# Chapter 4. Binary Data Representation and Binary Arithmetic

Save this PDF as:

Size: px
Start display at page:

Download "Chapter 4. Binary Data Representation and Binary Arithmetic"

## Transcription

1 Christian Jacob Chapter 4 Binary Data Representation and Binary Arithmetic 4.1 Binary Data Representation 4.2 Important Number Systems for Computers Number System Basics Useful Number Systems for Computers Decimal Number System Binary Number System Octal Number System Hexadecimal Number System Comparison of Number Systems 4.3 Powers of 2 Chapter Overview

2 Christian Jacob 4.4 Conversions Between Number Systems Conversion of Natural Numbers Conversion of Rational Numbers 4.5 Binary Logic 4.6 Binary Arithmetic 4.7 Negative Numbers and Complements Problems of Signed Number Representation Complement ((B-1)-Complement) Complement (B-Complement) 4.8 Floating Point Numbers Mantissa and Exponent Floating Point Arithmetics Limited Precision 4.9 Representation of Strings Representation of Characters Representation of Strings Chapter Overview

3 Page 3 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.1 Binary Data Representation Bit: smallest unit of information yes / no, on / off, L / 0, 1 / 0, 5V / 0V Byte: group of 8 bits --> 2 8 = 256 different states Word: the number of bits (word length) which can be processed by a computer in a single step (e.g., 32 or 64) --> machine dependent Representation: n n First Back TOC Prev Next Last

4 Page 4 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob The word size in any given computer is fixed. Example: 16-bit word every word (memory location) can hold a 16-bit pattern, with each bit either 0 or 1. How many distinct patterns are there in a 16-bit word? Each bit has 2 possible values: 0 or 1 1 bit has 2 distinct patterns: 0, 1 With 2 bits, each position has 2 possible values: 00, 01, 10, = 4 distinct bit patterns First Back TOC Prev Next Last

5 Page 5 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob With 3 bits, again each position can be either 0 or 1: = 8 distinct bit patterns In general, for n bits (a word of length n) we have 2 n distinct bit patterns. NOTE: What these bit patterns mean depends entirely on the context in which the patterns are used. First Back TOC Prev Next Last

6 Page 6 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.2 Important Number Systems for Computers Number System Basics Number systems, as we use them, consist of - a basic set Z of digits (or letters); example: Z = { 012,,,, 9} - a base B = Z (how many digits); example: B = Z = 10 A number is a linear sequence of digits. The value of a digit at a specific position depends on its assigned meaning and on its position. The value of a number N is the sum of these values. Number: N = B d n 1 d n 2 d 1 d 0 with word length n Value: N B = n 1 d i B i = d n 1 B n 1 + d n 2 B n d 1 B 1 + d 0 B 0 i = 0 First Back TOC Important Number Systems for Computers Prev Next Last

7 Page 7 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Useful Number Systems for Computers Name Base Digits dual binary 2 0, 1 octal 8 0, 1, 2, 3, 4, 5, 6, 7 decimal 10 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 hexadecimal sedecimal 16 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F First Back TOC Important Number Systems for Computers Prev Next Last

8 Page 8 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Decimal Number System Z = { ,,,,,,,,, }; B = 10 Each position to the left of a digit increases by a power of 10. Each position to the right of a digit decreases by a power of 10. Example: = = First Back TOC Important Number Systems for Computers Prev Next Last

9 Page 9 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Binary Number System Z = { 01, }; B = 2 Each position to the left of a digit increases by a power of 2. Each position to the right of a digit decreases by a power of 2. Example: = = = First Back TOC Important Number Systems for Computers Prev Next Last

10 Page 10 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Counting in Binary Decimal Dual Decimal Dual First Back TOC Important Number Systems for Computers Prev Next Last

11 Page 11 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Octal Number System Z = { ,,,,,,, }; B = 8 Each position to the left of a digit increases by a power of 8. Each position to the right of a digit decreases by a power of 8. Example: = = = First Back TOC Important Number Systems for Computers Prev Next Last

12 Page 12 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Counting in Octal Decimal Octal Decimal Octal First Back TOC Important Number Systems for Computers Prev Next Last

13 Page 13 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Hexadecimal Number System Z = { A,,,,,,,,,,, B, C, D, E, F} ; B = 16 Each position to the left of a digit increases by a power of 16. Each position to the right of a digit decreases by a power of 16. Example: FB40A 16 = = = 1,029, First Back TOC Important Number Systems for Computers Prev Next Last

14 Page 14 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Counting in Hexadecimal Decimal Hexadecimal Decimal Hexadecimal A 26 1 A 11 0 B 27 1 B 12 0 C 28 1 C 13 0 D 29 1 D 14 0 E 30 1 E 15 0 F 31 1 F First Back TOC Important Number Systems for Computers Prev Next Last

15 Page 15 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Comparison of Number Systems Decimal Dual Octal Hexadecimal Decimal Dual Octal Hexadecimal A A B B C C D D E E F F First Back TOC Important Number Systems for Computers Prev Next Last

16 Page 16 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.3 Powers of 2 N 2 N N 2 N , , , ,048, ,097, ,194, ,388, ,777, ,554, ,108, , ,217, , ,435, , ,870, , ,073,741, , ,147,483, , ,294,967, , ,589,934,592 First Back TOC Powers of 2 Prev Next Last

17 Page 17 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.4 Conversions Between Number Systems Conversion of Natural Numbers Base 10 Base B Conversion We use the metod of iterated division to convert a number of base 10 into a number N B of base B. N Divide the number N 10 by B: whole number N & remainder R. - Take N as the next number to be divided by B. - Keep R as the next left-most digit of the new number N B. 2. If N is zero then STOP, else set N 10 = N and go to step 1. First Back TOC Conversions Between Number Systems Prev Next Last

18 Page 18 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Example: =? 8 Number to be converted: N 10 = 1020; target base B = 8 Step 1: 1020 : 8 = 127 Remainder: = 1016 Step 2: 127 : 8 = 15 Remainder: = 120 Step 3: 15 : 8 = 1 Remainder: = 8 Step 4: 1 : 8 = 0 Remainder: = First Back TOC Conversions Between Number Systems Prev Next Last

19 Page 19 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Base B Base 10 Conversion We use the metod of iterated multiplication to convert a number of base B into a number N 10 of base 10. N B We simply add the contribution of each digit. Example: 1F2 16 =? 10 Number to be converted: N 16 = 1F2; target base B = 10 1F2 16 = = F2 16 = First Back TOC Conversions Between Number Systems Prev Next Last

20 Page 20 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Conversions between Numbers 2 n and 2 m Conversion 2 n 2 1 n = 3: octal dual: Replace 1 octal digit by 3 binary digits. Example: = n = 4: hexadecimal dual Replace 1 hexadecimal digit by 4 binary digits. Example: 1 F 2 16 = First Back TOC Conversions Between Number Systems Prev Next Last

21 Page 21 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Conversion n n = 3: dual octal: Replace 3 binary digits by 1 octal digit. Example: = = n = 4: hexadecimal dual Replace 4 binary digits by 1 hexadecimal digit. Example: = = 3 1 F 2 16 First Back TOC Conversions Between Number Systems Prev Next Last

22 Page 22 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Conversion 2 n 2 m 2 n m Example: octal hexadecimal = octal to binary: = rearrange binary: = = 2 C E 9 16 First Back TOC Conversions Between Number Systems Prev Next Last

23 Page 23 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Conversion of Rational Numbers Note: An exact conversion of rational numbers R 10 R B or R B R 10 is not always possible. We want: R 10 = R B + ε, where ε is a sufficiently small number. Base B Base 10 Conversion Example: =? = = = = First Back TOC Conversions Between Number Systems Prev Next Last

24 Page 24 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Base 10 Base B Conversion Example: =? 2 with k = 9 bit precision Step i N Operation R z = = = = = = = = = ( i) = ε Multiplication! First Back TOC Conversions Between Number Systems Prev Next Last

25 Page 25 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.5 Binary Logic Logical AND ( x y) AND Logical OR ( x y) OR First Back TOC Binary Logic Prev Next Last

26 Page 26 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Logical NOT ( x, negation) x x Logical XOR (exclusive OR) XOR Logical NAND (negated AND) NAND First Back TOC Binary Logic Prev Next Last

27 Page 27 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Examples: AND NAND OR XOR First Back TOC Binary Logic Prev Next Last

28 Page 28 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.6 Binary Arithmetic Elementary Rules for addition, subtraction, and multiplication Addition Subtraction Multiplication Operation Result Carry First Back TOC Binary Arithmetic Prev Next Last

29 Page 29 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Examples: Addition: Subtraction: carry: carry: First Back TOC Binary Arithmetic Prev Next Last

30 Page 30 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Multiplication: carry: carry: carry: First Back TOC Binary Arithmetic Prev Next Last

31 Page 31 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.7 Negative Numbers and Complements Integer Numbers: A possible representation sign value Number = sign & value sign = 0 +, positive number sign = 1 -, negative number Range of integer numbers for n bits: - 2 n n-1-1 First Back TOC Negative Numbers and Complements Prev Next Last

32 Page 32 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Example (1): n = 3 Sign Value Number ? Negative zero? Range: = = = First Back TOC Negative Numbers and Complements Prev Next Last

33 Page 33 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Problems of Signed Number Representation Given 2 binary numbers: one positive, one negative (base 2) 12 (base 10) (= ) : 1-1 = 1 + (-1) = 0? (= ) Clearly this won t work! Solution: 1-Complement and 2-Complement First Back TOC Negative Numbers and Complements Prev Next Last

34 Page 34 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Complement ((B-1)-Complement) We represent negative numbers by negating the positive representation of the number (and vice versa): any 1 is changed to a 0 any 0 is changed to a 1 Example: Binary Decimal One s Complement Decimal ? Problem! First Back TOC Negative Numbers and Complements Prev Next Last

35 Page 35 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Complement (B-Complement) Positive numbers are represented as usual. Negative numbers are created by - negating the positive value and - adding one (this avoids the negative zero). Examples: Binary Decimal Two s Complement Decimal = = = = ( ) = 1 (1 1110) = First Back TOC Negative Numbers and Complements Prev Next Last

36 Page 36 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Addition Number ring for 5-digit 2-complement negative numbers positive numbers Subtraction First Back TOC Negative Numbers and Complements Prev Next Last

37 Page 37 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob With the 2-complement, subtractions can be performed as simple additions: Example: 5 digit precision, base 10 Example 1 Example 2 B = Complement 7: (-7) B-1 : (-7) B : Addition 14: (-7) B : : : (-13) B-1 : (-13) B : : (-13) B : : First Back TOC Negative Numbers and Complements Prev Next Last

38 Page 38 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.8 Floating Point Numbers Mantissa and Exponent Recall the general form of numbers expressed in scientific notation: x Printed out by a computer these numbers usually look like this: e0 6.02e23 This representation saves space and reduces redundancy. Hence, the usual format for a 32-bit real (floating point) number is: mantissa = 21 bits exponent = 11 bits First Back TOC Floating Point Numbers Prev Next Last

39 Page 39 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Floating Point Arithmetics Example: Operands Compare Exponents Adjust Exponents and Mantissae e 1 - e 2 = Result: Addition: 1. Identify mantissae and exponents 2. Compare exponents 3. Adjust exponents if necessary 4. Adjust the shorter mantissa 5. Add mantissae 6. Normalize mantissa 7. Adjust exponent First Back TOC Floating Point Numbers Prev Next Last

40 Page 40 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Example: Multiplication 1. Multiply of mantissae 2. Normalize mantissae 3. Adjust exponents 4. Add exponents Example: ( ) ( ) Step 1: multiplication of the mantissae: = First Back TOC Floating Point Numbers Prev Next Last

41 Page 41 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Step 1 (cont.): step-by-step multiplication of the mantissae Step 2: Add exponents 5 + (-3) + (-1) = 1 Result: (with 3 digits of precision) in the mantissa) First Back TOC Floating Point Numbers Prev Next Last

42 Page 42 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob Limited Precision Note: Not all values can be represented! Example: - mantissa: 2 decimal digits - exponent: 1 decimal digit Sample number: = 7400 What is the next higher value? = 7500 What about values 7400 < x < 7500? They cannot be represented!!! Remedy, but not a perfect solution: Increase the number of mantissa digits. First Back TOC Floating Point Numbers Prev Next Last

43 Page 43 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.9 Representation of Strings Representation of Characters Usually: single characters are represented by 1 byte different standards: ASCII 1, EBCDIC 2 Currently: internationalization with Unicode a single character is represented by 2 bytes 1. ASCII: American Standard Code for Information Interchange 2. EBCDIC: Extended Binary Coded Decimal Interchange Code (derived from punch card standards) First Back TOC Representation of Strings Prev Next Last

44 Page 44 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob ASCII Character Encoding D C D C D C D C D C D C D C D C 0 NUL 16 DLE 32 SP P p 1 SCH 17 DC1 33! A 81 Q 97 a 113 q 2 STX 18 DC B 82 R 98 b 114 r 3 ETX 19 DC3 35 # C 83 S 99 c 115 s 4 EOT 20 DC4 36 \$ D 84 T 100 d 116 t 5 ENQ 21 NAK 37 % E 85 U 101 e 117 u 6 ACK 22 SYN 38 & F 86 V 102 f 118 v 7 BEL 23 ETB G 87 W 103 g 119 w 8 BS 24 CAN 40 ( H 88 X 104 h 120 x 9 HT 25 EM 41 ) I 89 Y 105 i 121 y 10 LF 26 SUB 42 * 58 : 74 J 90 Z 106 j 122 z 11 VT 27 ESC ; 75 K 91 [ 107 k 123 { 12 FF 28 FS 44, 60 < 76 L 92 \ 108 l CR 29 GS = 77 M 93 ] 109 m 125 } 14 S0 30 RS > 78 N 94 ^ 110 n 126 ~ 15 S1 31 US 47 / 63? 79 O 95 _ 111 o 127 DEL Compare the Unicode character encodings First Back TOC Representation of Strings Prev Next Last

45 Page 45 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob 4.10 Representation of Strings Strings are represented by sequences of characters ( usually: byte sequence) 1. Fixed length strings (example: length = 7) P e t e r SP SP C o m p u t e r 2. Variable length strings n-1 n P e t e r NUL 3. Encoding with string length at the beginning n-1 n 5 P e t e r First Back TOC Representation of Strings Prev Next Last

46 Page 46 Chapter 4: Binary Data Representation and Arithmetic Christian Jacob With ASCII charcter encoding string comparisons are possible: AA < AD because: < AA < Aa < A1 < AA < Hence, sorting of strings becomes easy: Compare their numerical representations! First Back TOC Representation of Strings Prev Next Last

### Memory is implemented as an array of electronic switches

Memory Structure Memory is implemented as an array of electronic switches Each switch can be in one of two states 0 or 1, on or off, true or false, purple or gold, sitting or standing BInary digits (bits)

### 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

### 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

### Chapter 5. Binary, octal and hexadecimal numbers

Chapter 5. Binary, octal and hexadecimal numbers A place to look for some of this material is the Wikipedia page http://en.wikipedia.org/wiki/binary_numeral_system#counting_in_binary Another place that

### ASCII Code. Numerous codes were invented, including Émile Baudot's code (known as Baudot

ASCII Code Data coding Morse code was the first code used for long-distance communication. Samuel F.B. Morse invented it in 1844. This code is made up of dots and dashes (a sort of binary code). It was

### Chapter 1. Binary, octal and hexadecimal numbers

Chapter 1. Binary, octal and hexadecimal numbers This material is covered in the books: Nelson Magor Cooke et al, Basic mathematics for electronics (7th edition), Glencoe, Lake Forest, Ill., 1992. [Hamilton

### 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

### Network Working Group Request for Comments: 20 October 16, 1969

Network Working Group Request for Comments: 20 Vint Cerf UCLA October 16, 1969 ASCII format for Network Interchange For concreteness, we suggest the use of standard 7-bit ASCII embedded in an 8 bit byte

### 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

### 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

### = 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

### 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

### URL encoding uses hex code prefixed by %. Quoted Printable encoding uses hex code prefixed by =.

ASCII = American National Standard Code for Information Interchange ANSI X3.4 1986 (R1997) (PDF), ANSI INCITS 4 1986 (R1997) (Printed Edition) Coded Character Set 7 Bit American National Standard Code

### Digital Logic Design. Introduction

Digital Logic Design Introduction A digital computer stores data in terms of digits (numbers) and proceeds in discrete steps from one state to the next. The states of a digital computer typically involve

### Section 5.3: GS1-128 Symbology Specifications

Section 5.3: Table of Contents 5.3.1 Symbology Characteristics...3 5.3.1.1 GS1-128 Symbology Characteristics...3 5.3.2 GS1-128 Bar Code Symbol Structure...4 5.3.3 GS1-128 Symbology Character Assignments...5

### 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

### Xi2000 Series Configuration Guide

U.S. Default Settings Sequence Reset Scanner Xi2000 Series Configuration Guide Auto-Sense Mode ON UPC-A Convert to EAN-13 OFF UPC-E Lead Zero ON Save Changes POS-X, Inc. 2130 Grant St. Bellingham, WA 98225

### 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

### 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

### 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

### The ASCII Character Set

The ASCII Character Set The American Standard Code for Information Interchange or ASCII assigns values between 0 and 255 for upper and lower case letters, numeric digits, punctuation marks and other symbols.

### 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

### 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

### 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

### ASCII ENCODED ENGLISH (CCSD0002)

TMG 8/92 Consultative Committee for Space Data Systems RECOMMENDATION FOR SPACE DATA SYSTEM STANDARDS ASCII ENCODED ENGLISH (CCSD0002) CCSDS 643.0-B-1 BLUE BOOK AUTHORITY Issue: Blue Book, Issue 1 Date:

### 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

### BAR CODE 39 ELFRING FONTS INC.

ELFRING FONTS INC. BAR CODE 39 This package includes 18 versions of a bar code 39 font in scalable TrueType and PostScript formats, a Windows utility, Bar39.exe, that helps you make bar codes, and Visual

### 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

### 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...

### Representação de Caracteres

Representação de Caracteres IFBA Instituto Federal de Educ. Ciencia e Tec Bahia Curso de Analise e Desenvolvimento de Sistemas Introdução à Ciência da Computação Prof. Msc. Antonio Carlos Souza Coletânea

### 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

### 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

### Symbols in subject lines. An in-depth look at symbols

An in-depth look at symbols What is the advantage of using symbols in subject lines? The age of personal emails has changed significantly due to the social media boom, and instead, people are receving

### Voyager 9520/40 Voyager GS9590 Eclipse 5145

Voyager 9520/40 Voyager GS9590 Eclipse 5145 Quick Start Guide Aller à www.honeywellaidc.com pour le français. Vai a www.honeywellaidc.com per l'italiano. Gehe zu www.honeywellaidc.com für Deutsch. Ir a

### 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

### This is great when speed is important and relatively few words are necessary, but Max would be a terrible language for writing a text editor.

Dealing With ASCII ASCII, of course, is the numeric representation of letters used in most computers. In ASCII, there is a number for each character in a message. Max does not use ACSII very much. In the

### Binary Numbers. Binary Octal Hexadecimal

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

### 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

### 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

### 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

### 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

### BARCODE READER V 2.1 EN USER MANUAL

BARCODE READER V 2.1 EN USER MANUAL INSTALLATION OF YOUR DEVICE PS-2 Connection RS-232 Connection (need 5Volts power supply) 1 INSTALLATION OF YOUR DEVICE USB Connection 2 USING THIS MANUAL TO SETUP YOUR

### BI-300. Barcode configuration and commands Manual

BI-300 Barcode configuration and commands Manual 1. Introduction This instruction manual is designed to set-up bar code scanner particularly to optimize the function of BI-300 bar code scanner. Terminal

### 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

### Binary math. Resources and methods for learning about these subjects (list a few here, in preparation for your research):

Binary math This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,

### DEBT COLLECTION SYSTEM ACCOUNT SUBMISSION FILE

CAPITAL RESOLVE LTD. DEBT COLLECTION SYSTEM ACCOUNT SUBMISSION FILE (DCS-ASF1107-7a) For further technical support, please contact Clive Hudson (IT Dept.), 01386 421995 13/02/2012 Account Submission File

### 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

### 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

### 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

### 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

### 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

### 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,

### MEP Y9 Practice Book A

1 Base Arithmetic 1.1 Binary Numbers We normally work with numbers in base 10. In this section we consider numbers in base 2, often called binary numbers. In base 10 we use the digits 0, 1, 2, 3, 4, 5,

### 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

### 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

### 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

### CHAPTER 8 BAR CODE CONTROL

CHAPTER 8 BAR CODE CONTROL CHAPTER 8 BAR CODE CONTROL - 1 CONTENTS 1. INTRODUCTION...3 2. PRINT BAR CODES OR EXPANDED CHARACTERS... 4 3. DEFINITION OF PARAMETERS... 5 3.1. Bar Code Mode... 5 3.2. Bar Code

### 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

### Create!form Barcodes. User Guide

Create!form Barcodes User Guide Barcodes User Guide Version 6.3 Copyright Bottomline Technologies, Inc. 2008. All Rights Reserved Printed in the United States of America Information in this document is

### ESPA 4.4.4 Nov 1984 PROPOSAL FOR SERIAL DATA INTERFACE FOR PAGING EQUIPMENT CONTENTS 1. INTRODUCTION 2. CHARACTER DESCRIPTION

PROPOSAL FOR SERIAL DATA INTERFACE FOR PAGING EQUIPMENT CONTENTS 1. INTRODUCTION 2. CHARACTER DESCRIPTION 2.1 CHARACTER STRUCTURE 2.2 THE CHARACTER SET 2.3 CONTROL CHARACTERS 2.3.1 Transmission control

### 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,

### Char Dec Oct Hex Char Dec Oct Hex Char Dec Oct Hex Char Dec Oct Hex

Table of ASCII Characters This table lists the ASCII characters and their decimal, octal and hexadecimal numbers. Characters which appear as names in parentheses (e.g., (nl)) are non-printing characters.

### 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

### 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

### 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

### 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

### 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

### 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.

### 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

### ASCII Driver Manual 2800/2900

ASCII Driver Manual 2800/2900 Compact Operator Interface Terminal with ASCII communication drivers Quartech Corporation 15923 Angelo Drive Macomb Township, Michigan 48042-4050 Phone: (586) 781-0373 FAX:

### 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

### INTERNATIONAL STANDARD

INTERNATIONAL STANDARD ISO/IEC 18004 First edition 2000-06-15 Information technology Automatic identification and data capture techniques Bar code symbology QR Code Technologies de l'information Techniques

### Command Emulator STAR Line Mode Command Specifications

Line Thermal Printer Command Emulator STAR Line Mode Command Specifications Revision 0.01 Star Micronics Co., Ltd. Special Products Division Table of Contents 1. Command Emulator 2 1-1) Command List 2

### 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

### TELOCATOR ALPHANUMERIC PROTOCOL (TAP)

TELOCATOR ALPHANUMERIC PROTOCOL (TAP) Version 1.8 February 4, 1997 TABLE OF CONTENTS 1.0 Introduction...1 2.0 TAP Operating Environment...1 3.0 Recommended Sequence Of Call Delivery From An Entry Device...2

### 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

### 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. ! 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

### 1. Convert the following base 10 numbers into 8-bit 2 s complement notation 0, -1, -12

C5 Solutions 1. Convert the following base 10 numbers into 8-bit 2 s complement notation 0, -1, -12 To Compute 0 0 = 00000000 To Compute 1 Step 1. Convert 1 to binary 00000001 Step 2. Flip the bits 11111110

### 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

### 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

### 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

### Characters & Strings Lesson 1 Outline

Outline 1. Outline 2. Numeric Encoding of Non-numeric Data #1 3. Numeric Encoding of Non-numeric Data #2 4. Representing Characters 5. How Characters Are Represented #1 6. How Characters Are Represented

### Character Codes for Modern Computers

Character Codes for Modern Computers This lecture covers the standard ways in which characters are stored in modern computers. There are five main classes of characters. 1. Alphabetic characters: upper

### MIPS Assembly Language Programming CS50 Discussion and Project Book. Daniel J. Ellard

MIPS Assembly Language Programming CS50 Discussion and Project Book Daniel J. Ellard September, 1994 Contents 1 Data Representation 1 1.1 Representing Integers........................... 1 1.1.1 Unsigned

### The first 32 characters in the ASCII-table are unprintable control codes and are used to control peripherals such as printers.

The following ASCII table contains both ASCII control characters, ASCII printable characters and the extended ASCII character set ISO 8859-1, also called ISO Latin1 The first 32 characters in the ASCII-table

### Barcode Magstripe. Decoder & Scanner. Programming Manual

Barcode Magstripe Decoder & Scanner Programming Manual CONTENTS Getting Started... 2 Setup Procedures... 3 Setup Flow Chart...4 Group 0 : Interface Selection... 5 Group 1 : Device Selection for keyboard

### 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,

### 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.,

### 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

### 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,

### 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

### 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

### ASCII Code - The extended ASCII table

ASCII Code - The extended ASCII table ASCII stands for American Standard Code for Information Interchange. It's a 7-bit character code where every single bit represents a unique character. On this webpage

### 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

### Counting in base 10, 2 and 16

Counting in base 10, 2 and 16 1. Binary Numbers A super-important fact: (Nearly all) Computers store all information in the form of binary numbers. Numbers, characters, images, music files --- all of these

### 7-Bit coded Character Set

Standard ECMA-6 6 th Edition - December 1991 Reprinted in electronic form in August 1997 Standardizing Information and Communication Systems 7-Bit coded Character Set Phone: +41 22 849.60.00 - Fax: +41