Number Systems. Decimal numbers. Binary numbers. Chapter 1 <1> 8's column. 1000's column. 2's column. 4's column
|
|
|
- Gregory Dixon
- 5 years ago
- Views:
Transcription
1 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 = Binary numbers = Chapter 1 <1>
2 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 Binary numbers = five thousands three hundreds seven tens four ones = = one eight one four no two one one Chapter 1 <2>
3 Powers of Two 2 0 = 2 1 = 2 2 = 2 3 = 2 4 = 2 5 = 2 6 = 2 7 = 2 8 = 2 9 = 2 10 = 2 11 = 2 12 = 2 13 = 2 14 = 2 15 = Chapter 1 <3>
4 Powers of Two 2 0 = = = = = = = = = = = = = = = = Handy to memorize up to 2 9 Chapter 1 <4>
5 Number Conversion Decimal to binary conversion: Convert to decimal Decimal to binary conversion: Convert to binary Chapter 1 <5>
6 Number Conversion Decimal to binary conversion: Convert to decimal = Decimal to binary conversion: Convert to binary = Chapter 1 <6>
7 Binary Values and Range N-digit decimal number How many values? Range? Example: 3-digit decimal number: N-bit binary number How many values? Range: Example: 3-digit binary number: Chapter 1 <7>
8 Binary Values and Range N-digit decimal number How many values? 10 N Range? [0, 10 N - 1] Example: 3-digit decimal number: 10 3 = 1000 possible values Range: [0, 999] N-bit binary number How many values? 2 N Range: [0, 2 N - 1] Example: 3-digit binary number: 2 3 = 8 possible values Range: [0, 7] = [000 2 to ] Chapter 1 <8>
9 Hexadecimal Numbers Hex Digit Decimal Equivalent Binary Equivalent A 10 B 11 C 12 D 13 E 14 F 15 Chapter 1 <9>
10 Hexadecimal Numbers Hex Digit Decimal Equivalent Binary Equivalent A B C D E F Chapter 1 <10>
11 Hexadecimal Numbers Base 16 Shorthand for binary Chapter 1 <11>
12 Hexadecimal to Binary Conversion Hexadecimal to binary conversion: Convert 4AF 16 (also written 0x4AF) to binary Hexadecimal to decimal conversion: Convert 0x4AF to decimal Chapter 1 <12>
13 Hexadecimal to Binary Conversion Hexadecimal to binary conversion: Convert 4AF 16 (also written 0x4AF) to binary Hexadecimal to decimal conversion: Convert 4AF 16 to decimal = Chapter 1 <13>
14 Bits, Bytes, Nibbles Bits Bytes & Nibbles Bytes most significant bit byte nibble CEBF9AD7 most significant byte least significant bit least significant byte Chapter 1 <14>
15 Large Powers of Two 2 10 = 1 kilo 1000 (1024) 2 20 = 1 mega 1 million (1,048,576) 2 30 = 1 giga 1 billion (1,073,741,824) Chapter 1 <15>
16 Estimating Powers of Two What is the value of 2 24? How many values can a 32-bit variable represent? Chapter 1 <16>
17 Estimating Powers of Two What is the value of 2 24? million How many values can a 32-bit variable represent? billion Chapter 1 <17>
18 Addition Decimal carries Binary + 11 carries Chapter 1 <18>
19 Binary Addition Examples Add the following 4-bit binary numbers Add the following 4-bit binary numbers Chapter 1 <19>
20 Binary Addition Examples Add the following 4-bit binary numbers Add the following 4-bit binary numbers Overflow! Chapter 1 <20>
21 Overflow Digital systems operate on a fixed number of bits Overflow: when result is too big to fit in the available number of bits See previous example of Chapter 1 <21>
22 Signed Binary Numbers Sign/Magnitude Numbers Two s Complement Numbers Chapter 1 <22>
23 Sign/Magnitude Numbers 1 sign bit, N-1 magnitude bits Sign bit is the most significant (left-most) bit A: a, a, a, a, a Positive number: sign bit = 0 Negative number: sign bit = 1 Example, 4-bit sign/mag representations of ± 6: +6 = - 6 = Range of an N-bit sign/magnitude number: A N 1 N n 2 an 1 ( 1) a 2 i 0 i i Chapter 1 <23>
24 Sign/Magnitude Numbers 1 sign bit, N-1 magnitude bits Sign bit is the most significant (left-most) bit A: a, a, a, a, a Positive number: sign bit = 0 Negative number: sign bit = 1 Example, 4-bit sign/mag representations of ± 6: +6 = = 1110 Range of an N-bit sign/magnitude number: [-(2 N-1-1), 2 N-1-1] A N 1 N n 2 an 1 ( 1) a 2 i 0 i i Chapter 1 <24>
25 Sign/Magnitude Numbers Problems: Addition doesn t work, for example : (wrong!) Two representations of 0 (± 0): Chapter 1 <25>
26 Two s Complement Numbers Don t have same problems as sign/magnitude numbers: Addition works Single representation for 0 Chapter 1 <26>
27 Two s Complement Numbers R Msb has value of -2 N-1 n 2 n 1 i n 1 2 i2 i 0 A a a Most positive 4-bit number: Most negative 4-bit number: The most significant bit still indicates the sign (1 = negative, 0 = positive) Range of an N-bit two s comp number: Chapter 1 <27>
28 Two s Complement Numbers Msb has value of -2 N-1 n 2 n 1 i n 1 2 i2 i 0 A a a Most positive 4-bit number: 0111 Most negative 4-bit number: 1000 The most significant bit still indicates the sign (1 = negative, 0 = positive) Range of an N-bit two s comp number: [-(2 N-1 ), 2 N-1-1] Chapter 1 <28>
29 Taking the Two s Complement Flip the sign of a two s complement number Method: 1. Invert the bits 2. Add 1 Example: Flip the sign of 3 10 = Chapter 1 <29>
30 Taking the Two s Complement Flip the sign of a two s complement number Method: 1. Invert the bits 2. Add 1 Example: Flip the sign of 3 10 = = Chapter 1 <30>
31 Two s Complement Examples Take the two s complement of 6 10 = What is the decimal value of ? Chapter 1 <31>
32 Two s Complement Examples Take the two s complement of 6 10 = = What is the decimal value of the two s complement number ? = 7 10, so = Chapter 1 <32>
33 Two s Complement Addition Add 6 + (-6) using two s complement numbers Add using two s complement numbers Chapter 1 <33>
34 Two s Complement Addition Add 6 + (-6) using two s complement numbers Add using two s complement numbers Chapter 1 <34>
35 Increasing Bit Width Extend number from N to M bits (M > N) : Sign-extension Zero-extension Copyright 2012 Elsevier Chapter 1 <35>
36 Sign-Extension Sign bit copied to msb s Number value is same Example 1: 4-bit representation of 3 = bit sign-extended value: Example 2: 4-bit representation of -5 = bit sign-extended value: Chapter 1 <36>
37 Zero-Extension Zeros copied to msb s Value changes for negative numbers Example 1: 4-bit value = = bit zero-extended value: = 3 10 Example 2: 4-bit value = 1011 = bit zero-extended value: = Chapter 1 <37>
38 Number System Comparison Number System Range Unsigned [0, 2 N -1] Sign/Magnitude [-(2 N-1-1), 2 N-1-1] Two s Complement [-2 N-1, 2 N-1-1] For example, 4-bit representation: Unsigned Two's Complement Sign/Magnitude Chapter 1 <38>
39 Number Systems Numbers we can represent using binary representations Positive numbers Unsigned binary Negative numbers Two s complement Sign/magnitude numbers What about fractions? Chapter 1 <39>
40 Numbers with Fractions Two common notations: Fixed-point: binary point fixed Floating-point: binary point floats to the right of the most significant 1 Chapter 1 <40>
41 Fixed-Point Numbers 6.75 using 4 integer bits and 4 fraction bits: = 6.75 Binary point is implied The number of integer and fraction bits must be agreed upon beforehand Chapter 1 <41>
42 Fixed-Point Number Example Represent using 4 integer bits and 4 fraction bits. Chapter 1 <42>
43 Fixed-Point Number Example Represent using 4 integer bits and 4 fraction bits Chapter 1 <43>
44 Signed Fixed-Point Numbers Representations: Sign/magnitude Two s complement Example: Represent using 4 integer and 4 fraction bits Sign/magnitude: Two s complement: Chapter 1 <44>
45 Signed Fixed-Point Numbers Representations: Sign/magnitude Two s complement Example: Represent using 4 integer and 4 fraction bits Sign/magnitude: Two s complement: : Invert bits: Add 1 to lsb: Chapter 1 <45>
46 Floating-Point Numbers Binary point floats to the right of the most significant 1 Similar to decimal scientific notation For example, write in scientific notation: 273 = In general, a number is written in scientific notation as: ± M B E M = mantissa B = base E = exponent In the example, M = 2.73, B = 10, and E = 2 Chapter 1 <46>
47 Floating-Point Numbers 1 bit 8 bits 23 bits Sign Exponent Mantissa Example: represent the value using a 32-bit floating point representation We show three versions final version is called the IEEE 754 floating-point standard Chapter 1 <47>
48 Floating-Point Representation 1 1. Convert decimal to binary (don t reverse steps 1 & 2!): = Write the number in binary scientific notation : = Fill in each field of the 32-bit floating point number: The sign bit is positive (0) The 8 exponent bits represent the value 7 The remaining 23 bits are the mantissa 1 bit 8 bits 23 bits Sign Exponent Mantissa Note: There should be an extra 0 bit on the right of the Mantissa. Chapter 1 <48>
49 Floating-Point Representation 2 First bit of the mantissa is always 1: = = So, no need to store it: implicit leading 1 Store just fraction bits in 23-bit field 1 bit 8 bits 23 bits Sign Exponent Fraction Chapter 1 <49>
50 Floating-Point Representation 3 Biased exponent: bias = 127 ( ) Biased exponent = bias + exponent Exponent of 7 is stored as: = 134 = 0x The IEEE bit floating-point representation of bit 8 bits 23 bits Sign Biased Exponent Fraction in hexadecimal: 0x Chapter 1 <50>
51 Floating-Point Example Write in floating point (IEEE 754) Chapter 1 <51>
52 Floating-Point Example Write in floating point (IEEE 754) 1. Convert decimal to binary: = Write in binary scientific notation: Fill in fields: Sign bit: 1 (negative) 8 exponent bits: ( ) = 132 = fraction bits: bit 8 bits 23 bits Sign Exponent Fraction in hexadecimal: 0xC Chapter 1 <52>
53 Floating-Point: Special Cases Number Sign Exponent Fraction 0 X NaN X non-zero Chapter 1 <53>
54 Floating-Point Precision Single-Precision: 32-bit 1 sign bit, 8 exponent bits, 23 fraction bits bias = 127 Double-Precision: 64-bit 1 sign bit, 11 exponent bits, 52 fraction bits bias = 1023 Chapter 1 <54>
55 Floating-Point: Rounding Overflow: number too large to be represented Underflow: number too small to be represented Rounding modes: Down Up Toward zero To nearest Example: round ( ) to only 3 fraction bits Down: Up: Toward zero: To nearest: (1.625 is closer to than 1.5 is) Chapter 1 <55>
56 Floating-Point Addition 1. Extract exponent and fraction bits 2. Prepend leading 1 to fraction to form mantissa 3. Compare exponents 4. Shift smaller mantissa if necessary 5. Add mantissas 6. Normalize mantissa and adjust exponent if necessary 7. Round result 8. Assemble exponent and fraction back into floating-point format Chapter 1 <56>
57 Floating-Point Addition Example Add the following floating-point numbers: 0x3FC x Chapter 1 <57>
58 Floating-Point Addition Example 1. Extract exponent and fraction bits 1 bit 8 bits 23 bits Sign Exponent Fraction 1 bit 8 bits 23 bits Sign Exponent Fraction For first number (N1): S = 0, E = 127, F =.1 For second number (N2): S = 0, E = 128, F = Prepend leading 1 to form mantissa N1: 1.1 N2: Chapter 1 <58>
59 Floating-Point Addition Example 3. Compare exponents = -1, so shift N1 right by 1 bit 4. Shift smaller mantissa if necessary shift N1 s mantissa: 1.1 >> 1 = 0.11 ( 2 1 ) 5. Add mantissas Chapter 1 <59>
60 Floating Point Addition Example 6. Normalize mantissa and adjust exponent if necessary = Round result No need (fits in 23 bits) 8. Assemble exponent and fraction back into floating-point format S = 0, E = = 129 = , F = bit 8 bits 23 bits Sign Exponent Fraction in hexadecimal: 0x Chapter 1 <60>
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
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,
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
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
Chapter 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
CS 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
Floating 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
Floating Point Numbers
Floating Point Numbers Summer 8 Fractional numbers Fractional numbers fixed point Floating point numbers the IEEE 7 floating point standard Floating point operations Rounding modes CMPE Summer 8 Slides
Inf2C - 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
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
Introduction 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
In this lesson you will learn: how to add and multiply positive binary integers how to work with signed binary numbers using two s complement how fixed and floating point numbers are used to represent
Data 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
Chapter 5. Digital Design and Computer Architecture, 2 nd Edition. David Money Harris and Sarah L. Harris. Chapter 5 <1>
Chapter 5 Digital Design and Computer Architecture, 2 nd Edition David Money Harris and Sarah L. Harris Chapter 5 Chapter 5 :: Topics Introduction Arithmetic Circuits umber Systems Sequential Building
Topic Notes: Bits and Bytes and Numbers
Computer Science 220 Assembly Language & Comp Architecture Siena College Fall 2010 Topic Notes: Bits and Bytes and Numbers Binary Basics At least some of this will be review, but we will go over it for
Computer (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,
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.
15213 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
Floating Point. The World is Not Just Integers. Programming languages support numbers with fraction
1 Floating Point The World is Not Just Integers Programming languages support numbers with fraction Called floating-point numbers Examples: 3.14159265 (π) 2.71828 (e) 0.000000001 or 1.0 10 9 (seconds in
Chapter 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
EE 109 Unit 19. IEEE 754 Floating Point Representation Floating Point Arithmetic
1 EE 109 Unit 19 IEEE 754 Floating Point Representation Floating Point Arithmetic 2 Floating Point Used to represent very small numbers (fractions) and very large numbers Avogadro s Number: +6.0247 * 10
Topic Notes: Bits and Bytes and Numbers
Computer Science 220 Assembly Language & Comp Architecture Siena College Fall 2011 Topic Notes: Bits and Bytes and Numbers Binary Basics At least some of this will be review for most of you, but we start
Number 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
l l l l l l l Base 2; each digit is 0 or 1 l Each bit in place i has value 2 i l Binary representation is used in computers
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 Computer Architecture What do computers do? Manipulate stored information
Signed 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,
CO212 Lecture 10: Arithmetic & Logical Unit
CO212 Lecture 10: Arithmetic & Logical Unit Shobhanjana Kalita, Dept. of CSE, Tezpur University Slides courtesy: Computer Architecture and Organization, 9 th Ed, W. Stallings Integer Representation For
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
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
UNIT 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
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
Number Systems. Binary Numbers. Appendix. Decimal notation represents numbers as powers of 10, for example
Appendix F Number Systems Binary Numbers Decimal notation represents numbers as powers of 10, for example 1729 1 103 7 102 2 101 9 100 decimal = + + + There is no particular reason for the choice of 10,
Chapter 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
FLOATING 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
CMPSCI 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
Digital Logic & Computer Design CS Professor Dan Moldovan Spring 2010
Digital Logic & Computer Design CS 434 Professor Dan Moldovan Spring 2 Copyright 27 Elsevier 5- Chapter 5 :: Digital Building Blocks Digital Design and Computer Architecture David Money Harris and Sarah
Question 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)
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
A 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
Chapter 4: Data Representations
Chapter 4: Data Representations Integer Representations o unsigned o sign-magnitude o one's complement o two's complement o bias o comparison o sign extension o overflow Character Representations Floating
1. 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
Course Schedule. CS 221 Computer Architecture. Week 3: Plan. I. Hexadecimals and Character Representations. Hexadecimal Representation
Course Schedule CS 221 Computer Architecture Week 3: Information Representation (2) Fall 2001 W1 Sep 11- Sep 14 Introduction W2 Sep 18- Sep 21 Information Representation (1) (Chapter 3) W3 Sep 25- Sep
Data 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
Under the Hood: Data Representation. Computer Science 104 Lecture 2
Under the Hood: Data Representation Computer Science 104 Lecture 2 Admin Piazza, Sakai Up Everyone should have access Homework 1 Posted Due Feb 6 PDF or Plain Text Only: No Word or RTF Recommended: Learn
C 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
Number 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
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
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 ) +(
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
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
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
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
IT 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
Computer 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
Computer Science 324 Computer Architecture Mount Holyoke College Fall Topic Notes: Bits and Bytes and Numbers
Computer Science 324 Computer Architecture Mount Holyoke College Fall 2007 Topic Notes: Bits and Bytes and Numbers Number Systems Much of this is review, given the 221 prerequisite Question: how high can
1.1. Unit 1. Integer Representation
1.1 Unit 1 Integer Representation 1.2 Skills & Outcomes You should know and be able to apply the following skills with confidence Convert an unsigned binary number to and from decimal Understand the finite
3.5 Floating Point: Overview
3.5 Floating Point: Overview Floating point (FP) numbers Scientific notation Decimal scientific notation Binary scientific notation IEEE 754 FP Standard Floating point representation inside a computer
Numeric 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
The ALU consists of combinational logic. Processes all data in the CPU. ALL von Neuman machines have an ALU loop.
CS 320 Ch 10 Computer Arithmetic The ALU consists of combinational logic. Processes all data in the CPU. ALL von Neuman machines have an ALU loop. Signed integers are typically represented in sign-magnitude
Floating Point Numbers. Lecture 9 CAP
Floating Point Numbers Lecture 9 CAP 3103 06-16-2014 Review of Numbers Computers are made to deal with numbers What can we represent in N bits? 2 N things, and no more! They could be Unsigned integers:
Chapter 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
Basic Definition INTEGER DATA. Unsigned Binary and Binary-Coded Decimal. BCD: Binary-Coded Decimal
Basic Definition REPRESENTING INTEGER DATA Englander Ch. 4 An integer is a number which has no fractional part. Examples: -2022-213 0 1 514 323434565232 Unsigned and -Coded Decimal BCD: -Coded Decimal
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
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
Signed Multiplication Multiply the positives Negate result if signs of operand are different
Another Improvement Save on space: Put multiplier in product saves on speed: only single shift needed Figure: Improved hardware for multiplication Signed Multiplication Multiply the positives Negate result
Floating point. Today! IEEE Floating Point Standard! Rounding! Floating Point Operations! Mathematical properties. Next time. !
Floating point Today! IEEE Floating Point Standard! Rounding! Floating Point Operations! Mathematical properties Next time! The machine model Chris Riesbeck, Fall 2011 Checkpoint IEEE Floating point Floating
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:
Numerical Representations On The Computer: Negative And Rational Numbers
Numerical Representations On The Computer: Negative And Rational Numbers How are negative and rational numbers represented on the computer? How are subtractions performed by the computer? Subtraction In
T02 Tutorial Slides for Week 2
T02 Tutorial Slides for Week 2 ENEL 353: Digital Circuits Fall 2017 Term Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary 19 September, 2017
unused unused unused unused unused unused
BCD numbers. In some applications, such as in the financial industry, the errors that can creep in due to converting numbers back and forth between decimal and binary is unacceptable. For these applications
Integer 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
ECE232: 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
Floating-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
Floating-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
CHAPTER 5: Representing Numerical Data
CHAPTER 5: Representing Numerical Data The Architecture of Computer Hardware and Systems Software & Networking: An Information Technology Approach 4th Edition, Irv Englander John Wiley and Sons 2010 PowerPoint
CS61c Midterm Review (fa06) Number representation and Floating points From your friendly reader
CS61c Midterm Review (fa06) Number representation and Floating points From your friendly reader Number representation (See: Lecture 2, Lab 1, HW#1) KNOW: Kibi (2 10 ), Mebi(2 20 ), Gibi(2 30 ), Tebi(2
Floating 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
Numerical Representations On The Computer: Negative And Rational Numbers
Numerical Representations On The Computer: Negative And Rational Numbers How are negative and rational numbers represented on the computer? How are subtractions performed by the computer? Subtraction In
CSCI 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
Representing numbers on the computer. Computer memory/processors consist of items that exist in one of two possible states (binary states).
Representing numbers on the computer. Computer memory/processors consist of items that exist in one of two possible states (binary states). These states are usually labeled 0 and 1. Each item in memory
3 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
ECE 372 Microcontroller Design Assembly Programming Arrays. ECE 372 Microcontroller Design Assembly Programming Arrays
Assembly Programming Arrays Assembly Programming Arrays Array For Loop Example: unsigned short a[]; for(j=; j
Number 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
Operations On Data CHAPTER 4. (Solutions to Odd-Numbered Problems) Review Questions
CHAPTER 4 Operations On Data (Solutions to Odd-Numbered Problems) Review Questions 1. Arithmetic operations interpret bit patterns as numbers. Logical operations interpret each bit as a logical values
Number Representations
Number Representations times XVII LIX CLXX -XVII D(CCL)LL DCCC LLLL X-X X-VII = DCCC CC III = MIII X-VII = VIIIII-VII = III 1/25/02 Memory Organization Viewed as a large, single-dimension array, with an
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
Set 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
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:
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 0 (0 volts) and one for logic
Semester Transition Point. EE 109 Unit 11 Binary Arithmetic. Binary Arithmetic ARITHMETIC
1 2 Semester Transition Point EE 109 Unit 11 Binary Arithmetic At this point we are going to start to transition in our class to look more at the hardware organization and the low-level software that is
Floating 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
Floating Point. CSE 351 Autumn Instructor: Justin Hsia
Floating Point CSE 351 Autumn 2017 Instructor: Justin Hsia Teaching Assistants: Lucas Wotton Michael Zhang Parker DeWilde Ryan Wong Sam Gehman Sam Wolfson Savanna Yee Vinny Palaniappan http://xkcd.com/571/
SIGNED AND UNSIGNED SYSTEMS
EE 357 Unit 1 Fixed Point Systems and Arithmetic Learning Objectives Understand the size and systems used by the underlying HW when a variable is declared in a SW program Understand and be able to find
Data Representation in Computer Memory
Data Representation in Computer Memory Data Representation in Computer Memory Digital computer stores the data in the form of binary bit sequences. Binary number system has two symbols: 0 and 1, called
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
Real Numbers finite subset real numbers floating point numbers Scientific Notation fixed point numbers
Real Numbers We have been studying integer arithmetic up to this point. We have discovered that a standard computer can represent a finite subset of the infinite set of integers. The range is determined
Foundations 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
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
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 (01) Digital Systems and Binary Numbers By: Dr. Ahmed ElShafee
١ Lecture (01) Digital Systems and Binary Numbers By: Dr. Ahmed ElShafee Digital systems Digital systems are used in communication, business transactions, traffic control, spacecraft guidance, medical
Floating 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