# Numbering systems. Dr Abu Arqoub

Size: px
Start display at page:

Transcription

1 Numbering systems The decimal numbering system is widely used, because the people Accustomed (معتاد) to use the hand fingers in their counting. But with the development of the computer science another numbering systems are founded. Such as: Binary, Octal, Hexadecimal, or any base r > 2. Numbering systems classification Positional numbering systems: that depends on the position of the digit in the number. Which means that the same digit can take different values according to his position [decimal, Binary, Octal, Hexadecimal] None positional numbering systems: that don t use digit weights in the numbers. [ROMAN numbering system] 1

2 Numbering Systems Radix The base or the radix of any numbering system is equal to the number of symbols that used to represent its numbers The decimal number system is said to be of Base (Radix), 10 because it uses 10 digits [ 0,1,2,.,9] The Binary number system is said to be of Base (Radix), 2 because it uses 2 digits [ 1,0 ] The Octal number system is said to be of Base (Radix), 8 because it uses 8 digits [0,1,2,3.,7] The Hexadecimal number system is said to be of Base (Radix), 16 because it uses 16 digits [0,1,2,3..,A,B,C,D,E,F] 2

3 decimal numbers The decimal number representation X= (d j d j-1.d 3 d 2 d 1 d 0.d -1 d -2 d -3, d -(k-1) d -k ) 10, where dj (digit): 0,1,2.9, j ( position): 3,2,1,0,-1,-2,-3.. X as a decimal number with j digits can be represented as X=10 j *d j +10 j-1 *d j *d *d *d *d *d (k-1) *d -(k-1) +10 -k *d -k the integer part of the number is represented by an ascending positive power The fraction part of the number is represented by an ascending negative power 3

4 4

5 Positional representation rule for numbers All positional numbering systems are the same in that they are depends on the wait of the digits in the number, and different only in the base. So we can use the same method of representing decimal numbers to be accepted by any other numbering system. As follows: N= i=n i= m a i R i Where : R- base of the system, ai- the digits of the number, m- the number of the fraction digits part, n- number of the integer part digits - 1 5

6 Binary system Is of base 2 and has two digits (0,1) It is the system that used by the computer It is represented in the electronic circuits as two states: on 1 off 0 6

7 Binary Digits (Bits) Only 2 states possible On 1 Electronic pulse present Positive magnetic field Off 0 Electronic pulse absent Negative magnetic field } Human readable symbols } Inside the computer s memory (RAM) } Permanently stored on disks On Off Pitted Not Pitted } Permanently stored on CD-ROM 7 Light Pulse No Light Pulse } Fiber Optic Cable

8 Binary representation Using the previous rule we can represent any binary number as x=.b 2 *2 2 +b 1 *2 1 +b 0 *2 0 +b -1 *2-1 +b -2 *2-2.. Where: b i =0,1 8

9 Most significant bit least significant bit 9

10 Octal system The base of this system is 8. Its symbols are (0,1,2.7) Using the previous rule we can represent any octal number as x=.o 2 *8 2 +o 1 *8 1 +o 0 *8 0 +o -1 *8-1 +o -2 *8-2.. Where: o i =0,1,2,3,4,5,6,7 10

11 11

12 Hexadecimal system The base of this system is 16 it has 16 digits(0,1,2.9,a,b,c,d,e,f) A=10,B=11,C=12,D=13,E=14,F=15 Using the previous rule we can represent any hexadecimal number as x=.h 2 *16 2 +H 1 *16 1 +H 0 *16 0 +H -1 *16-1 +H -2 * Where: H i =0,1,2,3,..9, A,B,C,D,E,F 12

13 13

14 Number base conversions Base R to decimal conversion A number expressed in base R can be converted to its decimal equivalent by using the Positional representation rule [ multiplying each coefficient with the corresponding power of R and adding] N= i=n i= m a i R i 14

15 Binary to decimal Number base conversions 15

16 Number base conversions Octal To Decimal Conversion Examples: (312.1) 8 =3*8 2 +1*8 1 +2*8 0 +1*8-1 =(3X64)+(8)+(2)+(1/8)=( ) 10 (752) 8 =7*8 2 +5*8 1 +2*8 0 =(7X64)+(5X8)+(2X1)=(490) 10 16

17 Number base conversions Hexadecimal to Decimal Conversion 17

18 Number base conversions Decimal to base R conversion The conversions from decimal to any base R number system is more convenient if the number is separated into an integer part and a fraction part and the conversion of each part is done separately. Example (d 3 d 2 d 1 d 0.d -1 d -2 d -3 ) 10 ( ) r (d 3 d 2 d 1 d 0 ) 10 ( ) r (.d -1 d -2 d -3 ) 10 ( ) r Integer part divided by base R Fraction part multiplied by base R 18

19 19 From decimal to binary We use the remainder method as follows: a) Integer part 1. The integer part of the number is divided by 2 2. The reminder calculated 3. The quotient resulted from step 1 divided by 2 4. The reminder calculated 5. The previous steps are repeated until the quotient becomes equal to 0 6. The required binary number is the collection of reminders ordered from the last reminder to the first one

20 From decimal to binary b) fraction part Converting fraction part is done using the multiplication instead of division as follows: 1. multiply the number by 2 2. Get the integer part from the result obtained from step 1 3. Multiply the fraction obtained in step1 by 2 4. Get the integer part from the result obtained from step 3 5. Repeat the above steps until we have the fraction part of the multiplication equals to 0 or we reached the required precision 6. The required number is the collection of integer digits obtained after each multiplication. 20

21 Examples ( ) 2 21

22 Examples Example : Convert the following decimal numbers to Binary numbers 1- (29)10 2-(53)10 3-(41)10 1: Division Remainder 29/2 1 LSB 14/2 0 7/2 1 3/2 1 1/2 1 MSB 0 (29)10 =(11101)2 2: (53)10 Division Remainder 53/2 1 LSB 26/2 0 13/2 1 6 /2 0 3 /2 1 1 /2 1 MSB 0 (53)10 =(110101)2 3 : (41)10 41/2 1 LSB 20/2 0 10/2 0 5 /2 1 2 /2 0 1 /2 1 MSB 0 22 (41)10 =(101001)2

23 Examples Convert the following decimal number to a binary ( ) 10 23

24 Examples Example: Convert the following decimal number into binary Number 1: ( ) x2 = MSB x 2 = x 2 = x 2 = x 2 = x 2 = LSB ( )10 =( )2 2: ( )10 =(41)10 + (0.625 )10 from the above example (41)10 is represented by (101001)2 (0.625 )10 is x 2 = MSB 0.25 x 2 = x 2 = LSB (0.625 )10 = (101)2 so ( )10 = ( )2 while 24

25 Decimal to octal Example: Convert the following decimal number into octal Number ( ) 10 (231.54) 10 25

26 Decimal to hexadecimal Example: Convert the following decimal number into hexadecimal Number ( ) 10 26

27 Binary to octal or hexadecimal and vise versa Each octal digit corresponds to 3 binary digits Base 8 = 2 3 Each hexadecimal digit corresponds to 4 binary digits Base 16=2 4 27

28 Binary to octal Group each 3 binary digits from the radix point 28

29 Binary to octal 29

30 Binary to hexadecimal Group each 4 binary digits from the radix point 30

32 Octal or hexadecimal to binary Each octal digit is converted to 3 binary digits according to the table 32

33 Octal or hexadecimal to binary Each hexadecimal digit is converted to 4 binary digits according to the table 33

34 Examples 34

35 Examples 35

36 Signed Number Representation : Any signed number can be represented using one of the following methods: A) Signed (true) and Magnitude Representation (SAM) B) complements R s complement (R-1) s complement Where R is the base of the system 36

37 Signed Number Representation : A) Signed (true) and Magnitude Representation (SAM): the leftmost bit (MSB) is used as a sign bit,where 0 indicates a positive integer and R-1 indicates a negative integer. The rest of the number is the magnitude sign =0 : positive = Sign bit magnitude sign =1 : Negative =

38 Signed Number Representation : Examples Represent the following numbers using the SAM method, where each number is of 6 digits, one of them for the sign (+ 67) (-67) (-2E) 16 (+ 4F) 16 F 0002E F 38

39 Signed Number Representation : B) Complements form Complements are used in digital computers for simplifying the subtraction and for logical manipulations There are two types of complements for each base R system: 1. r s complements 2. (r-1) s complements, r >= 2 base number 39

40 R s complement Given a positive number N in base r with an integer part of m digits, the r s complement of N is defined as r s complement( N)= r m -N, N#0 Example: 0, N=0 the 10 s complement of (25.639) 10 = (10 2 ) 10 - (25.639) 10 = the 8 s complement of ( ) 8 = (8 3 ) 10 - ( ) 8 = (45.655) 8 The 2 s complement of ( ) 2 =(2 5 ) 10 - ( ) 2 = =( ) 2 40

41 (R-1) s complement Given a positive number N in base r with an integer part of m digits, and a fraction part of k digits, the (r-1) s complement of N is defined as (R-1) s complement( N)= r m -r -k -N, no meter if n=0 Examples: The 9 s complement of (25.639) =100-1/ =(74.360) 10 The 9 s complement (25) = 74 The 7 s complement of ( ) 8 (8 3 ) 10 -(8-3 ) 10 -( ) 8 =(46.645) 8 The 1 s complement of ( ) 2 (2 7 ) ( ) 2 = =(11) 10 =( ) 2 41

42 Basic rules for binary addition and subtraction 42

43 1 s Complement Change each 0 to 1 and each 1 to 0 Example: find the 1 s complement of the following binary numbers 0101 the 1 s complement is the 1 s complement is 43

44 2 s complement 2 s complement= 1 s complement +1 Example: find the 2 s complement of the following binary numbers s complement s complement= 1 s complement + 1 = = s complement s complement= 1 s complement +1= =

45 45

46 Binary Arithmetic Unsigned Numbers 1- Binary Addition: Example: Carries Carries Carries Carries overflow

47 Unsigned Binary Addition Example 4 bit numbers can represent numbers from 0 to The result 18 is not in the range [0,15], And is too big. Consequently, Overflow happens here. So the carry bit from adding the two most significant bits represents a results that overflows.

48 Binary Subtraction I- Using 2 s complement: 1:- Find the 2's complement for the number with negative sign 2:- Add the 2 s complement to the second number. 3:- if the result is with a last carry (carry =1) then the result is positive discard the last carry, otherwise (carry=0) the result is negative and it is in the 2 s complement form ( get the 2's complement of the result and put a negative sign in front of the number). Example 1: carry =1 result positive =01 48

49 Using 2 s complement Example 2: carry=1 positive result=0100 Example 3: carry=0 Negative result (2 s complement form) s comp = -1 49

50 Using 2 s complement Example 4: carry=0 negative result (2 s complement form) s comp =

51 II- Using 1 s complement II- Using 1 s complement: - Get the 1's complement for the number with the negative sign - add the 1 s complement to the second number - if an end carry occur (carry =1 (end around carry)) add it to the least significant bit and the answer is positive. - otherwise (carry=0) : the answer is negative in the 1 s complement form (get the 1's complement of the result and put a negative sign in front of the number) Example 1: carry=1 (end around carry) positive result=0100=4 51

52 Example 2: II- Using 1 s complement carry=0 negative result in1 s complement form 1 s complement of 110 is 001=-1 52

53 Signed Binary Addition Example signed 4 bit numbers (2's complement) can represent numbers between -7 to 7 The extra carry from the most significant bit has no meaning 53

54 Signed Binary Addition Example signed 4 bit numbers (2's complement) can represent numbers between -7 to 7 if two numbers with the same sign (either positive or negative) are added and the result has the opposite sign, an overflow has occurred. 54

55 Binary Multiplication There are two methods fo multiplication 1. Shift and add 2. Repeated addition 1. shift and add the multiplication in the binary system is the same as the multiplication in the decimal system. by shift and add. 0x0=0 0x1=0 1x0=0 1x1=1 Example: x 2 x

56 Binary Multiplication 2. Binary Multiplication by Repeated Addition Repeated addition - add the second number to itself the number of times represented by the first multiplicand. Set result to 0 Repeat Add second multiplicand to result Reduce first multiplicand by 1 Until first multiplicand = 0 Result is correct STOP 56

57 Binary Multiplication Example x 2 x 10 x 10 x 10 x 10 x 10 x 10 x stop 57

58 Binary division There are two methods for binary division 1. shift subtract 2.repeated subtraction 1. shift subtract Set quotient to 0 Align leftmost digits in dividend and divisor Repeat If that portion of the dividend above the divisor is greater than or equal to the divisor Then subtract divisor from that portion of the dividend and Concatenate 1 to the right hand end of the quotient Else concatenate 0 to the right hand end of the quotient Shift the divisor one place right Until dividend is less than the divisor quotient is correct, dividend is remainder STOP 58

59 Binary division. 59

60 Binary division 2. Binary Division by Repeated Subtraction Set quotient to zero (quotient as counter) Repeat while dividend is greater than or equal to divisor Subtract divisor from dividend Add 1 to quotient End of repeat block quotient is correct, dividend is remainder STOP 60

61 Binary division 1111/11 61

62 Binary division 111 /10 62

63 Binary codes When numbers, letters, or words are represented by a special group of symbols, this is called encoding. And the group of symbols is called a code When a decimal number is represented by its equivalent binary number, we call it binary coding Ex. decimal binary code

64 Base 4 code Base 4 : 0, 1, 2, 3 Binary code: 00, 01, 10, 11 decimal base

65 Binary Coded Decimal code In computing and electronic systems, binary-coded decimal (BCD) (sometimes called natural binary-coded decimal, NBCD) is an encoding for decimal numbers in which each digit is represented by its own binary sequence Its main virtue is that it allows easy conversion to decimal digits for printing or display, and allows faster decimal calculations. Its drawbacks are a small increase in the complexity of circuits needed to implement mathematical operations Many non-integral values, such as decimal 0.2, have an infinite place-value representation in binary ( ) but have a finite place-value in binary-coded decimal (0.0010). Consequently a system based on binary-coded decimal representations of decimal fractions avoids errors representing and calculating such values. 65

66 BCD The BIOS in many personal computers stores the date and time in BCD because the MC6818 real-time clock chip used in the original IBM PC AT motherboard provided the time encoded in BCD. This form is easily converted into ASCII for display. In BCD, a digit is usually represented by four bits which, in general, represent the values /digits / characters

67 BCD

68 Addition in BCD The same as binary rules, but some times the result need some correction if we get a number greater than 9. Example: illegal code > 9 To be legal code we correct it by adding =

69 Subtraction in BCD The same rules as binary with correction 6 if the subtrahend > Minuend. Example: 52-24= illegal code >9, so subtract

70 Excess-3 code Excess-3 binary-coded decimal (XS-3), also called biased منحازة التمثيلrepresentation or Excess-N, is a numeral system used on some older computers that uses a pre-specified number N as a biasing value. It is a way to represent values with a balanced number of positive and negative numbers. In XS-3, numbers are represented as decimal digits, and each digit is represented by four bits as the BCD value plus 3 (the "excess" amount): 70

71 Excess-3 code 71

72 Gray code The reflected binary code, also known as Gray code after Frank Gray, is a binary numeral system where two successive values differ in only one bit. The reflected binary code was originally designed to prevent spurious output from electromechanical switches. Today, Gray codes are widely used to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. 72

73 73

74 From binary to Gray The most significant bit of the binary code is the same most significant bit in Gray code Add each two successive digits from the left to produce new gray digit, carry is discarded. Example (1100) 2 =( ) g (1 ) g ---- (10 ) g -----(101 ) g (1010) g 74

75 From gray to binary The most significant bit of the gray code is the same most significant bit in binary code Add this digit to the next digit in the gray cod,( diagonally addition) for all bits. Example ( ) g =( ) 2 (1 ) (11 ) (110 ) (1101 ) (11010 ) ( ) 75

76 Alpha- Numerical code (ASCII) American Standard Code for Information and Interchange. Many applications of digital computers require handling of data that consist not only of numbers, but also of letters. Numbers : 0-9 (10 numbers) Small letters : a-z(26 letters) Capital letters : A Z (26 letters) Special characters :?,!, #,.. (32 characters) all of 94 elements 2 6 =64, 2 7 =128 7 bit 76

77 ASCII-table 77

### BINARY SYSTEM. Binary system is used in digital systems because it is:

CHAPTER 2 CHAPTER CONTENTS 2.1 Binary System 2.2 Binary Arithmetic Operation 2.3 Signed & Unsigned Numbers 2.4 Arithmetic Operations of Signed Numbers 2.5 Hexadecimal Number System 2.6 Octal Number System

### Number Systems and Conversions UNIT 1 NUMBER SYSTEMS & CONVERSIONS. Number Systems (2/2) Number Systems (1/2) Iris Hui-Ru Jiang Spring 2010

Contents Number systems and conversion Binary arithmetic Representation of negative numbers Addition of two s complement numbers Addition of one s complement numbers Binary s Readings Unit.~. UNIT NUMBER

### Chapter 1 Review of Number Systems

1.1 Introduction Chapter 1 Review of Number Systems Before the inception of digital computers, the only number system that was in common use is the decimal number system which has a total of 10 digits

### CHW 261: Logic Design

CHW 261: Logic Design Instructors: Prof. Hala Zayed Dr. Ahmed Shalaby http://www.bu.edu.eg/staff/halazayed14 http://bu.edu.eg/staff/ahmedshalaby14# Slide 1 Slide 2 Slide 3 Digital Fundamentals CHAPTER

### CHAPTER 2 (b) : AND CODES

DKT 122 / 3 DIGITAL SYSTEMS 1 CHAPTER 2 (b) : NUMBER SYSTEMS OPERATION AND CODES m.rizal@unimap.edu.my sitizarina@unimap.edu.my DECIMAL VALUE OF SIGNED NUMBERS SIGN-MAGNITUDE: Decimal values of +ve & -ve

### Digital Fundamentals. CHAPTER 2 Number Systems, Operations, and Codes

Digital Fundamentals CHAPTER 2 Number Systems, Operations, and Codes Decimal Numbers The decimal number system has ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 The decimal numbering system has a base of

### Korea University of Technology and Education

MEC52 디지털공학 Binary Systems Jee-Hwan Ryu School of Mechanical Engineering Binary Numbers a 5 a 4 a 3 a 2 a a.a - a -2 a -3 base or radix = a n r n a n- r n-...a 2 r 2 a ra a - r - a -2 r -2...a -m r -m

### Digital Fundamentals

Digital Fundamentals Tenth Edition Floyd Chapter 2 2009 Pearson Education, Upper 2008 Pearson Saddle River, Education NJ 07458. All Rights Reserved Decimal Numbers The position of each digit in a weighted

### Module 2: Computer Arithmetic

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

### 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

### Lecture 2: Number Systems

Lecture 2: Number Systems Syed M. Mahmud, Ph.D ECE Department Wayne State University Original Source: Prof. Russell Tessier of University of Massachusetts Aby George of Wayne State University Contents

### MACHINE LEVEL REPRESENTATION OF DATA

MACHINE LEVEL REPRESENTATION OF DATA CHAPTER 2 1 Objectives Understand how integers and fractional numbers are represented in binary Explore the relationship between decimal number system and number systems

### Chapter 1. Digital Systems and Binary Numbers

Chapter 1. Digital Systems and Binary Numbers Tong In Oh 1 1.1 Digital Systems Digital age Characteristic of digital system Generality and flexibility Represent and manipulate discrete elements of information

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

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

### Logic Circuits I ECE 1411 Thursday 4:45pm-7:20pm. Nathan Pihlstrom.

Logic Circuits I ECE 1411 Thursday 4:45pm-7:20pm Nathan Pihlstrom www.uccs.edu/~npihlstr My Background B.S.E.E. from Colorado State University M.S.E.E. from Colorado State University M.B.A. from UCCS Ford

### Chap 1. Digital Computers and Information

Chap 1. Digital Computers and Information Spring 004 Overview Digital Systems and Computer Systems Information Representation Number Systems [binary, octal and hexadecimal] Arithmetic Operations Base Conversion

### MC1601 Computer Organization

MC1601 Computer Organization Unit 1 : Digital Fundamentals Lesson1 : Number Systems and Conversions (KSB) (MCA) (2009-12/ODD) (2009-10/1 A&B) Coverage - Lesson1 Shows how various data types found in digital

### Positional Number System

Positional Number System A number is represented by a string of digits where each digit position has an associated weight. The weight is based on the radix of the number system. Some common radices: Decimal.

### Digital Systems and Binary Numbers

Digital Systems and Binary Numbers Mano & Ciletti Chapter 1 By Suleyman TOSUN Ankara University Outline Digital Systems Binary Numbers Number-Base Conversions Octal and Hexadecimal Numbers Complements

### D I G I T A L C I R C U I T S E E

D I G I T A L C I R C U I T S E E Digital Circuits Basic Scope and Introduction This book covers theory solved examples and previous year gate question for following topics: Number system, Boolean algebra,

### Lecture (02) Operations on numbering systems

Lecture (02) Operations on numbering systems By: Dr. Ahmed ElShafee ١ Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I Complements of a number Complements are used in digital computers to simplify

### DIGITAL SYSTEM FUNDAMENTALS (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) COURSE / CODE NUMBER SYSTEM

COURSE / CODE DIGITAL SYSTEM FUNDAMENTALS (ECE 421) DIGITAL ELECTRONICS FUNDAMENTAL (ECE 422) NUMBER SYSTEM A considerable subset of digital systems deals with arithmetic operations. To understand the

### Digital Systems and Binary Numbers

Digital Systems and Binary Numbers ( 范倫達 ), Ph. D. Department of Computer Science National Chiao Tung University Taiwan, R.O.C. Spring, 2018 ldvan@cs.nctu.edu.tw http://www.cs.nctu.edu.tw/~ldvan/ Outline

### CS 121 Digital Logic Design. Chapter 1. Teacher Assistant. Hadeel Al-Ateeq

CS 121 Digital Logic Design Chapter 1 Teacher Assistant Hadeel Al-Ateeq Announcement DON T forgot to SIGN your schedule OR you will not be allowed to attend next lecture. Communication Office hours (8

### World Inside a Computer is Binary

C Programming 1 Representation of int data World Inside a Computer is Binary C Programming 2 Decimal Number System Basic symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Radix-10 positional number system. The radix

### Ms Sandhya Rani Dash UNIT 2: NUMBER SYSTEM AND CODES. 1.1 Introduction

Ms Sandhya Rani Dash UNIT 2: NUMBER SYSTEM AND CODES Structure 2.1 Introduction 2.2 Objectives 2.3 Binary Numbers 2.3.1 Binary-to-Decimal conversion 2.3.2 Decimal-to-Binary Conversion 2.4 Octal Numbers

### Organisasi Sistem Komputer

LOGO Organisasi Sistem Komputer OSK 8 Aritmatika Komputer 1 1 PT. Elektronika FT UNY Does the calculations Arithmetic & Logic Unit Everything else in the computer is there to service this unit Handles

### Digital Systems and Binary Numbers

Digital Systems and Binary Numbers Prof. Wangrok Oh Dept. of Information Communications Eng. Chungnam National University Prof. Wangrok Oh(CNU) 1 / 51 Overview 1 Course Summary 2 Binary Numbers 3 Number-Base

### Chapter 3: Number Systems and Codes. Textbook: Petruzella, Frank D., Programmable Logic Controllers. McGraw Hill Companies Inc.

Chapter 3: Number Systems and Codes Textbook: Petruzella, Frank D., Programmable Logic Controllers. McGraw Hill Companies Inc., 5 th edition Decimal System The radix or base of a number system determines

### DLD VIDYA SAGAR P. potharajuvidyasagar.wordpress.com. Vignana Bharathi Institute of Technology UNIT 1 DLD P VIDYA SAGAR

UNIT I Digital Systems: Binary Numbers, Octal, Hexa Decimal and other base numbers, Number base conversions, complements, signed binary numbers, Floating point number representation, binary codes, error

### Computer Sc. & IT. Digital Logic. Computer Sciencee & Information Technology. 20 Rank under AIR 100. Postal Correspondence

GATE Postal Correspondence Computer Sc. & IT 1 Digital Logic Computer Sciencee & Information Technology (CS) 20 Rank under AIR 100 Postal Correspondence Examination Oriented Theory, Practice Set Key concepts,

### CHAPTER 2 Data Representation in Computer Systems

CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 37 2.2 Positional Numbering Systems 38 2.3 Decimal to Binary Conversions 38 2.3.1 Converting Unsigned Whole Numbers 39 2.3.2 Converting

### Review of Number Systems

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 digital computer.

### Moodle WILLINGDON COLLEGE SANGLI. ELECTRONICS (B. Sc.-I) Introduction to Number System

Moodle 1 WILLINGDON COLLEGE SANGLI ELECTRONICS (B. Sc.-I) Introduction to Number System E L E C T R O N I C S Introduction to Number System and Codes Moodle developed By Dr. S. R. Kumbhar Department of

### DIGITAL SYSTEM DESIGN

DIGITAL SYSTEM DESIGN UNIT I: Introduction to Number Systems and Boolean Algebra Digital and Analog Basic Concepts, Some history of Digital Systems-Introduction to number systems, Binary numbers, Number

### 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

### Binary Systems and Codes

1010101010101010101010101010101010101010101010101010101010101010101010101010101010 1010101010101010101010101010101010101010101010101010101010101010101010101010101010 1010101010101010101010101010101010101010101010101010101010101010101010101010101010

### COMP Overview of Tutorial #2

COMP 1402 Winter 2008 Tutorial #2 Overview of Tutorial #2 Number representation basics Binary conversions Octal conversions Hexadecimal conversions Signed numbers (signed magnitude, one s and two s complement,

### COMPUTER ARITHMETIC (Part 1)

Eastern Mediterranean University School of Computing and Technology ITEC255 Computer Organization & Architecture COMPUTER ARITHMETIC (Part 1) Introduction The two principal concerns for computer arithmetic

### Binary Codes. Dr. Mudathir A. Fagiri

Binary Codes Dr. Mudathir A. Fagiri Binary System The following are some of the technical terms used in binary system: Bit: It is the smallest unit of information used in a computer system. It can either

### Logic and Computer Design Fundamentals. Chapter 1 Digital Computers and Information

Logic and Computer Design Fundamentals Chapter 1 Digital Computers and Information Overview Digital Systems and Computer Systems Information Representation Number Systems [binary, octal and hexadecimal]

### Chapter 2. Data Representation in Computer Systems

Chapter 2 Data Representation in Computer Systems Chapter 2 Objectives Understand the fundamentals of numerical data representation and manipulation in digital computers. Master the skill of converting

### 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 ) +(

### Computer Organization

Computer Organization Register Transfer Logic Number System Department of Computer Science Missouri University of Science & Technology hurson@mst.edu 1 Decimal Numbers: Base 10 Digits: 0, 1, 2, 3, 4, 5,

### Number Systems CHAPTER Positional Number Systems

CHAPTER 2 Number Systems Inside computers, information is encoded as patterns of bits because it is easy to construct electronic circuits that exhibit the two alternative states, 0 and 1. The meaning of

### CHAPTER 2 Data Representation in Computer Systems

CHAPTER 2 Data Representation in Computer Systems 2.1 Introduction 37 2.2 Positional Numbering Systems 38 2.3 Decimal to Binary Conversions 38 2.3.1 Converting Unsigned Whole Numbers 39 2.3.2 Converting

### Number System. Introduction. Decimal Numbers

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

### in this web service Cambridge University Press

978-0-51-85748- - Switching and Finite Automata Theory, Third Edition Part 1 Preliminaries 978-0-51-85748- - Switching and Finite Automata Theory, Third Edition CHAPTER 1 Number systems and codes This

### MYcsvtu Notes DATA REPRESENTATION. Data Types. Complements. Fixed Point Representations. Floating Point Representations. Other Binary Codes

DATA REPRESENTATION Data Types Complements Fixed Point Representations Floating Point Representations Other Binary Codes Error Detection Codes Hamming Codes 1. DATA REPRESENTATION Information that a Computer

### CHAPTER TWO. Data Representation ( M.MORRIS MANO COMPUTER SYSTEM ARCHITECTURE THIRD EDITION ) IN THIS CHAPTER

1 CHAPTER TWO Data Representation ( M.MORRIS MANO COMPUTER SYSTEM ARCHITECTURE THIRD EDITION ) IN THIS CHAPTER 2-1 Data Types 2-2 Complements 2-3 Fixed-Point Representation 2-4 Floating-Point Representation

### Decimal & Binary Representation Systems. Decimal & Binary Representation Systems

Decimal & Binary Representation Systems Decimal & binary are positional representation systems each position has a value: d*base i for example: 321 10 = 3*10 2 + 2*10 1 + 1*10 0 for example: 101000001

### COMPUTER ORGANIZATION AND ARCHITECTURE

COMPUTER ORGANIZATION AND ARCHITECTURE For COMPUTER SCIENCE COMPUTER ORGANIZATION. SYLLABUS AND ARCHITECTURE Machine instructions and addressing modes, ALU and data-path, CPU control design, Memory interface,

### Chapter 5: Computer Arithmetic. In this chapter you will learn about:

Slide 1/29 Learning Objectives In this chapter you will learn about: Reasons for using binary instead of decimal numbers Basic arithmetic operations using binary numbers Addition (+) Subtraction (-) Multiplication

### CS & IT Conversions. Magnitude 10,000 1,

CS & IT Conversions There are several number systems that you will use when working with computers. These include decimal, binary, octal, and hexadecimal. Knowing how to convert between these number systems

### Digital Fundamentals

Digital Fundamentals Tenth Edition Floyd Chapter 1 Modified by Yuttapong Jiraraksopakun Floyd, Digital Fundamentals, 10 th 2008 Pearson Education ENE, KMUTT ed 2009 Analog Quantities Most natural quantities

### Excerpt from: Stephen H. Unger, The Essence of Logic Circuits, Second Ed., Wiley, 1997

Excerpt from: Stephen H. Unger, The Essence of Logic Circuits, Second Ed., Wiley, 1997 APPENDIX A.1 Number systems and codes Since ten-fingered humans are addicted to the decimal system, and since computers

### COE 202: Digital Logic Design Number Systems Part 2. Dr. Ahmad Almulhem ahmadsm AT kfupm Phone: Office:

COE 0: Digital Logic Design Number Systems Part Dr. Ahmad Almulhem Email: ahmadsm AT kfupm Phone: 860-7554 Office: -34 Objectives Arithmetic operations: Binary number system Other number systems Base Conversion

### SE311: Design of Digital Systems

SE311: Design of Digital Systems Lecture 3: Complements and Binary arithmetic Dr. Samir Al-Amer (Term 041) SE311_Lec3 (c) 2004 AL-AMER ١ Outlines Complements Signed Numbers Representations Arithmetic Binary

### Arithmetic Processing

CS/EE 5830/6830 VLSI ARCHITECTURE Chapter 1 Basic Number Representations and Arithmetic Algorithms Arithmetic Processing AP = (operands, operation, results, conditions, singularities) Operands are: Set

### Learning Objectives. Binary over Decimal. In this chapter you will learn about:

Ref Page Slide 1/29 Learning Objectives In this chapter you will learn about: Reasons for using binary instead of decimal numbers Basic arithmetic operations using binary numbers Addition (+) Subtraction

### Internal Data Representation

Appendices This part consists of seven appendices, which provide a wealth of reference material. Appendix A primarily discusses the number systems and their internal representation. Appendix B gives information

### Data Representation COE 301. Computer Organization Prof. Muhamed Mudawar

Data Representation COE 30 Computer Organization Prof. Muhamed Mudawar College of Computer Sciences and Engineering King Fahd University of Petroleum and Minerals Presentation Outline Positional Number

### 9/3/2015. Data Representation II. 2.4 Signed Integer Representation. 2.4 Signed Integer Representation

Data Representation II CMSC 313 Sections 01, 02 The conversions we have so far presented have involved only unsigned numbers. To represent signed integers, computer systems allocate the high-order bit

### Number Systems Prof. Indranil Sen Gupta Dept. of Computer Science & Engg. Indian Institute of Technology Kharagpur Number Representation

Number Systems Prof. Indranil Sen Gupta Dept. of Computer Science & Engg. Indian Institute of Technology Kharagpur 1 Number Representation 2 1 Topics to be Discussed How are numeric data items actually

### carry in carry 1101 carry carry

Chapter Binary arithmetic Arithmetic is the process of applying a mathematical operator (such as negation or addition) to one or more operands (the values being operated upon). Binary arithmetic works

### Chapter 10 - Computer Arithmetic

Chapter 10 - Computer Arithmetic Luis Tarrataca luis.tarrataca@gmail.com CEFET-RJ L. Tarrataca Chapter 10 - Computer Arithmetic 1 / 126 1 Motivation 2 Arithmetic and Logic Unit 3 Integer representation

### Chapter 5 : Computer Arithmetic

Chapter 5 Computer Arithmetic Integer Representation: (Fixedpoint representation): An eight bit word can be represented the numbers from zero to 255 including = 1 = 1 11111111 = 255 In general if an nbit

### Kinds Of Data CHAPTER 3 DATA REPRESENTATION. Numbers Are Different! Positional Number Systems. Text. Numbers. Other

Kinds Of Data CHAPTER 3 DATA REPRESENTATION Numbers Integers Unsigned Signed Reals Fixed-Point Floating-Point Binary-Coded Decimal Text ASCII Characters Strings Other Graphics Images Video Audio Numbers

### Computer Organization

Computer Organization It describes the function and design of the various units of digital computers that store and process information. It also deals with the units of computer that receive information

### 1010 2?= ?= CS 64 Lecture 2 Data Representation. Decimal Numbers: Base 10. Reading: FLD Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

CS 64 Lecture 2 Data Representation Reading: FLD 1.2-1.4 Decimal Numbers: Base 10 Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Example: 3271 = (3x10 3 ) + (2x10 2 ) + (7x10 1 ) + (1x10 0 ) 1010 10?= 1010 2?= 1

### NUMBER SYSTEMS AND CODES

C H A P T E R 69 Learning Objectives Number Systems The Decimal Number System Binary Number System Binary to Decimal Conversion Binary Fractions Double-Dadd Method Decimal to Binary Conversion Shifting

### Number representations

Number representations Number bases Three number bases are of interest: Binary, Octal and Hexadecimal. We look briefly at conversions among them and between each of them and decimal. Binary Base-two, or

### CPE 323 REVIEW DATA TYPES AND NUMBER REPRESENTATIONS IN MODERN COMPUTERS

CPE 323 REVIEW DATA TYPES AND NUMBER REPRESENTATIONS IN MODERN COMPUTERS Aleksandar Milenković The LaCASA Laboratory, ECE Department, The University of Alabama in Huntsville Email: milenka@uah.edu Web:

### Number Systems Base r

King Fahd University of Petroleum & Minerals Computer Engineering Dept COE 2 Fundamentals of Computer Engineering Term 22 Dr. Ashraf S. Hasan Mahmoud Rm 22-44 Ext. 724 Email: ashraf@ccse.kfupm.edu.sa 3/7/23

### COMP2121: Microprocessors and Interfacing. Number Systems

COMP2121: Microprocessors and Interfacing Number Systems http://www.cse.unsw.edu.au/~cs2121 Lecturer: Hui Wu Session 2, 2017 1 1 Overview Positional notation Decimal, hexadecimal, octal and binary Converting

### CPE 323 REVIEW DATA TYPES AND NUMBER REPRESENTATIONS IN MODERN COMPUTERS

CPE 323 REVIEW DATA TYPES AND NUMBER REPRESENTATIONS IN MODERN COMPUTERS Aleksandar Milenković The LaCASA Laboratory, ECE Department, The University of Alabama in Huntsville Email: milenka@uah.edu Web:

### (+A) + ( B) + (A B) (B A) + (A B) ( A) + (+ B) (A B) + (B A) + (A B) (+ A) (+ B) + (A - B) (B A) + (A B) ( A) ( B) (A B) + (B A) + (A B)

COMPUTER ARITHMETIC 1. Addition and Subtraction of Unsigned Numbers The direct method of subtraction taught in elementary schools uses the borrowconcept. In this method we borrow a 1 from a higher significant

### Slide Set 1. for ENEL 339 Fall 2014 Lecture Section 02. Steve Norman, PhD, PEng

Slide Set 1 for ENEL 339 Fall 2014 Lecture Section 02 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2014 ENEL 353 F14 Section

### ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-278: Digital Logic Design Fall Notes - Unit 4. hundreds.

ECE-78: Digital Logic Design Fall 6 UNSIGNED INTEGER NUMBERS Notes - Unit 4 DECIMAL NUMBER SYSTEM A decimal digit can take values from to 9: Digit-by-digit representation of a positive integer number (powers

### 10.1. Unit 10. Signed Representation Systems Binary Arithmetic

0. Unit 0 Signed Representation Systems Binary Arithmetic 0.2 BINARY REPRESENTATION SYSTEMS REVIEW 0.3 Interpreting Binary Strings Given a string of s and 0 s, you need to know the representation system

### ELECTRICAL AND COMPUTER ENGINEERING DEPARTMENT, OAKLAND UNIVERSITY ECE-2700: Digital Logic Design Winter Notes - Unit 4. hundreds.

UNSIGNED INTEGER NUMBERS Notes - Unit 4 DECIMAL NUMBER SYSTEM A decimal digit can take values from to 9: Digit-by-digit representation of a positive integer number (powers of ): DIGIT 3 4 5 6 7 8 9 Number:

### Divide: Paper & Pencil

Divide: Paper & Pencil 1001 Quotient Divisor 1000 1001010 Dividend -1000 10 101 1010 1000 10 Remainder See how big a number can be subtracted, creating quotient bit on each step Binary => 1 * divisor or

### ECE 20B, Winter Purpose of Course. Introduction to Electrical Engineering, II. Administration

ECE 20B, Winter 2003 Introduction to Electrical Engineering, II Instructor: Andrew B Kahng (lecture) Email: abk@eceucsdedu Telephone: 858-822-4884 office, 858-353-0550 cell Office: 3802 AP&M Lecture: TuThu

### Chapter 5: Computer Arithmetic

Slide 1/29 Learning Objectives Computer Fundamentals: Pradeep K. Sinha & Priti Sinha In this chapter you will learn about: Reasons for using binary instead of decimal numbers Basic arithmetic operations

### Positional notation Ch Conversions between Decimal and Binary. /continued. Binary to Decimal

Positional notation Ch.. /continued Conversions between Decimal and Binary Binary to Decimal - use the definition of a number in a positional number system with base - evaluate the definition formula using

### Representing Information. Bit Juggling. - Representing information using bits - Number representations. - Some other bits - Chapters 1 and 2.3,2.

Representing Information 0 1 0 Bit Juggling 1 1 - Representing information using bits - Number representations 1 - Some other bits 0 0 - Chapters 1 and 2.3,2.4 Motivations Computers Process Information

### Computer Arithmetic. Appendix A Fall 2003 Lec.03-58

Computer Arithmetic Appendix A 18-347 Fall 2003 Lec.03-58 Intro Computer Arithmetic Computers use the binary system Easy to implement in electronics: 1 is 1V, 0 is 0V Easy to implement with switches (transistors!)

### Lecture 8: Addition, Multiplication & Division

Lecture 8: Addition, Multiplication & Division Today s topics: Signed/Unsigned Addition Multiplication Division 1 Signed / Unsigned The hardware recognizes two formats: unsigned (corresponding to the C

### Rui Wang, Assistant professor Dept. of Information and Communication Tongji University.

Data Representation ti and Arithmetic for Computers Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Questions What do you know about

### CS 101: Computer Programming and Utilization

CS 101: Computer Programming and Utilization Jul-Nov 2017 Umesh Bellur (cs101@cse.iitb.ac.in) Lecture 3: Number Representa.ons Representing Numbers Digital Circuits can store and manipulate 0 s and 1 s.

### Agenda EE 224: INTRODUCTION TO DIGITAL CIRCUITS & COMPUTER DESIGN. Lecture 1: Introduction. Go over the syllabus 3/31/2010

// EE : INTRODUCTION TO DIGITAL CIRCUITS & COMPUTER DESIGN Lecture : Introduction /9/ Avinash Kodi, kodi@ohio.edu Agenda Go over the syllabus Introduction ti to Digital it Systems // Why Digital Systems?

### DATA REPRESENTATION. By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region. Based on CBSE curriculum Class 11. Neha Tyagi, KV 5 Jaipur II Shift

DATA REPRESENTATION Based on CBSE curriculum Class 11 By- Neha Tyagi PGT CS KV 5 Jaipur II Shift, Jaipur Region Neha Tyagi, KV 5 Jaipur II Shift Introduction As we know that computer system stores any

### COMP2611: Computer Organization. Data Representation

COMP2611: Computer Organization Comp2611 Fall 2015 2 1. Binary numbers and 2 s Complement Numbers 3 Bits: are the basis for binary number representation in digital computers What you will learn here: How

### Chapter 2. Data Representation in Computer Systems

Chapter 2 Data Representation in Computer Systems Chapter 2 Objectives Understand the fundamentals of numerical data representation and manipulation in digital computers. Master the skill of converting

### 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

### COMPUTER ARCHITECTURE AND ORGANIZATION. Operation Add Magnitudes Subtract Magnitudes (+A) + ( B) + (A B) (B A) + (A B)

Computer Arithmetic Data is manipulated by using the arithmetic instructions in digital computers. Data is manipulated to produce results necessary to give solution for the computation problems. The Addition,

### Chapter 2 Data Representations

Computer Engineering Chapter 2 Data Representations Hiroaki Kobayashi 4/21/2008 4/21/2008 1 Agenda in Chapter 2 Translation between binary numbers and decimal numbers Data Representations for Integers

### M1 Computers and Data

M1 Computers and Data Module Outline Architecture vs. Organization. Computer system and its submodules. Concept of frequency. Processor performance equation. Representation of information characters, signed