number representation
play

Number representation A number can be represented in binary in many - PowerPoint PPT Presentation

Feb 08, 2023 •319 likes •1k views

Number representation A number can be represented in binary in many ways. The most common number types to be represented are: Integers, positive integers one's complement, two's complement, sign-magnitude Decimal numbers with fixed

  1. Number representation A number can be represented in binary in many ways. The most common number types to be represented are: • Integers, positive integers one's complement, two's complement, sign-magnitude • Decimal numbers with fixed increment fixed-point numbers • Decimal numbers with expanded range floating-point numbers William Sandqvist william@kth.se

  2. Integers Positive Integer: -2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 0 =1 ⋅ 2 6 + 1 ⋅ 2 5 + 1 ⋅ 2 3 + 1 ⋅ 2 2 + 1 ⋅ 2 0 = 109 0 1 1 0 1 1 0 1 Negative Integer: -2 7 2 6 2 5 2 4 2 3 2 2 2 1 2 0 = -1 ⋅ 2 7 +1 ⋅ 2 6 + 1 ⋅ 2 5 + 1 ⋅ 2 3 + 1 ⋅ 2 2 + 1 ⋅ 2 0 = -19 1 1 1 0 1 1 0 1 William Sandqvist william@kth.se

  3. Multiplication of two positive integers 0 0 1 0 2 *11 * 1 0 1 1 0 0 1 0 Shift, add 0 0 1 0 multiplicand or 0 0 0 0 0 + 0 0 1 0 =22 0 0 1 0 1 1 0 William Sandqvist william@kth.se

  4. Multiplication with signbit 0 0 1 0 2 * 1 0 1 1 -5 0 0 1 0 The sign has a negative weight! => Take the two-complement! 0 0 1 0 0 0 0 0 + 1 1 1 0 1 1 1 0 1 1 0 -10 William Sandqvist william@kth.se

  5. The signbit the other way around 1 0 1 1 -5 * 0 0 1 0 2 0 0 0 0 Signextension! 1 1 1 0 1 1 (here to 7 bit’s) 0 0 0 0 + 0 0 0 0 1 1 1 0 1 1 0 -10 William Sandqvist william@kth.se

  6. Multiplication of two negative integers 1 0 1 1 -5 * 1 0 1 1 -5 1 1 1 1 0 1 1 Signextension! 1 1 1 0 1 1 0 0 0 0 The sign has a negative weight! => Take the two-complement! + 0 1 0 1 0 0 1 1 0 0 1 +25 Answer 7-bit with sign William Sandqvist william@kth.se

  7. Or so we make it easy for us Use only positive numbers in the multiplication Convert to positive numbers Keep track of the result's sign + ⋅ + ⇒ + + ⋅ − ⇒ − ( ) ( ) ( ) ( ) ( ) ( ) − ⋅ + ⇒ − − ⋅ − ⇒ + ( ) ( ) ( ) ( ) ( ) ( ) Two-complement for negative numbers if necessary William Sandqvist william@kth.se

  8. A simple hardware solution … A N-1 A N-2 A 0 Sign A Sign A F correct 0 1 Sign B 0 0 1 ... Sign B 1 1 0 0 0 0 F correct =Sign A ⊕ Sign B ... HA HA HA The correction is done by inverting the bits, And 2’s complement of Product, when correction is needed. add 1 William Sandqvist william@kth.se

  9. Multiplication (two positive integers) ⋅ = ( a a a a ) ( b b b b ) ( q q q q q q q q ) 3 2 1 0 3 2 1 0 7 6 5 4 3 2 1 0 AND AND FA, carry AND FA, carry AND FA, carry William Sandqvist william@kth.se

  10. Multiplication (two positive integers) ⋅ = ( a a a a ) ( b b b b ) ( q q q q q q q q ) 3 2 1 0 3 2 1 0 7 6 5 4 3 2 1 0 AND AND FA, carry AND FA, carry AND FA, carry William Sandqvist william@kth.se

  11. Multiplication (two positive integers) ⋅ = ( a a a a ) ( b b b b ) ( q q q q q q q q ) 3 2 1 0 3 2 1 0 7 6 5 4 3 2 1 0 AND AND FA, carry AND FA, carry AND FA, carry William Sandqvist william@kth.se

  12. Multiplication (two positive integers) ⋅ = ( a a a a ) ( b b b b ) ( q q q q q q q q ) 3 2 1 0 3 2 1 0 7 6 5 4 3 2 1 0 AND AND FA, carry AND FA, carry AND FA, carry William Sandqvist william@kth.se

  13. Multiplication (two positive integers) ⋅ = ( a a a a ) ( b b b b ) ( q q q q q q q q ) 3 2 1 0 3 2 1 0 7 6 5 4 3 2 1 0 AND AND FA, carry AND FA, carry AND FA, carry William Sandqvist william@kth.se

  14. ⋅ = ( a a a a ) ( b b b b ) ( q q q q q q q q ) 3 2 1 0 3 2 1 0 7 6 5 4 3 2 1 0 Multiplication (two positive integers) a a a a a a a a Shift of the 3 3 0 0 b 2 2 1 1 0 b multiplicator is 1 done by passing it a b step to the left 2 a a a a 3 0 2 1 of the grid for each level b a 3 a a a 3 2 1 0 12 FA for 4 × 4 bits q q q q q q q q 7 6 3 0 5 4 2 1 William Sandqvist william@kth.se

  15. ⋅ = ( a a a a ) ( b b b b ) ( q q q q q q q q ) 3 2 1 0 3 2 1 0 7 6 5 4 3 2 1 0 Multiplication (two positive integers) a a a a a a a a 3 3 0 0 b 2 2 1 1 0 b T MUL ~(3*N-4)*T FA 1 b 2 a a a a 3 0 2 1 b a 3 a a a 3 2 1 0 Carry is delayed trough 8 q q q q q q q q stages! 7 6 3 0 5 4 2 1 William Sandqvist william@kth.se

  16. Could the bit’s be added in some other order? William Sandqvist william@kth.se

  17. Quickie question Ripple Carry-adder: a b a b a b a b C C C C 3 3 0 0 2 2 1 1 in 3 in 2 in 1 in 0 C C C C s s s s out 2 out 0 out 3 out 1 1 3 0 2 • What will happen if one confuses the lines a b C in ? a a b a b a b C b 3 0 0 2 2 1 1 in 0 3 C C C Here someone in 3 in 2 in 1 has completly confused the lines! C C C C s s s s out 0 out 2 out 3 out 1 1 0 3 2 b) Don’t care! a) Disaster! Discuss your answer with your bench neighbor! William Sandqvist william@kth.se

  18. Quickie question a a b a b a b C b 3 0 0 2 2 1 1 in 0 3 C C C in 3 in 2 in 1 C C C C s s s s out 0 out 2 out 3 out 1 1 0 3 2 b) Don’t care! • The sum bit equals ” odd parity” of the input bits. • Carry out bit equals the ” majority value” of the inputbits. In neither case the bits order is of any significance. William Sandqvist william@kth.se

  19. Could the bit’s be added in some other order? In this way q 2 will get the same value but with a Carry is fed different bit directly to order! the second row! William Sandqvist william@kth.se

  20. ⋅ = ( a a a a ) ( b b b b ) ( q q q q q q q q ) 3 2 1 0 3 2 1 0 7 6 5 4 3 2 1 0 A quicker solution - Carry-Save Multiplier (BV: page 311) Figure 5.45 Multiplier carry-save array. William Sandqvist william@kth.se

  21. ⋅ = ( a a a a ) ( b b b b ) ( q q q q q q q q ) 3 2 1 0 3 2 1 0 7 6 5 4 3 2 1 0 A quicker solution - Carry-Save Multiplier (BV: page 311) T MUL ~(2*N-2)*T FA Reduces the delay because the carry is fed directly to the next row! Carry is delayed through 6 stages! Figure 5.45 Multiplier carry-save array. William Sandqvist william@kth.se

  22. ⋅ = ( a a a a ) ( b b b b ) ( q q q q q q q q ) 3 2 1 0 3 2 1 0 7 6 5 4 3 2 1 0 A quicker solution - Carry-Save Multiplier (BV: page 311) Extra tip : A fast Carry- look ahead adder here! Figure 5.45 Multiplier carry-save array. William Sandqvist william@kth.se

  23. William Sandqvist william@kth.se

  24. Multiplication with the constant 2 0101*2 = 1010 (5*2=10) 1010*2 = 10100 (-6*2=-12) 01010101*2 = 010101010 (85*2=190) 10010101*2 = 100101010 (-107*2=-214) As with multiplication with 10 in our base 10: 63*10 = 630, -63*10 = -630 William Sandqvist william@kth.se

  25. Multiplication with 2 n 0101*2 = 1010 (5*2=10) 0101*2 2 = 10100 (5*4=20) 0101*2 3 = 101000 (5*8=40) 0101*2 4 = 1010000 (5*16=80) compare multiplication with exponents of 10 in our base 10: 6*10 = 60, 6*100 = 600, 6*1000 = 6000 William Sandqvist william@kth.se

  26. Multiplication with 2 n A multiplication with 2 n can be done by shifting all bits n steps to the left and to fill in with zeros 13 × 8 can be calculated by shifting (01011) three bits to the right Result: 01011000 corresponds to (104) 10 Note that you need more bits to represent the results! William Sandqvist william@kth.se

  27. Barrel-shifter B 0 0 1 2 3 B 1 A 0 A 1 Multiplexor A 2 A 3 0 0 0 0 0 0 B 0 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 MUX B 1 Multiplication with (2 0 , 2 1 , 2 2 , 2 3 ) – 1, 2, 4, 8 = ⋅ ( B B ) ( S S S S S S S ) ( A A A A ) 2 1 0 6 5 4 3 2 1 0 3 2 1 0 William Sandqvist william@kth.se

  28. Division between two positive integers (BV page 693 fig 10.21 b) 0 1 0 1 a r 11/2 = 5 = + q 1 0 1 0 1 1 b b - 1 0 0 0 1 - 0 0 Reminder = 1 0 1 1 - 1 0 1 William Sandqvist william@kth.se

  29. a r = + A little more detail … q b b 0 1 0 1 11/2 = 5 1 0 0 1 0 1 1 - 0 0 1 0 - 1 0 0 0 1 - 0 0 Reminder = 1 0 1 1 - 1 0 1 William Sandqvist william@kth.se

  30. A little more detail … 11 1 a r = + = + 5 q b b 2 2 a : 0 1 0 1 1 r : q : b : 1 0 William Sandqvist william@kth.se

  31. A little more detail … 11 1 a r = + = + 5 q b b 2 2 a : 0 1 0 1 1 r : q : 0 b : -1 0 William Sandqvist william@kth.se

  32. A little more detail … 11 1 a r = + = + 5 q b b 2 2 a : 0 1 0 1 1 r : q : 0 0 b : -1 0 William Sandqvist william@kth.se

  33. A little more detail … 11 1 a r = + = + 5 q b b 2 2 a : 0 1 0 1 1 r : q : 0 0 1 b : -1 0 William Sandqvist william@kth.se

  34. A little more detail … 11 1 a r = + = + 5 q b b 2 2 a : 0 0 0 1 1 r : q : 0 0 1 0 b : -1 0 William Sandqvist william@kth.se

  35. A little more detail … 11 1 a r = + = + 5 q b b 2 2 a : 0 0 0 1 1 r : q : 0 0 1 0 1 b : -1 0 William Sandqvist william@kth.se

  36. A little more detail … 11 1 a r = + = + 5 q b b 2 2 0 0 0 0 1 r : 1 q : 0 0 1 0 1 William Sandqvist william@kth.se

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Lecture 1 Number Representation CS 230 - Spring 2020 1-1 Number Representation Radix

Lecture 1 Number Representation CS 230 - Spring 2020 1-1 Number Representation Radix

CS 230 Introduction to Computers and Computer Systems Lecture 1 Number Representation CS 230 - Spring 2020 1-1 Number Representation Radix representation radix also know as base Writing natural numbers using a finite alphabet

975 views • 72 slides
Number representation in Java Scientific notation Overview topics Binary representation of

Number representation in Java Scientific notation Overview topics Binary representation of

Number representation in Java DOS, HOM Its about interpretation Number representation in Java Scientific notation Overview topics Binary representation of Thijs Dorssers, Pieter van den Hombergh integers Binary representation of

524 views • 10 slides
Microprocessors & Interfacing Data Representation Number representation

Microprocessors & Interfacing Data Representation Number representation

Lecture Overview Microprocessor Hardware Structures Microprocessors & Interfacing Data Representation Number representation Instruction Set Basics of Computing with Microprocessor Systems Lecturer : Dr. Annie Guo S2,

822 views • 13 slides
Number representation A number can be represented in binary in many ways. The most common number

Number representation A number can be represented in binary in many ways. The most common number

Number representation A number can be represented in binary in many ways. The most common number types to be represented are: Integers, positive integers one-complement, two-complement, sign-magnitude Decimal real numbers with a fixed

979 views • 72 slides
COMP 1402 Winter 2008 Tutorial #3 Overview of Tutorial #3 Review of Number Representation

COMP 1402 Winter 2008 Tutorial #3 Overview of Tutorial #3 Review of Number Representation

COMP 1402 Winter 2008 Tutorial #3 Overview of Tutorial #3 Review of Number Representation Creating Archives (TAR utility) Compling C Programs Creating Makefiles Emacs text editor 1 Review of Number Representation

370 views • 14 slides
Floating point representation (Unsigned) Fixed-point representation The numbers are stored with a

Floating point representation (Unsigned) Fixed-point representation The numbers are stored with a

Floating point representation (Unsigned) Fixed-point representation The numbers are stored with a fixed number of bits for the integer part and a fixed number of bits for the fractional part. Suppose we have 8 bits to store a real number, where

906 views • 39 slides
Real Number Representation 1 Topics Terminology IEEE standard for floating-point

Real Number Representation 1 Topics Terminology IEEE standard for floating-point

Real Number Representation 1 Topics Terminology IEEE standard for floating-point representation Floating point arithmetic Limitations 2 Terminology All digits in a number following any leading zeros are significant

1.24k views • 99 slides
Systems Number Representation Shankar Balachandran* Associate Professor, CSE Department Indian

Systems Number Representation Shankar Balachandran* Associate Professor, CSE Department Indian

Spring 2015 Week 9 Module 48 Digital Circuits and Systems Number Representation Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras *Currently a Visiting Professor at IIT Bombay Positional Number

160 views • 12 slides
ECE 0142 Computer Organization Lecture 2 Number Representation Slides adapted from Ron Hoelzeman

ECE 0142 Computer Organization Lecture 2 Number Representation Slides adapted from Ron Hoelzeman

ECE 0142 Computer Organization Lecture 2 Number Representation Slides adapted from Ron Hoelzeman 1 Data Representation Computers manipulate discrete digital quantities - that is 1s or 0s Example representations of data Numeric

1.01k views • 38 slides
Number Representation Number Representation CS10001: Programming & Data Structures

Number Representation Number Representation CS10001: Programming & Data Structures

Number Representation Number Representation CS10001: Programming & Data Structures Programming & Data Structures CS10001: Pallab Dasgupta Dasgupta Pallab Professor, Dept. of Computer Sc. & Engg Engg., ., Professor, Dept. of

724 views • 41 slides
Integer Representation Bits, binary numbers, and bytes Fixed-width representation of integers:

Integer Representation Bits, binary numbers, and bytes Fixed-width representation of integers:

Wellesley CS 240 Integer Representation Bits, binary numbers, and bytes Fixed-width representation of integers: unsigned and signed Modular arithmetic and overflow 1 positional number representation 2 4 0 = 2 x 10 2 + 4 x 10 1 + 0 x 10 0

373 views • 23 slides
Summary of previous lecture number representation: usually twos complement, but other

Summary of previous lecture number representation: usually twos complement, but other

Summary of previous lecture number representation: usually twos complement, but other representations possible choice depends on hardware size, performance, development time, availability of design tools etc ALU: develop

596 views • 32 slides
2 4 0 Integer Representation 100 10 1 weight 10 2 10 1 10 0 position 2 1 0 Base

2 4 0 Integer Representation 100 10 1 weight 10 2 10 1 10 0 position 2 1 0 Base

positional number representation Wellesley CS 240 = 2 x 10 2 + 4 x 10 1 + 0 x 10 0 2 4 0 Integer Representation 100 10 1 weight 10 2 10 1 10 0 position 2 1 0 Base determines: Bits, binary numbers, and bytes Fixed-width

444 views • 5 slides
IBM Watson example https://www.youtube.com/watch?v=DywO4zksfXw VOLUME 56, NUMBER 3/4, MAY/JUL.

IBM Watson example https://www.youtube.com/watch?v=DywO4zksfXw VOLUME 56, NUMBER 3/4, MAY/JUL.

Knowledge Representation Knowledge Representation IBM Watson example https://www.youtube.com/watch?v=DywO4zksfXw VOLUME 56, NUMBER 3/4, MAY/JUL. 2012 Journal of Research and Development Including IBM Systems Journal Knowledge Representation

278 views • 15 slides
SET PARTITIONING Denis Khryashchev INTRODUCTION INTRODUCTION PROBLEM STATEMENT NUMBER OF

SET PARTITIONING Denis Khryashchev INTRODUCTION INTRODUCTION PROBLEM STATEMENT NUMBER OF

SET PARTITIONING Denis Khryashchev INTRODUCTION INTRODUCTION PROBLEM STATEMENT NUMBER OF PARTITIONS MATHEMATICAL STATEMENT MATRIX REPRESENTATION MATRIX REPRESENTATION MATRIX REPRESENTATION MATRIX REPRESENTATION MATRIX REPRESENTATION

961 views • 30 slides
Unit 1 Integer Representation 1.2 Skills & Outcomes You should know and be able to apply

Unit 1 Integer Representation 1.2 Skills & Outcomes You should know and be able to apply

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 number of

463 views • 44 slides
Variable precision computing: Applications and challenges David H. Bailey Lawrence Berkeley

Variable precision computing: Applications and challenges David H. Bailey Lawrence Berkeley

Variable precision computing: Applications and challenges David H. Bailey Lawrence Berkeley Natlional Laboratory (retired) University of California, Davis, Department of Computer Science This talk is available at:

525 views • 29 slides
Fixed and Floating-point Numbers Eric McCreath Fractional binary numbers Remember how the

Fixed and Floating-point Numbers Eric McCreath Fractional binary numbers Remember how the

Fixed and Floating-point Numbers Eric McCreath Fractional binary numbers Remember how the meaning of the digits in a binary number is defined: Note the binary radix point For example, the binary number: means 2 Binary Converting a

449 views • 7 slides
Floating Point Basics Normally write: mantissa * 2 exponent Alternately, sign * mantissa *

Floating Point Basics Normally write: mantissa * 2 exponent Alternately, sign * mantissa *

III. Floating Point Representation Floating Point numbers contain the following components 1) Mantissa 2) Mantissa sign (optional): Some method to allow a signed mantissa such as: a) an explicit sign bit, or b) another bit to allow a signed

314 views • 3 slides
Outline Lattices and fixed points 1 The varieties D n 2 Results 3 2 2/17 Lattices and fixed

Outline Lattices and fixed points 1 The varieties D n 2 Results 3 2 2/17 Lattices and fixed

Lattices and fixed points The varieties D n Results Fixed-point theory in the varieties D n Sabine Frittella and Luigi Santocanale Laboratoire dInformatique Fondamentale de Marseille, France May 1, 2014 RAMiCS 2014, Marienstatt im

520 views • 17 slides
Towards an Optimizing Compiler for Numerical Kernels joint work with: Heiko Becker, Anastasiia

Towards an Optimizing Compiler for Numerical Kernels joint work with: Heiko Becker, Anastasiia

Photo by Sebastian Bruggisser Towards an Optimizing Compiler for Numerical Kernels joint work with: Heiko Becker, Anastasiia Izycheva, Debasmita Lohar, Viktor Kuncak, Magnus Myreen, Sylvie Putot, Eric Goubault, Helmut Seidl Eva Darulova

849 views • 51 slides
Automating the Verification of Floating-point Algorithms Guillaume Melquiond Inria Saclay

Automating the Verification of Floating-point Algorithms Guillaume Melquiond Inria Saclay

Introduction Interval+Error Advanced Gappa Conclusion Automating the Verification of Floating-point Algorithms Guillaume Melquiond Inria Saclay Ile-de-France LRI, Universit e Paris Sud, CNRS 2014-07-18 Guillaume Melquiond

1.55k views • 126 slides
Optimizing MPC for robust and scalable integer and floating-point arithmetic Liisi Kerik * Peeter

Optimizing MPC for robust and scalable integer and floating-point arithmetic Liisi Kerik * Peeter

Optimizing MPC for robust and scalable integer and floating-point arithmetic Liisi Kerik * Peeter Laud * Jaak Randmets * * Cybernetica AS University of Tartu, Institute of Computer Science January 30, 2016 Introduction Secure

588 views • 23 slides
Interest Rate Hedging Best Practices & Pitfalls to Avoid March 14, 2017 @ SVAFP &

Interest Rate Hedging Best Practices & Pitfalls to Avoid March 14, 2017 @ SVAFP &

3/16/2017 Interest Rate Hedging Best Practices & Pitfalls to Avoid March 14, 2017 @ SVAFP & INTEREST RATE Ruth Hardie rhardie@hedgetrackers.com 1 3/16/2017 Agenda Risk Mitigation Scenarios focused on Fixing variable

454 views • 16 slides

More recommend