAPTER V NUMBER SYSTEMS AND ARITHMETIC

CHAPTER V NUMBER SYSTEMS AND ARITHMETIC CHAPTER V-1 CHAPTER V CHAPTER V NUMBER SYSTEMS AND ARITHMETIC CHAPTER V-2 NUMBER SYSTEMS RADIX-R REPRESENTATION Decimal number expansion 73625 10 = ( 7 10 4 ) + ( 3 10 3 ) + ( 6 10 2 ) + ( 2 10 1 ) +(

More information

Number Systems, Base Conversions, and Computer Data Representation

Number Systems, Base Conversions, and Computer Data Representation , Base Conversions, and Computer Data Representation Decimal and Binary Numbers When we write decimal (base 10) numbers, we use a positional notation system. Each digit is multiplied by an appropriate

More information

Two s Complement Arithmetic

Two s Complement Arithmetic Two s Complement Arithmetic We now address the issue of representing integers as binary strings in a computer. There are four formats that have been used in the past; only one is of interest to us. The

More information

Activity 1: Bits and Bytes

Activity 1: Bits and Bytes ICS3U (Java): Introduction to Computer Science, Grade 11, University Preparation Activity 1: Bits and Bytes The Binary Number System Computers use electrical circuits that include many transistors and

More information

Numeral Systems. The number twenty-five can be represented in many ways: Decimal system (base 10): 25 Roman numerals:

Numeral Systems. The number twenty-five can be represented in many ways: Decimal system (base 10): 25 Roman numerals: Numeral Systems Which number is larger? 25 8 We need to distinguish between numbers and the symbols that represent them, called numerals. The number 25 is larger than 8, but the numeral 8 above is larger

More information

Digital Logic. The Binary System is a way of writing numbers using only the digits 0 and 1. This is the method used by the (digital) computer.

Digital Logic. The Binary System is a way of writing numbers using only the digits 0 and 1. This is the method used by the (digital) computer. Digital Logic 1 Data Representations 1.1 The Binary System The Binary System is a way of writing numbers using only the digits 0 and 1. This is the method used by the (digital) computer. The system we

More information

Computer is a binary digital system. Data. Unsigned Integers (cont.) Unsigned Integers. Binary (base two) system: Has two states: 0 and 1

Computer is a binary digital system. Data. Unsigned Integers (cont.) Unsigned Integers. Binary (base two) system: Has two states: 0 and 1 Computer Programming Programming Language Is telling the computer how to do something Wikipedia Definition: Applies specific programming languages to solve specific computational problems with solutions

More information

Unit 2: Number Systems, Codes and Logic Functions

Unit 2: Number Systems, Codes and Logic Functions Unit 2: Number Systems, Codes and Logic Functions Introduction A digital computer manipulates discrete elements of data and that these elements are represented in the binary forms. Operands used for calculations

More information

1 Number System (Lecture 1 and 2 supplement)

1 Number System (Lecture 1 and 2 supplement) 1 Number System (Lecture 1 and 2 supplement) By Dr. Taek Kwon Many different number systems perhaps from the prehistoric era have been developed and evolved. Among them, binary number system is one of

More information

Chapter 1: Digital Systems and Binary Numbers

Chapter 1: Digital Systems and Binary Numbers Chapter 1: Digital Systems and Binary Numbers Digital age and information age Digital computers general purposes many scientific, industrial and commercial applications Digital systems telephone switching

More information

Decimal-to-Binary Conversion. Computer & Microprocessor Architecture HCA103. Repeated Division-by-2 Method. Repeated Multiplication-by-2 Method

Decimal-to-Binary Conversion. Computer & Microprocessor Architecture HCA103. Repeated Division-by-2 Method. Repeated Multiplication-by-2 Method Decimal-to-Binary Conversion Computer & Microprocessor Architecture HCA103 Computer Arithmetic Algorithm 1 Step 1: Break the number in two parts: Whole number and fraction part. Step 2: Repeated Division-by-2

More information

Base Conversion written by Cathy Saxton

Base Conversion written by Cathy Saxton Base Conversion written by Cathy Saxton 1. Base 10 In base 10, the digits, from right to left, specify the 1 s, 10 s, 100 s, 1000 s, etc. These are powers of 10 (10 x ): 10 0 = 1, 10 1 = 10, 10 2 = 100,

More information

Number Systems! Why Bits (Binary Digits)?!

Number Systems! Why Bits (Binary Digits)?! Number Systems Why Bits (Binary Digits)? Computers are built using digital circuits Inputs and outputs can have only two values True (high voltage) or false (low voltage) Represented as and Can represent

More information

Bits, Data Types, and Operations. University of Texas at Austin CS310H - Computer Organization Spring 2010 Don Fussell

Bits, Data Types, and Operations. University of Texas at Austin CS310H - Computer Organization Spring 2010 Don Fussell Bits, Data Types, and Operations University of Texas at Austin CS3H - Computer Organization Spring 2 Don Fussell How do we represent data in a computer? At the lowest level, a computer is an electronic

More information

Number Systems and Base Conversions

Number Systems and Base Conversions Number Systems and Base Conversions As you know, the number system that we commonly use is the decimal or base- 10 number system. That system has 10 digits, 0 through 9. While it's very convenient for

More information

ORG ; ZERO. Introduction To Computing

ORG ; ZERO. Introduction To Computing Dec 0 Hex 0 Bin 00000000 ORG ; ZERO Introduction To Computing OBJECTIVES this chapter enables the student to: Convert any number from base 2, base 10, or base 16 to any of the other two bases. Add and

More information

LSN 2 Number Systems. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology

LSN 2 Number Systems. ECT 224 Digital Computer Fundamentals. Department of Engineering Technology LSN 2 Number Systems Department of Engineering Technology LSN 2 Decimal Number System Decimal number system has 10 digits (0-9) Base 10 weighting system... 10 5 10 4 10 3 10 2 10 1 10 0. 10-1 10-2 10-3

More information

Imperial College London Department of Computing

Imperial College London Department of Computing Imperial College London Department of Computing Architecture SELF-STUDY NOTES January 2016 Integers and Characters (1) Self-Study Welcome to the Computer Architecture course. These notes cover basic topics

More information

Binary Numbers. Binary Numbers. Wolfgang Schreiner Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria

Binary Numbers. Binary Numbers. Wolfgang Schreiner Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria Binary Numbers Wolfgang Schreiner Research Institute for Symbolic Computation (RISC) Johannes Kepler University, Linz, Austria Wolfgang.Schreiner@risc.uni-linz.ac.at http://www.risc.uni-linz.ac.at/people/schreine

More information

Digital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur

Digital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Digital System Design Prof. D Roychoudhry Department of Computer Science and Engineering Indian Institute of Technology, Kharagpur Lecture - 04 Digital Logic II May, I before starting the today s lecture

More information

Binary Representation and Computer Arithmetic

Binary Representation and Computer Arithmetic Binary Representation and Computer Arithmetic The decimal system of counting and keeping track of items was first created by Hindu mathematicians in India in A.D. 4. Since it involved the use of fingers

More information

Chapter 3 DATA REPRESENTATION. 3.1 Character Representation

Chapter 3 DATA REPRESENTATION. 3.1 Character Representation Chapter 3 DATA REPRESENTATION Binary codes are used to represent both characters and numbers inside computers. Moreover, the binary codes used to represent numbers must be consistent with the arithmetic

More information

Chapter 4. Binary Data Representation and Binary Arithmetic

Chapter 4. Binary Data Representation and Binary Arithmetic Christian Jacob Chapter 4 Binary Data Representation and Binary Arithmetic 4.1 Binary Data Representation 4.2 Important Number Systems for Computers 4.2.1 Number System Basics 4.2.2 Useful Number Systems

More information

Chapter 2: Number Systems

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

More information

Assembly Language for Intel-Based Computers, 4 th Edition. Chapter 1: Basic Concepts

Assembly Language for Intel-Based Computers, 4 th Edition. Chapter 1: Basic Concepts Assembly Language for Intel-Based Computers, 4 th Edition Kip R. Irvine Chapter 1: Basic Concepts Slides prepared by Kip R. Irvine Revision date: 07/21/2002 Chapter corrections (Web) Assembly language

More information

Number Systems and Data Representation CS221

Number Systems and Data Representation CS221 Number Systems and Data Representation CS221 Inside today s computers, data is represented as 1 s and 0 s. These 1 s and 0 s might be stored magnetically on a disk, or as a state in a transistor, core,

More information

THE ISLAMIC UNIVERSITY OF GAZA ENGINEERING FACULTY DEPARTMENT OF COMPUTER ENGINEERING DIGITAL LOGIC DESIGN DISCUSSION ECOM Eng. Huda M.

THE ISLAMIC UNIVERSITY OF GAZA ENGINEERING FACULTY DEPARTMENT OF COMPUTER ENGINEERING DIGITAL LOGIC DESIGN DISCUSSION ECOM Eng. Huda M. THE ISLAMIC UNIVERSITY OF GAZA ENGINEERING FACULTY DEPARTMENT OF COMPUTER ENGINEERING DIGITAL LOGIC DESIGN DISCUSSION ECOM 2012 Eng. Huda M. Dawoud September, 2015 1.1 List the octal and hexadecimal numbers

More information

CSI 333 Lecture 1 Number Systems

CSI 333 Lecture 1 Number Systems CSI 333 Lecture 1 Number Systems 1 1 / 23 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = 2 10 0 + 9 10 1 + 1 10 2 General: Digit sequence : d n 1 d n 2...

More information

Binary Numbers. X. Zhang Fordham Univ.

Binary Numbers. X. Zhang Fordham Univ. Binary Numbers X. Zhang Fordham Univ. 1 Numeral System! A way for expressing numbers, using symbols in a consistent manner.!! "11" can be interpreted differently:!! in the binary symbol: three!! in the

More information