Number Systems and. Data Representation

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Number Systems and Data Representation 1

2 Lecture Outline Number Systems Binary, Octal, Hexadecimal Representation of characters using codes Representation of Numbers Integer, Floating Point, Binary Coded Decimal Program Language and Data Types 2

3 Data Representation? Representation = Measurement Most things in the Real World actually exist as a single, continuously varying quantity Mass, Volume, Speed, Pressure, Temperature Easy to measure by representing it using a different thing that varies in the same way Eg. Pressure as the height of column of mercury or as voltage produced by a pressure transducer These are ANALOG measurements 3

4 Digital Representation Convert ANALOG to DIGITAL measurement by using a scale of units DIGITAL measurements In units a set of symbolic values - digits Values larger than any symbol in the set use sequence of digits Units, Tens, Hundreds Measured in discrete or whole units Difficult to measure something that is not a multiple of units in size. Eg Fractions 4

5 Analog vs. Digital representation 5

6 Data Representation Computers use digital representation Based on a binary system (uses on/off states to represent 2 digits). Many different types of data. Examples? ALL data (no matter how complex) must be represented in memory as binary digits (bits). 6

7 Number systems and computers Computers store all data as binary digits, but we may need to convert this to a number system we are familiar with. Computer programs and data are often represented (outside the computer) using octal and hexadecimal number systems because they are a short hand way of representing binary numbers. 7

8 Number Systems - Decimal The decimal system is a base-10 system. There are 10 distinct digits (0 to 9) to represent any quantity. For an n-digit number, the value that each digit represents depends on its weight or position. The weights are based on powers of = 1* * * *10 0 =

9 Number Systems - Binary The binary system is a base-2 system. There are 2 distinct digits (0 and 1) to represent any quantity. For an n-digit number, the value of a digit in each column depends on its position. The weights are based on powers of = 1* * * *2 0 =8+2+1 =

10 Number Systems - Octal Octal and hexadecimal systems provide a shorthand way to deal with the long strings of 1 s and 0 s in binary. Octal is base-8 system using the digits 0 to 7. To convert to decimal, you can again use a column weighted system = 7* * * *8 0 = An octal number can easily be converted to binary by replacing each octal digit with the corresponding group of 3 binary digits =

11 Number Systems - Hexadecimal Hexadecimal is a base-16 system. It contains the digits 0 to 9 and the letters A to F (16 digit values). The letters A to F represent the unit values 10 to 15. This system is often used in programming as a condensed form for binary numbers (0x00FF, 00FFh) To convert to decimal, use a weighted system with powers of

12 Number Systems - Hexadecimal Conversion to binary is done the same way as octal to binary conversions. This time though the binary digits are organised into groups of 4. Conversion from binary to hexadecimal involves breaking the bits into groups of 4 and replacing them with the hexadecimal equivalent. 12

13 Example #1 Value of 2001 in Binary, Octal and Hexadecimal 13

14 Example #2 Conversion: Binary Octal Hexadecimal 14

15 Decimal to Base N Conversions To convert from decimal to a different number base such as Octal, Binary or Hexadecimal involves repeated division by that number base Keep dividing until the quotient is zero Use the remainders in reverse order as the digits of the converted number 15

16 Example #3 Decimal to Binary 1492 (decimal) =??? (binary) Repeated Divide by 2 16

17 Base N to Decimal Conversions Multiply each digit by increasing powers of the base value and add the terms Example: =??? (decimal) 17

18 Data Representation Computers store everything as binary digits. So, how can we encode numbers, images, sound, text?? We need standard encoding systems for each type of data. Some standards evolve from proprietary products which became very popular. Other standards are created by official industry bodies where none previously existed. Some example encoding standards are? 18

19 Alphanumeric Data Alphanumeric data such as names and addresses are represented by assigning a unique binary code or sequence of bits to represent each character. As each character is entered from a keyboard (or other input device) it is converted into a binary code. Character code sets contain two types of characters: Printable (normal characters) Non-printable. Characters used as control codes. CTRL G (beep) CTRL Z (end of file) 19

20 Alphanumeric Codes There are 3 main coding methods in use: ASCII EBCDIC Unicode 20

21 ASCII 7-bit code (128 characters) has an extended 8-bit version used on PC s and non-ibm mainframes widely used to transfer data from one computer to another Examples: 21

22 EBCDIC An 8-bit code (256 characters) Different collating sequence to ASCII used on mainframe IBM machine Both ASCII and EBCDIC are 8 bit codes inadequate for representing all international characters Some European characters Most non-alphabetic languages eg Mandarin, Kanji, Arabic, etc 22

23 Unicode New 16 bit standard - can represent 65,536 characters Of which 49,000 have been defined 6400 reserved for private use 10,000 for future expansions Incorporates ASCII-7 Example - Java code: char letter = A ; char word[ ] = YES ; stores the values using Unicode characters Java VM uses 2 bytes to store one unicode character

24 Numeric Data Need to perform computations Need to represent only numbers Using ASCII coded digits is very inefficient Representation depends on nature of the data and processing requirements Display purposes only (no computations): CHAR PRINT Computation involving integers: INT COMPUTE 16 / 3 = 5 Computation involving fractions: FLOAT COMPUTE * =

25 Representing Numeric Data Stored within the computer using one of several different numeric representation systems Derived from the binary (base 2) number system. We can represent unsigned numbers from just using 8 bits Or in general we can represent values from 0 to 2 N -1 using N bits. The maximum value is restricted by the number of bits available (called Truncation or Overflow) However, most programming languages support manipulation of signed and fractional numbers. How can these be represented in binary form? 25

26 Representing Numeric Data Range of Values 0 to 2 N -1 in N bits N Range N Range 4 0 to to to to to to to to to to

27 Integer Representation UNSIGNED representing numbers from 0 upwards or SIGNED to allow for negatives. In the computer we only have binary digits, so to represent negative integers we need some sort of convention. Four conventions in use for representing negative integers are: Sign Magnitude 1 s Complement 2 s Complement Excess

28 Negative Integers Sign Magnitude Simplest form of representation In an n-bit word, the rightmost n-1 bits hold the magnitude of the integer Example: +6 in 8-bit representation is: in 8-bit representation is: Disadvantages arithmetic is difficult Two representations for zero

29 Binary Arithmetic Addition Table Digit Digit Sum Carry

30 Negative Integers One s (1 s) Complement Computers generally use a system called complementary representation to store negative integers. Two basic types - ones and twos complement, of which 2 s complement is the most widely used. The number range is split into two halves, to represent the positive and negative numbers. Negative numbers begin with 1, positive with 0. 30

31 Negative Integers One s (1 s) Complement To perform 1 s complement operation on a binary number, replace 1 s with 0 s and 0 s with 1 s (ie Complement it!) +6 represented by: represented by: Advantages: arithmetic is easier (cheaper/faster electronics) Fairly straightforward addition Add any carry from the Most Significant (left-most) Bit to Least Significant (right-most) Bit of the result For subtraction form 1 s complement of number to be subtracted and then add Disadvantages : still two representations for zero and (in 8-bit representation) 31

32 Negative Integers Two s (2 s) Complement To perform the 2 s complement operation on a binary number replace 1 s with 0 s and 0 s with 1 s (i.e. the one s complement of the number) add 1 +6 represented by: represented by: Advantages: Arithmetic is very straightforward End Around Carry is ignored only one representation for zero ( ) 32

33 Negative Integers Two s (2 s) Complement Two s Complement To convert an integer to 2 s complement»take the binary form of the number (6 as an 8-bit representation) »Flip the bits: (Find 1 s Complement)»Add (2 s complement of 6) Justification of representation: 6+(-6)=0? (6) (2 s complement of 6) (0) 33

34 Negative Integers Two s (2 s) Complement Properties of Two s Complement The 2 s comp of a 2 s comp is the original number (6) (2 s comp of 6) (2 s comp of 2 s comp of 6) The sign of a number is given by its MSB The bit patterns: represents zero 0nnnnnnn represents positive numbers 1nnnnnnn represents negative numbers 34

35 Negative Integers Two s (2 s) Complement Addition Addition is performed by adding corresponding bits ( 7) (+5) (12) Subtraction Subtraction is performed by adding the 2 s complement Ignore End-Around-Carry (12) (-5) ( 7) 35

36 Negative Integers Two s (2 s) Complement Interpretation of Negative Results ( 5) (-12) ( ) Result is negative MSB of result is 1 so it is a negative number in 2 s complement form Negative what? Take the 2 s comp of the result to find out since the 2 s comp of a 2 s comp is the original number Negative 7 the 2 s complement of is or

37 excess 128 representation excess 128 for 8-bit signed numbers (or excess 2 m-1 for m-bit numbers) Stored as the true value plus 128 eg =125 ( ) =154 ( ) Number in range 128 to +127 map to bit values 0 to 255 same as 2 s comp, but with sign bit reversed!! 37

38 Binary Fractions The Binary Point Digits on the left +ve powers of 2 Digits on the right ve powers of / 2 (0.5) 2 1 / 4 (0.25) 4 1 / 16 (0.125) 8 1 / 32 (0.0625)

39 Integer Overflow 39

40 Problem: word size is fixed, but addition can produce a result that is too large to fit in the number of bits available. This is called overflow. If two numbers of the same sign are added, but the result has the opposite sign then overflow has occurred Overflow can occur whether or not there is a carry Examples: ( +64) (-128) ( +65) ( -64) (-127) ( +64) 40

41 Floating Point Representation Fractional numbers, and very large or very small numbers can be represented with only a few digits by using scientific notation. For example: 976,000,000,000,000 = 9.76 * = 9.76 * This same approach can be used for binary numbers. A number represented by ±M*B ±E can be stored in a binary word with three fields: Sign - plus or minus Mantissa M (often called the significand) Exponent E (includes exponent sign) The base B is generally 2 and need not be stored. 41

42 Floating Point Representation Typical 32-bit Representation The first bit contains the sign The next 8 bits contain the exponent The remaining 23 bits contain the mantissa The more bits we use for the exponent, the larger the range of numbers available, but at the expense of precision. We still only have a total of 2 32 numbers that can be represented. A value from a calculation may have to be rounded to the nearest value that can be represented. Converting 5.75 to 32 bit IEEE format 5.75 (dec) = (bin) = *

43 Floating Point Representation The only way to increase both range and precision is to use more bits. with 32 bits, 2 32 numbers can be represented with 64 bits, 2 64 numbers can be represented Most microcomputers offer at least single precision (32 bit) and double precision (64 bit) numbers. Mainframes will have several larger floating point formats available. 43

44 Floating Point Representation Standards Several floating-point representations exist including: IBM System/370 VAX IEEE Standard 754 Overflow refers to values whose magnitude is too large to be represented. Underflow refers to numbers whose fractional magnitude is too small to be represented - they are then usually approximated by zero. 44

45 Floating Point Arithmetic (Not Examinable) Multiplication and division involve adding or subtracting exponents, and multiplying the mantissas much like for integer arithmetic. Addition and subtraction are more complicated as the operands must have the same exponent - this may involve shifting the radix point on one of the operands. 45

46 Binary Coded Decimal Scheme whereby each decimal digit is represented by its 4-bit binary code 7 = = Many CPUs provide arithmetic instructions for operating directly on BCD. However, calculations slower and more difficult. 46

47 Boolean Representation Boolean or logical data type is used to represent only two values: TRUE FALSE Although only one bit is needed, a single byte often used. It may be represented as: = FALSE FF 16 or Non-Zero = TRUE This data type is used with logical operators such as comparisons = > < 47

48 Programming languages and data types CPU will have instructions for dealing with limited set of data types (primitive data types). Usually these are: Char Boolean Integer Real Memory addresses Recent processors include special instructions to deal with multimedia data eg MMX extension 48

49 Data Representation All languages allow programmer to specify data as belonging to particular data types. Programmers can also define special user defined variable data types such as days_of_ week Software can combine primitive data types to form data structures such as strings, arrays, records, etc 49

50 Data Type Selection Consider the type of data and its use. Alphanumeric for text (eg. surname, subject name) Alphanumeric for numbers not used in calculations (eg. phone number, postcode) One of the numeric data types for numbers Binary integers for whole numbers signed or unsigned as appropriate Floating point for large numbers, fractions, or approximations in measurement Boolean for flags 50

51 (end) 51

52 52

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

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

198:211 Computer Architecture

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 1 Computer Architecture What do computers do? Manipulate stored

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

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

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

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

Numbering Systems. InThisAppendix...

G InThisAppendix... Introduction Binary Numbering System Hexadecimal Numbering System Octal Numbering System Binary Coded Decimal (BCD) Numbering System Real (Floating Point) Numbering System BCD/Binary/Decimal/Hex/Octal

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

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

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

CS 16: Assembly Language Programming for the IBM PC and Compatibles

CS 16: Assembly Language Programming for the IBM PC and Compatibles First, a little about you Your name Have you ever worked with/used/played with assembly language? If so, talk about it Why are you taking

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

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

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

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

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

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

Divide: Paper & Pencil. Computer Architecture ALU Design : Division and Floating Point. Divide algorithm. DIVIDE HARDWARE Version 1

Divide: Paper & Pencil Computer Architecture ALU Design : Division and Floating Point 1001 Quotient Divisor 1000 1001010 Dividend 1000 10 101 1010 1000 10 (or Modulo result) See how big a number can be

The Essentials of Computer Organization and Architecture. Linda Null and Julia Lobur Jones and Bartlett Publishers, 2003

The Essentials of Computer Organization and Architecture Linda Null and Julia Lobur Jones and Bartlett Publishers, 2003 Chapter 2 Instructor's Manual Chapter Objectives Chapter 2, Data Representation,

4 Operations On Data

4 Operations On Data 4.1 Source: Foundations of Computer Science Cengage Learning Objectives After studying this chapter, students should be able to: List the three categories of operations performed on

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...

Data Storage 3.1. Foundations of Computer Science Cengage Learning

3 Data Storage 3.1 Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: List five different data types used in a computer. Describe how

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

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

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

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

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

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

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

2010/9/19. Binary number system. Binary numbers. Outline. Binary to decimal

2/9/9 Binary number system Computer (electronic) systems prefer binary numbers Binary number: represent a number in base-2 Binary numbers 2 3 + 7 + 5 Some terminology Bit: a binary digit ( or ) Hexadecimal

Binary Division. Decimal Division. Hardware for Binary Division. Simple 16-bit Divider Circuit

Decimal Division Remember 4th grade long division? 43 // quotient 12 521 // divisor dividend -480 41-36 5 // remainder Shift divisor left (multiply by 10) until MSB lines up with dividend s Repeat until

This 3-digit ASCII string could also be calculated as n = (Data[2]-0x30) +10*((Data[1]-0x30)+10*(Data[0]-0x30));

Introduction to Embedded Microcomputer Systems Lecture 5.1 2.9. Conversions ASCII to binary n = 100*(Data[0]-0x30) + 10*(Data[1]-0x30) + (Data[2]-0x30); This 3-digit ASCII string could also be calculated

CS101 Lecture 11: Number Systems and Binary Numbers. Aaron Stevens 14 February 2011

CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 14 February 2011 1 2 1 3!!! MATH WARNING!!! TODAY S LECTURE CONTAINS TRACE AMOUNTS OF ARITHMETIC AND ALGEBRA PLEASE BE ADVISED THAT CALCULTORS

Section 1.4 Place Value Systems of Numeration in Other Bases

Section.4 Place Value Systems of Numeration in Other Bases Other Bases The Hindu-Arabic system that is used in most of the world today is a positional value system with a base of ten. The simplest reason

To convert an arbitrary power of 2 into its English equivalent, remember the rules of exponential arithmetic:

Binary Numbers In computer science we deal almost exclusively with binary numbers. it will be very helpful to memorize some binary constants and their decimal and English equivalents. By English equivalents

Lecture 11: Number Systems

Lecture 11: Number Systems Numeric Data Fixed point Integers (12, 345, 20567 etc) Real fractions (23.45, 23., 0.145 etc.) Floating point such as 23. 45 e 12 Basically an exponent representation Any number

CPEN 214 - Digital Logic Design Binary Systems

CPEN 4 - Digital Logic Design Binary Systems C. Gerousis Digital Design 3 rd Ed., Mano Prentice Hall Digital vs. Analog An analog system has continuous range of values A mercury thermometer Vinyl records

This Unit: Floating Point Arithmetic. CIS 371 Computer Organization and Design. Readings. Floating Point (FP) Numbers

This Unit: Floating Point Arithmetic CIS 371 Computer Organization and Design Unit 7: Floating Point App App App System software Mem CPU I/O Formats Precision and range IEEE 754 standard Operations Addition

Fractional Numbers. Fractional Number Notations. Fixed-point Notation. Fixed-point Notation

2 Fractional Numbers Fractional Number Notations 2010 - Claudio Fornaro Ver. 1.4 Fractional numbers have the form: xxxxxxxxx.yyyyyyyyy where the x es constitute the integer part of the value and the y

plc numbers - 13.1 Encoded values; BCD and ASCII Error detection; parity, gray code and checksums

plc numbers - 3. Topics: Number bases; binary, octal, decimal, hexadecimal Binary calculations; s compliments, addition, subtraction and Boolean operations Encoded values; BCD and ASCII Error detection;

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

Data Storage. Chapter 3. Objectives. 3-1 Data Types. Data Inside the Computer. After studying this chapter, students should be able to:

Chapter 3 Data Storage Objectives After studying this chapter, students should be able to: List five different data types used in a computer. Describe how integers are stored in a computer. Describe how

Bachelors of Computer Application Programming Principle & Algorithm (BCA-S102T)

Unit- I Introduction to c Language: C is a general-purpose computer programming language developed between 1969 and 1973 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating

Review 1/2. CS61C Characters and Floating Point. Lecture 8. February 12, Review 2/2 : 12 new instructions Arithmetic:

Review 1/2 CS61C Characters and Floating Point Lecture 8 February 12, 1999 Handling case when number is too big for representation (overflow) Representing negative numbers (2 s complement) Comparing signed

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

Primitive Data Types Summer 2010 Margaret Reid-Miller

Primitive Data Types 15-110 Summer 2010 Margaret Reid-Miller Data Types Data stored in memory is a string of bits (0 or 1). What does 1000010 mean? 66? 'B'? 9.2E-44? How the computer interprets the string

Computers. Hardware. The Central Processing Unit (CPU) CMPT 125: Lecture 1: Understanding the Computer

Computers CMPT 125: Lecture 1: Understanding the Computer Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 3, 2009 A computer performs 2 basic functions: 1.

Lecture 4 Representing Data on the Computer. Ramani Duraiswami AMSC/CMSC 662 Fall 2009

Lecture 4 Representing Data on the Computer Ramani Duraiswami AMSC/CMSC 662 Fall 2009 x = ±(1+f) 2 e 0 f < 1 f = (integer < 2 52 )/ 2 52-1022 e 1023 e = integer Effects of floating point Effects of floating

ECE 0142 Computer Organization. Lecture 3 Floating Point Representations

ECE 0142 Computer Organization Lecture 3 Floating Point Representations 1 Floating-point arithmetic We often incur floating-point programming. Floating point greatly simplifies working with large (e.g.,

Binary Representation

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

Digital Design. Assoc. Prof. Dr. Berna Örs Yalçın

Digital Design Assoc. Prof. Dr. Berna Örs Yalçın Istanbul Technical University Faculty of Electrical and Electronics Engineering Office Number: 2318 E-mail: siddika.ors@itu.edu.tr Grading 1st Midterm -

Restoring division. 2. Run the algorithm Let s do 0111/0010 (7/2) unsigned. 3. Find remainder here. 4. Find quotient here.

Binary division Dividend = divisor q quotient + remainder CS/COE447: Computer Organization and Assembly Language Given dividend and divisor, we want to obtain quotient (Q) and remainder (R) Chapter 3 We

ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-470/570: Microprocessor-Based System Design Fall 2014.

REVIEW OF NUMBER SYSTEMS Notes Unit 2 BINARY NUMBER SYSTEM In the decimal system, a decimal digit can take values from to 9. For the binary system, the counterpart of the decimal digit is the binary digit,

DNA Data and Program Representation. Alexandre David 1.2.05 adavid@cs.aau.dk

DNA Data and Program Representation Alexandre David 1.2.05 adavid@cs.aau.dk Introduction Very important to understand how data is represented. operations limits precision Digital logic built on 2-valued

Simulation & Synthesis Using VHDL

Floating Point Multipliers: Simulation & Synthesis Using VHDL By: Raj Kumar Singh - B.E. (Hons.) Electrical & Electronics Shivananda Reddy - B.E. (Hons.) Electrical & Electronics BITS, PILANI Outline Introduction

Binary. ! You are probably familiar with decimal

Arithmetic operations in assembly language Prof. Gustavo Alonso Computer Science Department ETH Zürich alonso@inf.ethz.ch http://www.inf.ethz.ch/department/is/iks/ Binary! You are probably familiar with

CS321. Introduction to Numerical Methods

CS3 Introduction to Numerical Methods Lecture Number Representations and Errors Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506-0633 August 7, 05 Number in

CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

Number Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi)

Conversions Dr. Sarita Agarwal (Acharya Narendra Dev College,University of Delhi) INTRODUCTION System- A number system defines a set of values to represent quantity. We talk about the number of people

NUMBER SYSTEMS. 1.1 Introduction

NUMBER SYSTEMS 1.1 Introduction There are several number systems which we normally use, such as decimal, binary, octal, hexadecimal, etc. Amongst them we are most familiar with the decimal number system.

= Chapter 1. The Binary Number System. 1.1 Why Binary?

Chapter The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base-0 system. When you

Fixed-Point Arithmetic

Fixed-Point Arithmetic Fixed-Point Notation A K-bit fixed-point number can be interpreted as either: an integer (i.e., 20645) a fractional number (i.e., 0.75) 2 1 Integer Fixed-Point Representation N-bit

HOMEWORK # 2 SOLUTIO

HOMEWORK # 2 SOLUTIO Problem 1 (2 points) a. There are 313 characters in the Tamil language. If every character is to be encoded into a unique bit pattern, what is the minimum number of bits required to

Binary Number System. 16. Binary Numbers. Base 10 digits: 0 1 2 3 4 5 6 7 8 9. Base 2 digits: 0 1

Binary Number System 1 Base 10 digits: 0 1 2 3 4 5 6 7 8 9 Base 2 digits: 0 1 Recall that in base 10, the digits of a number are just coefficients of powers of the base (10): 417 = 4 * 10 2 + 1 * 10 1

Number Systems I. CIS008-2 Logic and Foundations of Mathematics. David Goodwin. 11:00, Tuesday 18 th October

Number Systems I CIS008-2 Logic and Foundations of Mathematics David Goodwin david.goodwin@perisic.com 11:00, Tuesday 18 th October 2011 Outline 1 Number systems Numbers Natural numbers Integers Rational

1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal:

Exercises 1 - number representations Questions 1. Give the 16 bit signed (twos complement) representation of the following decimal numbers, and convert to hexadecimal: (a) 3012 (b) - 435 2. For each of

A single register, called the accumulator, stores the. operand before the operation, and stores the result. Add y # add y from memory to the acc

Other architectures Example. Accumulator-based machines A single register, called the accumulator, stores the operand before the operation, and stores the result after the operation. Load x # into acc

Binary Numbers Binary Octal Hexadecimal Binary Numbers COUNTING SYSTEMS UNLIMITED... Since you have been using the 10 different digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 all your life, you may wonder how

Introduction to Programming

Introduction to Programming SS 2012 Adrian Kacso, Univ. Siegen adriana.dkacsoa@duni-siegena.de Tel.: 0271/740-3966, Office: H-B 8406 Stand: April 25, 2012 Betriebssysteme / verteilte Systeme Introduction

Unsigned Conversions from Decimal or to Decimal and other Number Systems

Page 1 of 5 Unsigned Conversions from Decimal or to Decimal and other Number Systems In all digital design, analysis, troubleshooting, and repair you will be working with binary numbers (or base 2). It

Binary Numbering Systems

Binary Numbering Systems April 1997, ver. 1 Application Note 83 Introduction Binary numbering systems are used in virtually all digital systems, including digital signal processing (DSP), networking, and

CHAPTER TWO. 2.1 Unsigned Binary Counting. Numbering Systems

CHAPTER TWO Numbering Systems Chapter one discussed how computers remember numbers using transistors, tiny devices that act like switches with only two positions, on or off. A single transistor, therefore,

2 Number Systems 2.1. Foundations of Computer Science Cengage Learning

2 Number Systems 2.1 Foundations of Computer Science Cengage Learning 2.2 Objectives After studying this chapter, the student should be able to: Understand the concept of number systems. Distinguish between

Bits, bytes, and representation of information

Bits, bytes, and representation of information digital representation means that everything is represented by numbers only the usual sequence: something (sound, pictures, text, instructions,...) is converted

Binary, Hexadecimal, Octal, and BCD Numbers

23CH_PHCalter_TMSETE_949118 23/2/2007 1:37 PM Page 1 Binary, Hexadecimal, Octal, and BCD Numbers OBJECTIVES When you have completed this chapter, you should be able to: Convert between binary and decimal

Borland C++ Compiler: Operators

Introduction Borland C++ Compiler: Operators An operator is a symbol that specifies which operation to perform in a statement or expression. An operand is one of the inputs of an operator. For example,

The New IoT Standard: Any App for Any Device Using Any Data Format. Mike Weiner Product Manager, Omega DevCloud KORE Telematics

The New IoT Standard: Any App for Any Device Using Any Data Format Mike Weiner Product Manager, Omega DevCloud KORE Telematics About KORE The world s largest M2M/IoT services provider 12 Carriers Enterprise

6 3 4 9 = 6 10 + 3 10 + 4 10 + 9 10

Lesson The Binary Number System. Why Binary? The number system that you are familiar with, that you use every day, is the decimal number system, also commonly referred to as the base- system. When you

Integer multiplication

Integer multiplication Suppose we have two unsigned integers, A and B, and we wish to compute their product. Let A be the multiplicand and B the multiplier: A n 1... A 1 A 0 multiplicand B n 1... B 1 B

Today. Binary addition Representing negative numbers. Andrew H. Fagg: Embedded Real- Time Systems: Binary Arithmetic

Today Binary addition Representing negative numbers 2 Binary Addition Consider the following binary numbers: 0 0 1 0 0 1 1 0 0 0 1 0 1 0 1 1 How do we add these numbers? 3 Binary Addition 0 0 1 0 0 1 1

APPENDIX C The Hexadecimal Number System and Memory Addressing U nderstanding the number system and the coding system that computers use to store data and communicate with each other is fundamental to

Modbus Frequently Asked Questions WP-34-REV0-0609-1/7 The Answer to the 14 Most Frequently Asked Modbus Questions Exactly what is Modbus? Modbus is an open serial communications protocol widely used in

DIGITAL FUNDAMENTALS lesson 4 Number systems

The decimal number system In the decimal number system the base is 0. For example the number 4069 means 4x000 + 0x00 + 6x0 + 9x. The powers of 0 with the number 4069 in tabelvorm are: power 0 3 0 2 0 0

Pemrograman Dasar. Basic Elements Of Java

Pemrograman Dasar Basic Elements Of Java Compiling and Running a Java Application 2 Portable Java Application 3 Java Platform Platform: hardware or software environment in which a program runs. Oracle

The University of Nottingham

The University of Nottingham School of Computer Science A Level 1 Module, Autumn Semester 2007-2008 Computer Systems Architecture (G51CSA) Time Allowed: TWO Hours Candidates must NOT start writing their

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

The programming language C. sws1 1

The programming language C sws1 1 The programming language C invented by Dennis Ritchie in early 1970s who used it to write the first Hello World program C was used to write UNIX Standardised as K&C (Kernighan

Bits and Bytes. Computer Literacy Lecture 4 29/09/2008

Bits and Bytes Computer Literacy Lecture 4 29/09/2008 Lecture Overview Lecture Topics How computers encode information How to quantify information and memory How to represent and communicate binary data

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,

COMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012

Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about

Chapter 7 Lab - Decimal, Binary, Octal, Hexadecimal Numbering Systems

Chapter 7 Lab - Decimal, Binary, Octal, Hexadecimal Numbering Systems This assignment is designed to familiarize you with different numbering systems, specifically: binary, octal, hexadecimal (and decimal)

ASCII Characters. 146 CHAPTER 3 Information Representation. The sign bit is 1, so the number is negative. Converting to decimal gives

146 CHAPTER 3 Information Representation The sign bit is 1, so the number is negative. Converting to decimal gives 37A (hex) = 134 (dec) Notice that the hexadecimal number is not written with a negative

Paramedic Program Pre-Admission Mathematics Test Study Guide

Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page

> 2. Error and Computer Arithmetic

> 2. Error and Computer Arithmetic Numerical analysis is concerned with how to solve a problem numerically, i.e., how to develop a sequence of numerical calculations to get a satisfactory answer. Part

Today s topics. Digital Computers. More on binary. Binary Digits (Bits)

Today s topics! Binary Numbers! Brookshear.-.! Slides from Prof. Marti Hearst of UC Berkeley SIMS! Upcoming! Networks Interactive Introduction to Graph Theory http://www.utm.edu/cgi-bin/caldwell/tutor/departments/math/graph/intro

2 Number Systems. Source: Foundations of Computer Science Cengage Learning. Objectives After studying this chapter, the student should be able to:

2 Number Systems 2.1 Source: Foundations of Computer Science Cengage Learning Objectives After studying this chapter, the student should be able to: Understand the concept of number systems. Distinguish