Number Systems. Binary Numbers. Appendix. Decimal notation represents numbers as powers of 10, for example
|
|
- Stanley Ferguson
- 6 years ago
- Views:
Transcription
1 Appendix F Number Systems Binary Numbers Decimal notation represents numbers as powers of 10, for example decimal = There is no particular reason for the choice of 10, except that several historical number systems were derived from people s counting with their fingers. Other number systems, using a base of 12, 20, or 60, have been used by various cultures throughout human history. However, computers use a number system with base 2 because it is far easier to build electronic components that work with two values, which can be represented by a current being either off or on, than it would be to represent 10 different values of electrical signals. A number written in base 2 is also called a binary number. For example, binary = = + + =
2 2 APPENDIX F Number Systems For digits after the decimal point, use negative powers of binary = = = = In general, to convert a binary number into its decimal equivalent, simply evaluate the powers of 2 corresponding to digits with value 1, and add them up. Table 1 shows the first powers of 2. Table 1 Powers of Two Power Decimal Value , , , , , , ,536
3 APPENDIX F Number Systems 3 To convert a decimal integer into its binary equivalent, keep dividing the integer by 2, keeping track of the remainders. Stop when the number is 0. Then write the remainders as a binary number, starting with the last one. For example, = 50 remainder = 25 remainder = 12 remainder = 6 remainder = 3 remainder = 1remainder = 0 remainder 1 Therefore, 100decimal = binary. Conversely, to convert a fractional number less than 1 to its binary format, keep multiplying by 2. If the result is greater than 1, subtract 1. Stop when the number is 0. Then use the digits before the decimal points as the binary digits of the fractional part, starting with the first one. For example, 035. i 2 = i 2 = i 2 = i 2 = i 2 = i 2 = 0. 4 Here the pattern repeats. That is, the binary representation of 0.35 is To convert any floating-point number into binary, convert the whole part and the fractional part separately. Long, Short, Signed, and Unsigned Integers There are two important properties that characterize integer values in computers. These are the number of bits used in the representation, and whether the integers are considered to be signed or unsigned. Most computers you are likely to encounter use a 32-bit integer. However, the C++ language does not require this, and there have been machines that used 16-, 20-, 36-, or even 64-bit integers. There are times when it is useful to have integers of different sizes. The C++ language provides two modifiers that are used to declare such integers. A short int (or simply a short) is an integer that, on most
4 4 APPENDIX F Number Systems implementations, has fewer bits than an int. (The phrase on most implementations is necessary because the language definition only requires that a short integer have no more bits than a standard integer.) On most platforms that use a 32-bit integer, a short is 16 bits. At the other extreme are long integers. As you might expect, a long int (or simply a long) contains no fewer bits than a standard integer. At the present time most personal computers still use a 32-bit long, but processors that provide 64-bit longs have started to appear and will likely be more common in the future. A character (or char) is sometimes used as a very short (8-bit) integer. The C++ programmer therefore has the following hierarchy of integer sizes: Type char short int long Typical Size 8-bit 16-bit 32-bit 32- or 64-bit The sizeof operator can be used to tell how many bits your compiler assigns to each type. This operator takes a type as argument and returns the number of bytes each type requires. Multiplying the number of bytes by 8 will tell you the number of bits: cout << "Number of bytes for char " << sizeof(char) << " number of bits " << 8 * sizeof(char) << "\n"; cout << "Number of bytes for short " << sizeof(short) << " number of bits " << 8 * sizeof(short) << "\n"; cout << "Number of bytes for int " << sizeof(int) << " number of bits " << 8 * sizeof(int) << "\n"; cout << "Number of bytes for long " << sizeof(long) << " number of bits " << 8 * sizeof(long) << "\n"; If the only numbers you needed were positive, then the preceding discussion would be everything you needed to know. However, in most applications it is more useful to allow both positive and negative values, and so a more complicated encoding is necessary. This characteristic of an integer is declared using the modifiers signed and unsigned. An unsigned integer holds only positive values. An unsigned short int that is represented using 16 bits can maintain the values between 0 and 65,535 (that is, between zero and ). A 32-bit unsigned int can represent values between 0 and 4,294,967,295. If no modifier is provided, an integer is assumed to be signed. Allowing both positive and negative values requires changing the representation of an integer value. The details of this representation are described in the next section. However, an important feature is that allowing both positive and negative numbers requires setting aside one bit (the so-called sign bit) to indicate whether the number is positive or negative. This reduces the largest value that can be
5 APPENDIX F Number Systems 5 represented. The following table shows the range of values that can be represented using signed and unsigned integers of 8, 16, 32, and 64 bits. Integer Type Range of Values 8-bit signed 128 to bit unsigned 0 to bit signed 32,768 to 32, bit unsigned 0 to 65, bit signed 2,147,483,648 to 2,147,483, bit unsigned 0 to 4,294,967, bit signed 9,223,372,036,854,775,808 to 9,223,372,036,854,775, bit unsigned 0 to 18,446,744,073,709,551,615 Two s Complement Integers To represent negative integers, there are two common representations, called signed magnitude and two s complement. Signed magnitude notation is simple: use the leftmost bit for the sign (0 = positive, 1 = negative). For example, when using 8-bit numbers, 13 = signed magnitude However, building circuitry for adding numbers gets a bit more complicated when one has to take a sign bit into account. The two s complement representation solves this problem. To form the two s complement of a number, Flip all bits. Then add 1. For example, to compute 13 as an 8-bit value, first flip all bits of to get Then add 1: 13 = two s complement Now no special circuitry is required for adding two numbers. Just follow the normal rule for addition, with a carry to the next position if the sum of the digits and the prior carry is 2 or 3.
6 6 APPENDIX F Number Systems For example, But only the last 8 bits count, so +13 and 13 add up to 0, as they should. In particular, 1 has two s complement representation , with all bits set. The leftmost bit of a two s complement number is 0 if the number is positive or zero, 1 if it is negative. Two s complement notation with a given number of bits can represent one more negative number than positive numbers. For example, the 8-bit two s complement numbers range from 128 to This phenomenon is an occasional cause for a programming error. For example, consider the following code: short b =...; if (b < 0) b = -b; This code does not guarantee that b is nonnegative afterwards. If short values are 16 bits and b happens to be 32,768, then computing its negative again yields 32,768. (Try it out take (15 zeros), flip all bits, and add 1.) IEEE Floating-Point Numbers The Institute for Electrical and Electronics Engineering (IEEE) defines standards for floating-point representations in the IEEE-754 standard. Figure 1 shows how single-precision (float) and double-precision (double) values are decomposed into A sign bit An exponent A mantissa 1 bit 8 bit 23 bit sign biased exponent e mantissa (without leading 1) Single Precision 1 bit 11 bit 52 bit sign biased exponent e mantissa (without leading 1) Double Precision Figure 1 IEEE Floating-Point Representation
7 APPENDIX F Number Systems 7 Floating-point numbers use scientific notation, in which a number is represented as b b b b e In this representation, e is the exponent, and the digits b0. b1b2b3 form the mantissa. The normalized representation is the one where b 0 0. For example, 100 = = decimal binary binary Because in the binary number system the first bit of a normalized representation must be 1, it is not actually stored in the mantissa. Therefore, you always need to add it on to represent the actual value. For example, the mantissa is stored as The exponent part of the IEEE representation uses neither signed magnitude nor two s complement representation. Instead, a bias is added to the actual exponent. The bias is 127 for single-precision numbers and 1023 for double-precision numbers. For example, the exponent e = 6 would be stored as 133 in a single-precision number. Thus, 100 decimal = single-precision IEEE In addition, there are several special values. Among them are: Zero: biased exponent = 0, mantissa = 0. Infinity: biased exponent = , mantissa = ±0. NaN (not a number): biased exponent = , mantissa ±0.
8 8 APPENDIX F Number Systems Hexadecimal Numbers Because binary numbers can be hard to read for humans, programmers often use the hexadecimal number system, with base 16. The digits are denoted as 0, 1,, 9, A, B, C, D, E, F. (See Table 2.) Four binary digits correspond to one hexadecimal digit. That makes it easy to convert between binary and hexadecimal values. For example, binary = 3B1 hexadecimal In C++, hexadecimal integers are denoted with a 0x prefix, such as 0x3B1. Table 2 Hexadecimal Digits Hexadecimal Decimal Binary A B C D E F
Data Representation 1
1 Data Representation Outline Binary Numbers Adding Binary Numbers Negative Integers Other Operations with Binary Numbers Floating Point Numbers Character Representation Image Representation Sound Representation
More informationNumber Systems. Both numbers are positive
Number Systems Range of Numbers and Overflow When arithmetic operation such as Addition, Subtraction, Multiplication and Division are performed on numbers the results generated may exceed the range of
More informationCOMP2611: 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
More informationDeclaration. Fundamental Data Types. Modifying the Basic Types. Basic Data Types. All variables must be declared before being used.
Declaration Fundamental Data Types All variables must be declared before being used. Tells compiler to set aside an appropriate amount of space in memory to hold a value. Enables the compiler to perform
More informationFloating-point Arithmetic. where you sum up the integer to the left of the decimal point and the fraction to the right.
Floating-point Arithmetic Reading: pp. 312-328 Floating-Point Representation Non-scientific floating point numbers: A non-integer can be represented as: 2 4 2 3 2 2 2 1 2 0.2-1 2-2 2-3 2-4 where you sum
More informationSigned umbers. Sign/Magnitude otation
Signed umbers So far we have discussed unsigned number representations. In particular, we have looked at the binary number system and shorthand methods in representing binary codes. With m binary digits,
More informationCOMP 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,
More informationCS 265. Computer Architecture. Wei Lu, Ph.D., P.Eng.
CS 265 Computer Architecture Wei Lu, Ph.D., P.Eng. 1 Part 1: Data Representation Our goal: revisit and re-establish fundamental of mathematics for the computer architecture course Overview: what are bits
More informationUNIT 7A Data Representation: Numbers and Text. Digital Data
UNIT 7A Data Representation: Numbers and Text 1 Digital Data 10010101011110101010110101001110 What does this binary sequence represent? It could be: an integer a floating point number text encoded with
More informationIntroduction to Computers and Programming. Numeric Values
Introduction to Computers and Programming Prof. I. K. Lundqvist Lecture 5 Reading: B pp. 47-71 Sept 1 003 Numeric Values Storing the value of 5 10 using ASCII: 00110010 00110101 Binary notation: 00000000
More informationFloating Point Arithmetic
Floating Point Arithmetic CS 365 Floating-Point What can be represented in N bits? Unsigned 0 to 2 N 2s Complement -2 N-1 to 2 N-1-1 But, what about? very large numbers? 9,349,398,989,787,762,244,859,087,678
More informationChapter 03: Computer Arithmetic. Lesson 09: Arithmetic using floating point numbers
Chapter 03: Computer Arithmetic Lesson 09: Arithmetic using floating point numbers Objective To understand arithmetic operations in case of floating point numbers 2 Multiplication of Floating Point Numbers
More informationCS & 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
More informationCMPSCI 145 MIDTERM #1 Solution Key. SPRING 2017 March 3, 2017 Professor William T. Verts
CMPSCI 145 MIDTERM #1 Solution Key NAME SPRING 2017 March 3, 2017 PROBLEM SCORE POINTS 1 10 2 10 3 15 4 15 5 20 6 12 7 8 8 10 TOTAL 100 10 Points Examine the following diagram of two systems, one involving
More informationMACHINE 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
More information15213 Recitation 2: Floating Point
15213 Recitation 2: Floating Point 1 Introduction This handout will introduce and test your knowledge of the floating point representation of real numbers, as defined by the IEEE standard. This information
More informationNumber Systems. Decimal numbers. Binary numbers. Chapter 1 <1> 8's column. 1000's column. 2's column. 4's column
1's column 10's column 100's column 1000's column 1's column 2's column 4's column 8's column Number Systems Decimal numbers 5374 10 = Binary numbers 1101 2 = Chapter 1 1's column 10's column 100's
More informationBits, Words, and Integers
Computer Science 52 Bits, Words, and Integers Spring Semester, 2017 In this document, we look at how bits are organized into meaningful data. In particular, we will see the details of how integers are
More informationQuestion 4: a. We want to store a binary encoding of the 150 original Pokemon. How many bits do we need to use?
Question 4: a. We want to store a binary encoding of the 150 original Pokemon. How many bits do we need to use? b. What is the encoding for Pikachu (#25)? Question 2: Flippin Fo Fun (10 points, 14 minutes)
More informationFinal Labs and Tutors
ICT106 Fundamentals of Computer Systems - Topic 2 REPRESENTATION AND STORAGE OF INFORMATION Reading: Linux Assembly Programming Language, Ch 2.4-2.9 and 3.6-3.8 Final Labs and Tutors Venue and time South
More informationIntegers and Floating Point
CMPE12 More about Numbers Integers and Floating Point (Rest of Textbook Chapter 2 plus more)" Review: Unsigned Integer A string of 0s and 1s that represent a positive integer." String is X n-1, X n-2,
More informationCOMP2121: 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
More informationNumber Systems and Binary Arithmetic. Quantitative Analysis II Professor Bob Orr
Number Systems and Binary Arithmetic Quantitative Analysis II Professor Bob Orr Introduction to Numbering Systems We are all familiar with the decimal number system (Base 10). Some other number systems
More informationIntegers. N = sum (b i * 2 i ) where b i = 0 or 1. This is called unsigned binary representation. i = 31. i = 0
Integers So far, we've seen how to convert numbers between bases. How do we represent particular kinds of data in a certain (32-bit) architecture? We will consider integers floating point characters What
More informationECE 2020B Fundamentals of Digital Design Spring problems, 6 pages Exam Two Solutions 26 February 2014
Problem 1 (4 parts, 21 points) Encoders and Pass Gates Part A (8 points) Suppose the circuit below has the following input priority: I 1 > I 3 > I 0 > I 2. Complete the truth table by filling in the input
More informationA complement number system is used to represent positive and negative integers. A complement number system is based on a fixed length representation
Complement Number Systems A complement number system is used to represent positive and negative integers A complement number system is based on a fixed length representation of numbers Pretend that integers
More informationCHAPTER V NUMBER SYSTEMS AND ARITHMETIC
CHAPTER V-1 CHAPTER V CHAPTER V NUMBER SYSTEMS AND ARITHMETIC CHAPTER V-2 NUMBER SYSTEMS RADIX-R REPRESENTATION Decimal number expansion 73625 10 = ( 7 10 4 ) + ( 3 10 3 ) + ( 6 10 2 ) + ( 2 10 1 ) +(
More information10.1. 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
More informationOn a 64-bit CPU. Size/Range vary by CPU model and Word size.
On a 64-bit CPU. Size/Range vary by CPU model and Word size. unsigned short x; //range 0 to 65553 signed short x; //range ± 32767 short x; //assumed signed There are (usually) no unsigned floats or doubles.
More informationInf2C - Computer Systems Lecture 2 Data Representation
Inf2C - Computer Systems Lecture 2 Data Representation Boris Grot School of Informatics University of Edinburgh Last lecture Moore s law Types of computer systems Computer components Computer system stack
More informationIT 1204 Section 2.0. Data Representation and Arithmetic. 2009, University of Colombo School of Computing 1
IT 1204 Section 2.0 Data Representation and Arithmetic 2009, University of Colombo School of Computing 1 What is Analog and Digital The interpretation of an analog signal would correspond to a signal whose
More informationNumber 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
More informationECE232: Hardware Organization and Design
ECE232: Hardware Organization and Design Lecture 11: Floating Point & Floating Point Addition Adapted from Computer Organization and Design, Patterson & Hennessy, UCB Last time: Single Precision Format
More informationNumeric Encodings Prof. James L. Frankel Harvard University
Numeric Encodings Prof. James L. Frankel Harvard University Version of 10:19 PM 12-Sep-2017 Copyright 2017, 2016 James L. Frankel. All rights reserved. Representation of Positive & Negative Integral and
More informationCHAPTER 3 Expressions, Functions, Output
CHAPTER 3 Expressions, Functions, Output More Data Types: Integral Number Types short, long, int (all represent integer values with no fractional part). Computer Representation of integer numbers - Number
More informationComputer System and programming in C
1 Basic Data Types Integral Types Integers are stored in various sizes. They can be signed or unsigned. Example Suppose an integer is represented by a byte (8 bits). Leftmost bit is sign bit. If the sign
More informationFloating Point Arithmetic
Floating Point Arithmetic Clark N. Taylor Department of Electrical and Computer Engineering Brigham Young University clark.taylor@byu.edu 1 Introduction Numerical operations are something at which digital
More informationLESSON 5 FUNDAMENTAL DATA TYPES. char short int long unsigned char unsigned short unsigned unsigned long
LESSON 5 ARITHMETIC DATA PROCESSING The arithmetic data types are the fundamental data types of the C language. They are called "arithmetic" because operations such as addition and multiplication can be
More informationChapter 7. Basic Types
Chapter 7 Basic Types Dr. D. J. Jackson Lecture 7-1 Basic Types C s basic (built-in) types: Integer types, including long integers, short integers, and unsigned integers Floating types (float, double,
More informationC NUMERIC FORMATS. Overview. IEEE Single-Precision Floating-point Data Format. Figure C-0. Table C-0. Listing C-0.
C NUMERIC FORMATS Figure C-. Table C-. Listing C-. Overview The DSP supports the 32-bit single-precision floating-point data format defined in the IEEE Standard 754/854. In addition, the DSP supports an
More informationInteger Representation Floating point Representation Other data types
Chapter 2 Bits, Data Types & Operations Integer Representation Floating point Representation Other data types Why do Computers use Base 2? Base 10 Number Representation Natural representation for human
More informationFloating-Point Data Representation and Manipulation 198:231 Introduction to Computer Organization Lecture 3
Floating-Point Data Representation and Manipulation 198:231 Introduction to Computer Organization Instructor: Nicole Hynes nicole.hynes@rutgers.edu 1 Fixed Point Numbers Fixed point number: integer part
More informationC How to Program, 6/e by Pearson Education, Inc. All Rights Reserved.
C How to Program, 6/e 1992-2010 by Pearson Education, Inc. An important part of the solution to any problem is the presentation of the results. In this chapter, we discuss in depth the formatting features
More informationDRAM uses a single capacitor to store and a transistor to select. SRAM typically uses 6 transistors.
Data Representation Data Representation Goal: Store numbers, characters, sets, database records in the computer. What we got: Circuit that stores 2 voltages, one for logic 0 (0 volts) and one for logic
More informationChapter 3: Arithmetic for Computers
Chapter 3: Arithmetic for Computers Objectives Signed and Unsigned Numbers Addition and Subtraction Multiplication and Division Floating Point Computer Architecture CS 35101-002 2 The Binary Numbering
More information8/30/2016. In Binary, We Have A Binary Point. ECE 120: Introduction to Computing. Fixed-Point Representations Support Fractions
University of Illinois at Urbana-Champaign Dept. of Electrical and Computer Engineering ECE 120: Introduction to Computing Fixed- and Floating-Point Representations In Binary, We Have A Binary Point Let
More informationFoundations of Computer Systems
18-600 Foundations of Computer Systems Lecture 4: Floating Point Required Reading Assignment: Chapter 2 of CS:APP (3 rd edition) by Randy Bryant & Dave O Hallaron Assignments for This Week: Lab 1 18-600
More informationCS 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.
More information9/3/2015. 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
More informationChapter 2. Positional number systems. 2.1 Signed number representations Signed magnitude
Chapter 2 Positional number systems A positional number system represents numeric values as sequences of one or more digits. Each digit in the representation is weighted according to its position in the
More informationFloating Point Numbers
Floating Point Numbers Computer Systems Organization (Spring 2016) CSCI-UA 201, Section 2 Instructor: Joanna Klukowska Slides adapted from Randal E. Bryant and David R. O Hallaron (CMU) Mohamed Zahran
More informationFloating Point Numbers
Floating Point Numbers Computer Systems Organization (Spring 2016) CSCI-UA 201, Section 2 Fractions in Binary Instructor: Joanna Klukowska Slides adapted from Randal E. Bryant and David R. O Hallaron (CMU)
More informationChapter 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
More informationRui 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
More informationCHW 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
More informationChapter 3. Fundamental Data Types
Chapter 3. Fundamental Data Types Byoung-Tak Zhang TA: Hanock Kwak Biointelligence Laboratory School of Computer Science and Engineering Seoul National Univertisy http://bi.snu.ac.kr Variable Declaration
More informationWorld 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
More informationFloating Point Puzzles. Lecture 3B Floating Point. IEEE Floating Point. Fractional Binary Numbers. Topics. IEEE Standard 754
Floating Point Puzzles Topics Lecture 3B Floating Point IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties For each of the following C expressions, either: Argue that
More informationDecimal & 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
More informationFLOATING POINT NUMBERS
Exponential Notation FLOATING POINT NUMBERS Englander Ch. 5 The following are equivalent representations of 1,234 123,400.0 x 10-2 12,340.0 x 10-1 1,234.0 x 10 0 123.4 x 10 1 12.34 x 10 2 1.234 x 10 3
More informationComputer Systems Programming. Practice Midterm. Name:
Computer Systems Programming Practice Midterm Name: 1. (4 pts) (K&R Ch 1-4) What is the output of the following C code? main() { int i = 6; int j = -35; printf( %d %d\n,i++, ++j); i = i >
More informationNumber Systems Standard positional representation of numbers: An unsigned number with whole and fraction portions is represented as:
N Number Systems Standard positional representation of numbers: An unsigned number with whole and fraction portions is represented as: a n a a a The value of this number is given by: = a n Ka a a a a a
More information1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM
1. NUMBER SYSTEMS USED IN COMPUTING: THE BINARY NUMBER SYSTEM 1.1 Introduction Given that digital logic and memory devices are based on two electrical states (on and off), it is natural to use a number
More informationChapter 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
More informationBasic data types. Building blocks of computation
Basic data types Building blocks of computation Goals By the end of this lesson you will be able to: Understand the commonly used basic data types of C++ including Characters Integers Floating-point values
More informationToday. o main function. o cout object. o Allocate space for data to be used in the program. o The data can be changed
CS 150 Introduction to Computer Science I Data Types Today Last we covered o main function o cout object o How data that is used by a program can be declared and stored Today we will o Investigate the
More informationFloating Point (with contributions from Dr. Bin Ren, William & Mary Computer Science)
Floating Point (with contributions from Dr. Bin Ren, William & Mary Computer Science) Floating Point Background: Fractional binary numbers IEEE floating point standard: Definition Example and properties
More informationBinary Representations and Arithmetic
Binary Representations and Arithmetic 9--26 Common number systems. Base : decimal Base 2: binary Base 6: hexadecimal (memory addresses) Base 8: octal (obsolete computer systems) Base 64 (email attachments,
More informationSet Theory in Computer Science. Binary Numbers. Base 10 Number. What is a Number? = Binary Number Example
Set Theory in Computer Science Binary Numbers Part 1B Bit of This and a Bit of That What is a Number? Base 10 Number We use the Hindu-Arabic Number System positional grouping system each position is a
More informationData Representation. DRAM uses a single capacitor to store and a transistor to select. SRAM typically uses 6 transistors.
Data Representation Data Representation Goal: Store numbers, characters, sets, database records in the computer. What we got: Circuit that stores 2 voltages, one for logic ( volts) and one for logic (3.3
More informationNumber 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
More informationThese are reserved words of the C language. For example int, float, if, else, for, while etc.
Tokens in C Keywords These are reserved words of the C language. For example int, float, if, else, for, while etc. Identifiers An Identifier is a sequence of letters and digits, but must start with a letter.
More informationGroups of two-state devices are used to represent data in a computer. In general, we say the states are either: high/low, on/off, 1/0,...
Chapter 9 Computer Arithmetic Reading: Section 9.1 on pp. 290-296 Computer Representation of Data Groups of two-state devices are used to represent data in a computer. In general, we say the states are
More informationRepresentation of Non Negative Integers
Representation of Non Negative Integers In each of one s complement and two s complement arithmetic, no special steps are required to represent a non negative integer. All conversions to the complement
More informationData Representations & Arithmetic Operations
Data Representations & Arithmetic Operations Hiroaki Kobayashi 7/13/2011 7/13/2011 Computer Science 1 Agenda Translation between binary numbers and decimal numbers Data Representations for Integers Negative
More informationChapter Three. Arithmetic
Chapter Three 1 Arithmetic Where we've been: Performance (seconds, cycles, instructions) Abstractions: Instruction Set Architecture Assembly Language and Machine Language What's up ahead: Implementing
More informationFloating Point January 24, 2008
15-213 The course that gives CMU its Zip! Floating Point January 24, 2008 Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties class04.ppt 15-213, S 08 Floating
More informationIEEE Standard for Floating-Point Arithmetic: 754
IEEE Standard for Floating-Point Arithmetic: 754 G.E. Antoniou G.E. Antoniou () IEEE Standard for Floating-Point Arithmetic: 754 1 / 34 Floating Point Standard: IEEE 754 1985/2008 Established in 1985 (2008)
More informationChapter 2 Float Point Arithmetic. Real Numbers in Decimal Notation. Real Numbers in Decimal Notation
Chapter 2 Float Point Arithmetic Topics IEEE Floating Point Standard Fractional Binary Numbers Rounding Floating Point Operations Mathematical properties Real Numbers in Decimal Notation Representation
More informationInteger Data Types. Data Type. Data Types. int, short int, long int
Data Types Variables are classified according to their data type. The data type determines the kind of information that may be stored in the variable. A data type is a set of values. Generally two main
More informationNumbers and Computers. Debdeep Mukhopadhyay Assistant Professor Dept of Computer Sc and Engg IIT Madras
Numbers and Computers Debdeep Mukhopadhyay Assistant Professor Dept of Computer Sc and Engg IIT Madras 1 Think of a number between 1 and 15 8 9 10 11 12 13 14 15 4 5 6 7 12 13 14 15 2 3 6 7 10 11 14 15
More informationMath 340 Fall 2014, Victor Matveev. Binary system, round-off errors, loss of significance, and double precision accuracy.
Math 340 Fall 2014, Victor Matveev Binary system, round-off errors, loss of significance, and double precision accuracy. 1. Bits and the binary number system A bit is one digit in a binary representation
More informationComputer Architecture and System Software Lecture 02: Overview of Computer Systems & Start of Chapter 2
Computer Architecture and System Software Lecture 02: Overview of Computer Systems & Start of Chapter 2 Instructor: Rob Bergen Applied Computer Science University of Winnipeg Announcements Website is up
More informationECE 2030D Computer Engineering Spring problems, 5 pages Exam Two 8 March 2012
Instructions: This is a closed book, closed note exam. Calculators are not permitted. If you have a question, raise your hand and I will come to you. Please work the exam in pencil and do not separate
More informationCPS 104 Computer Organization and Programming Lecture-2 : Data representations,
CPS 104 Computer Organization and Programming Lecture-2 : Data representations, Sep. 1, 1999 Dietolf Ramm http://www.cs.duke.edu/~dr/cps104.html CPS104 Lec2.1 GK&DR Fall 1999 Data Representation Computers
More informationCSCI 402: Computer Architectures. Arithmetic for Computers (3) Fengguang Song Department of Computer & Information Science IUPUI.
CSCI 402: Computer Architectures Arithmetic for Computers (3) Fengguang Song Department of Computer & Information Science IUPUI 3.5 Today s Contents Floating point numbers: 2.5, 10.1, 100.2, etc.. How
More informationISA 563 : Fundamentals of Systems Programming
ISA 563 : Fundamentals of Systems Programming Variables, Primitive Types, Operators, and Expressions September 4 th 2008 Outline Define Expressions Discuss how to represent data in a program variable name
More informationVariables and Constants
HOUR 3 Variables and Constants Programs need a way to store the data they use. Variables and constants offer various ways to work with numbers and other values. In this hour you learn: How to declare and
More informationIntroduction to Scientific Computing Lecture 1
Introduction to Scientific Computing Lecture 1 Professor Hanno Rein Last updated: September 10, 2017 1 Number Representations In this lecture, we will cover two concept that are important to understand
More informationModule 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
More informationCSE 351: The Hardware/Software Interface. Section 2 Integer representations, two s complement, and bitwise operators
CSE 351: The Hardware/Software Interface Section 2 Integer representations, two s complement, and bitwise operators Integer representations In addition to decimal notation, it s important to be able to
More informationCHAPTER 1 Numerical Representation
CHAPTER 1 Numerical Representation To process a signal digitally, it must be represented in a digital format. This point may seem obvious, but it turns out that there are a number of different ways to
More informationECE 2020B Fundamentals of Digital Design Spring problems, 6 pages Exam Two 26 February 2014
Instructions: This is a closed book, closed note exam. Calculators are not permitted. If you have a question, raise your hand and I will come to you. Please work the exam in pencil and do not separate
More informationFinite arithmetic and error analysis
Finite arithmetic and error analysis Escuela de Ingeniería Informática de Oviedo (Dpto de Matemáticas-UniOvi) Numerical Computation Finite arithmetic and error analysis 1 / 45 Outline 1 Number representation:
More informationFloating Point Puzzles The course that gives CMU its Zip! Floating Point Jan 22, IEEE Floating Point. Fractional Binary Numbers.
class04.ppt 15-213 The course that gives CMU its Zip! Topics Floating Point Jan 22, 2004 IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties Floating Point Puzzles For
More informationComputer Architecture Chapter 3. Fall 2005 Department of Computer Science Kent State University
Computer Architecture Chapter 3 Fall 2005 Department of Computer Science Kent State University Objectives Signed and Unsigned Numbers Addition and Subtraction Multiplication and Division Floating Point
More informationComputer (Literacy) Skills. Number representations and memory. Lubomír Bulej KDSS MFF UK
Computer (Literacy Skills Number representations and memory Lubomír Bulej KDSS MFF UK Number representations? What for? Recall: computer works with binary numbers Groups of zeroes and ones 8 bits (byte,
More informationCPE 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:
More informationM1 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
More informationSystems I. Floating Point. Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties
Systems I Floating Point Topics IEEE Floating Point Standard Rounding Floating Point Operations Mathematical properties IEEE Floating Point IEEE Standard 754 Established in 1985 as uniform standard for
More information