number systems and arithmetic
play

Number Systems and Arithmetic Jason Mars Thursday, January 24, 13 - PowerPoint PPT Presentation

Sep 28, 2022 •575 likes •1.49k views

Number Systems and Arithmetic Jason Mars Thursday, January 24, 13 What do all those bits mean? bits (011011011100010 ....01) data instruction number text chars .............. R-format I-format ... integer floating point single precision

  1. Number Systems and Arithmetic Jason Mars Thursday, January 24, 13

  2. What do all those bits mean? bits (011011011100010 ....01) data instruction number text chars .............. R-format I-format ... integer floating point single precision double precision signed unsigned ... ... ... ... Thursday, January 24, 13

  3. Questions About Numbers • How do you represent • negative numbers? • fractions? • really large numbers? • really small numbers? • How do you • do arithmetic? • identify errors (e.g. overflow)? • What is an ALU and what does it look like? • ALU=arithmetic logic unit Thursday, January 24, 13

  4. Introduction to Binary Numbers • Consider a 4 bit binary number Decimal Binary Decimal Binary 0 0000 4 0100 1 0001 5 0101 2 0010 6 0110 3 0011 7 0111 • Examples of binary arithmetic 3 + 2 = 5 3 + 3 = 6 1 1 1 0 0 1 1 0 0 1 1 + 0 0 1 0 + 0 0 1 1 Thursday, January 24, 13

  5. Introduction to Binary Numbers • Consider a 4 bit binary number Decimal Binary Decimal Binary 0 0000 4 0100 1 0001 5 0101 2 0010 6 0110 3 0011 7 0111 • Examples of binary arithmetic 3 + 2 = 5 3 + 3 = 6 1 1 1 0 0 1 1 0 0 1 1 + 0 0 1 0 + 0 0 1 1 1 Thursday, January 24, 13

  6. Introduction to Binary Numbers • Consider a 4 bit binary number Decimal Binary Decimal Binary 0 0000 4 0100 1 0001 5 0101 2 0010 6 0110 3 0011 7 0111 • Examples of binary arithmetic 3 + 2 = 5 3 + 3 = 6 1 1 1 0 0 1 1 0 0 1 1 + 0 0 1 0 + 0 0 1 1 0 1 Thursday, January 24, 13

  7. Introduction to Binary Numbers • Consider a 4 bit binary number Decimal Binary Decimal Binary 0 0000 4 0100 1 0001 5 0101 2 0010 6 0110 3 0011 7 0111 • Examples of binary arithmetic 3 + 2 = 5 3 + 3 = 6 1 1 1 0 0 1 1 0 0 1 1 + 0 0 1 0 + 0 0 1 1 1 0 1 Thursday, January 24, 13

  8. Introduction to Binary Numbers • Consider a 4 bit binary number Decimal Binary Decimal Binary 0 0000 4 0100 1 0001 5 0101 2 0010 6 0110 3 0011 7 0111 • Examples of binary arithmetic 3 + 2 = 5 3 + 3 = 6 1 1 1 0 0 1 1 0 0 1 1 + 0 0 1 0 + 0 0 1 1 1 0 1 0 Thursday, January 24, 13

  9. Introduction to Binary Numbers • Consider a 4 bit binary number Decimal Binary Decimal Binary 0 0000 4 0100 1 0001 5 0101 2 0010 6 0110 3 0011 7 0111 • Examples of binary arithmetic 3 + 2 = 5 3 + 3 = 6 1 1 1 0 0 1 1 0 0 1 1 + 0 0 1 0 + 0 0 1 1 1 0 1 1 0 Thursday, January 24, 13

  10. Introduction to Binary Numbers • Consider a 4 bit binary number Decimal Binary Decimal Binary 0 0000 4 0100 1 0001 5 0101 2 0010 6 0110 3 0011 7 0111 • Examples of binary arithmetic 3 + 2 = 5 3 + 3 = 6 1 1 1 0 0 1 1 0 0 1 1 + 0 0 1 0 + 0 0 1 1 1 0 1 1 1 0 Thursday, January 24, 13

  11. Negative Numbers? • We would like a number system that provides • obvious representation of 0,1,2... • uses adder for addition • single value of 0 • equal coverage of positive and negative numbers • easy detection of sign • easy negation Thursday, January 24, 13

  12. Two’s Complement Representation • 2’s complement representation of negative numbers • Take the bitwise inverse and add 1 • Biggest 4-bit Binary Number: 7 Smallest 4-bit Binary Number: -8 Decimal Two � s Complement Binary -8 1000 -7 1001 -6 1010 -5 1011 -4 1100 -3 1101 -2 1110 -1 1111 0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 Thursday, January 24, 13

  13. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 Thursday, January 24, 13

  14. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 1 Thursday, January 24, 13

  15. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 1 Thursday, January 24, 13

  16. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 1 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 1 Thursday, January 24, 13

  17. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 1 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 0 1 Thursday, January 24, 13

  18. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 1 1 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 0 1 Thursday, January 24, 13

  19. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 1 1 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 0 0 1 Thursday, January 24, 13

  20. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 1 1 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 0 0 1 0 Thursday, January 24, 13

  21. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 1 1 1 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 0 0 1 0 Thursday, January 24, 13

  22. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 1 1 1 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 0 0 1 1 0 Thursday, January 24, 13

  23. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 1 1 1 1 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 0 0 1 1 0 Thursday, January 24, 13

  24. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 1 1 1 1 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 0 0 1 1 1 0 Thursday, January 24, 13

  25. Two’s Complement Arithmetic (Subtraction) Decimal 2 � s Complement Binary Decimal 2 � s Complement Binary 0 0000 -1 1111 1 0001 -2 1110 2 0010 -3 1101 3 0011 -4 1100 4 0100 -5 1011 5 0101 -6 1010 6 0110 -7 1001 7 0111 -8 1000 Examples: 7 - 6 = 7 + (- 6) = 1 3 - 5 = 3 + (- 5) = -2 1 1 1 1 0 1 1 1 0 0 1 1 + 1 0 1 0 + 1 0 1 1 0 0 0 1 1 1 1 0 Thursday, January 24, 13

Recommend

Summary Introduction Exotic Number Systems Basic number systems for Hardware Arithmetic

Summary Introduction Exotic Number Systems Basic number systems for Hardware Arithmetic

Summary Introduction Exotic Number Systems Basic number systems for Hardware Arithmetic Operators integer and fixed point floating point Arnaud Tisserand Exotic number systems logarithmic number system (LNS) CNRS, IRISA

534 views • 18 slides
Number Systems and Computer Arithmetic Counting to four billion two fingers at a time CSE 141,

Number Systems and Computer Arithmetic Counting to four billion two fingers at a time CSE 141,

Number Systems and Computer Arithmetic Counting to four billion two fingers at a time CSE 141, S2'06 Jeff Brown What do all those bits mean now? bits (011011011100010 ....01) data instruction number text chars .............. R-format

861 views • 46 slides
CHAPTER V NUMBER SYSTEMS AND ARITHMETIC R.M. Dansereau; v.1.0 NUMBER SYSTEMS INTRO. TO COMP.

CHAPTER V NUMBER SYSTEMS AND ARITHMETIC R.M. Dansereau; v.1.0 NUMBER SYSTEMS INTRO. TO COMP.

INTRO. TO COMP. ENG. CHAPTER V CHAPTER V-1 NUMBERS & ARITHMETIC CHAPTER V NUMBER SYSTEMS AND ARITHMETIC R.M. Dansereau; v.1.0 NUMBER SYSTEMS INTRO. TO COMP. ENG. NUMBER SYSTEMS CHAPTER V-2 RADIX-R REPRESENTATION NUMBERS &

384 views • 24 slides
Digital Design Discussion: Arithmetic Binary Arithmetic Floating-Point Arithmetic Binary

Digital Design Discussion: Arithmetic Binary Arithmetic Floating-Point Arithmetic Binary

Principles Of Digital Design Discussion: Arithmetic Binary Arithmetic Floating-Point Arithmetic Binary Arithmetic Same basic methodology as decimal arithmetic Important to know number representation Unsigned Signed

704 views • 11 slides
Faster arithmetic for number-theoretic transforms David Harvey University of New South Wales 7th

Faster arithmetic for number-theoretic transforms David Harvey University of New South Wales 7th

Faster arithmetic for number-theoretic transforms David Harvey University of New South Wales 7th October 2011, Macquarie University David Harvey Faster arithmetic for number-theoretic transforms Plan for talk 1. Review number-theoretic

353 views • 34 slides
Continued fractions and number systems: applications to correctly-rounded implementations of

Continued fractions and number systems: applications to correctly-rounded implementations of

Continued fractions and number systems: applications to correctly-rounded implementations of elementary functions and modular arithmetic. Mourad Gouicem PEQUAN Team, LIP6/UPMC Nancy, France May 28 th 2013 Table of Contents Continued fraction

753 views • 42 slides
Number Theory Arithmetic Fermats and Eulers Theorems Discrete Cryptography Logarithms

Number Theory Arithmetic Fermats and Eulers Theorems Discrete Cryptography Logarithms

Cryptography Number Theory Divisibility and Primes Modular Number Theory Arithmetic Fermats and Eulers Theorems Discrete Cryptography Logarithms Computationally Hard Problems School of Engineering and Technology CQUniversity

964 views • 67 slides
Number Theory A bit more depth. . . modular arithmetic primes Euclids algorithm

Number Theory A bit more depth. . . modular arithmetic primes Euclids algorithm

1 Number Theory A bit more depth. . . modular arithmetic primes Euclids algorithm Chinese remainder theorem Eulers totient function Eulers theorem 2 Modular Arithmetic m; n integers, n > 0

2.23k views • 15 slides
Arithmetic Division Distributed Arithmetic Newton Raphson Newton Raphson CORDIC Unsigned

Arithmetic Division Distributed Arithmetic Newton Raphson Newton Raphson CORDIC Unsigned

Advanced Digital IC Design Advanced Digital IC Design Number Representation Addition Multiplication Arithmetic Division Distributed Arithmetic Newton Raphson Newton Raphson CORDIC Unsigned Number Representation Example: Unsigned Number

899 views • 30 slides
i radix point (binary point) assumed to be to the right of i 0 the rightmost digit.

i radix point (binary point) assumed to be to the right of i 0 the rightmost digit.

Number Systems Binary: Computer Arithmetic Hexadecimal: Word Size: (Fixed) number of bits used to represent a number School of Computer Science G51CSA School of Computer Science G51CSA 1 2 Integer Representation Non Negative Integer

391 views • 5 slides
CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 8 February 2010 1 2 1 3

CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 8 February 2010 1 2 1 3

CS101 Lecture 11: Number Systems and Binary Numbers Aaron Stevens 8 February 2010 1 2 1 3 4 2 !!! MATH WARNING !!! TODAYS LECTURE CONTAINS TRACE AMOUNTS OF ARITHMETIC AND ALGEBRA PLEASE BE ADVISED THAT CALCULTORS WILL BE ALLOWED

529 views • 19 slides
Arithmetic, Logic and Control Instructions Systems Design & Programming CMPE 310 Intel

Arithmetic, Logic and Control Instructions Systems Design & Programming CMPE 310 Intel

Arithmetic, Logic and Control Instructions Systems Design & Programming CMPE 310 Intel Assembly Arithmetic Operations: Addition Subtraction Multiplication Division Comparison Negation Increment Decrement

441 views • 11 slides
Number Systems and Codes 1 Number Systems Decimal number: 123.45 = 1 10 2 + 2 10 1 + 3 10 0 + 4

Number Systems and Codes 1 Number Systems Decimal number: 123.45 = 1 10 2 + 2 10 1 + 3 10 0 + 4

Number Systems and Codes 1 Number Systems Decimal number: 123.45 = 1 10 2 + 2 10 1 + 3 10 0 + 4 10 -1 + 5 10 -2 Base b number: N = a q- 1 b q- 1 + + a 0 b 0 + + a -p b -p b >1, 0 <= a i <= b -1 Integer part: a q -1 a q -2

453 views • 12 slides
Reliable multiprecision arithmetic for number theory Fredrik Johansson LFANT seminar, IMB /

Reliable multiprecision arithmetic for number theory Fredrik Johansson LFANT seminar, IMB /

Reliable multiprecision arithmetic for number theory Fredrik Johansson LFANT seminar, IMB / INRIA 2014-09-16 Where I come from Malung (population 5000) in Dalarna, Sweden Things Ive done previously Software Since 2007: mpmath , a

948 views • 64 slides
The arithmetic dynamics of correspondences Patrick Ingram Colorado State University Silvermania

The arithmetic dynamics of correspondences Patrick Ingram Colorado State University Silvermania

The arithmetic dynamics of correspondences Patrick Ingram Colorado State University Silvermania 2015 Arithmetic dynamics (Arithmetic) dynamics Let K be a (number) field, X / K a variety, and f : X X a morphism. Describe O + f ( P ) = { P ,

554 views • 30 slides
Foundations of Computer Science Last Time Lecture 10 Number Theory Division and the Greatest

Foundations of Computer Science Last Time Lecture 10 Number Theory Division and the Greatest

Foundations of Computer Science Last Time Lecture 10 Number Theory Division and the Greatest Common Divisor Fundamental Theorem of Arithmetic 1 Why sums and reccurrences? Running times of programs. Cryptography and Modular Arithmetic RSA:

147 views • 4 slides
Frobenius Distributions Edgar Costa (MIT) Simons Collab. on Arithmetic Geometry, Number Theory,

Frobenius Distributions Edgar Costa (MIT) Simons Collab. on Arithmetic Geometry, Number Theory,

Frobenius Distributions Edgar Costa (MIT) Simons Collab. on Arithmetic Geometry, Number Theory, and Computation April 23rd, 2019 University of Washington Slides available at edgarcosta.org under Research factorization of f p x e.g.: f p x

995 views • 76 slides
ECE 645: Lecture 4 Number Representation Part 1 Fixed-Point Representations. Endianness.

ECE 645: Lecture 4 Number Representation Part 1 Fixed-Point Representations. Endianness.

ECE 645: Lecture 4 Number Representation Part 1 Fixed-Point Representations. Endianness. Required Reading B. Parhami, Computer Arithmetic: Algorithms and Hardware Design Chapter 1, Numbers and Arithmetic, Sections 1.1-1.6 Chapter 2,

1.09k views • 88 slides
Chapter 1 Analog vs. Digital Positional Number Systems Number Systems And Codes

Chapter 1 Analog vs. Digital Positional Number Systems Number Systems And Codes

Chapter Outline Chapter 1 Analog vs. Digital Positional Number Systems Number Systems And Codes Positional Number System Conversions Binary System operations Computer Application Encoding Techniques Flaxer Eli - ComputerAppl

447 views • 8 slides
CS 251 Fall 2019 CS 240 Principles of Programming Languages Foundations of Computer Systems

CS 251 Fall 2019 CS 240 Principles of Programming Languages Foundations of Computer Systems

CS 251 Fall 2019 CS 240 Principles of Programming Languages Foundations of Computer Systems Ben Wood Arithmetic Logic adders Arithmetic Logic Unit https://cs.wellesley.edu/~cs240/ Arithmetic Logic 1 ex Addition: 1-bit half adder A +

273 views • 13 slides
Introduction to Computer Arithmetic for Efficient Hardware Implementations Arnaud Tisserand

Introduction to Computer Arithmetic for Efficient Hardware Implementations Arnaud Tisserand

Introduction to Computer Arithmetic for Efficient Hardware Implementations Arnaud Tisserand CNRS, Lab-STICC CEA-SPEC Seminar, Nov. 2019 -- Babylonian Arithmetic Use of a positional number system with: primary radix 60 auxiliary radix

1.46k views • 108 slides
Fun With Number Systems Or: How Knowing Number Systems Can Help You Interview Better Finding the

Fun With Number Systems Or: How Knowing Number Systems Can Help You Interview Better Finding the

Fun With Number Systems Or: How Knowing Number Systems Can Help You Interview Better Finding the Odd Ball 1 2 9 10 11 3 3 2 1 10 11 9 Goal: Find the odd ball and whether it's heavier or lighter in three weighings. The Solution 5 6

2.23k views • 199 slides
Lecture 6 Arithmetic Operators 3. Hardware and Softw are 4. High Level Languages C has a

Lecture 6 Arithmetic Operators 3. Hardware and Softw are 4. High Level Languages C has a

1. Introduction 2. Binary Representation Lecture 6 Arithmetic Operators 3. Hardware and Softw are 4. High Level Languages C has a number of arithmetic operators which are used to 5. Standard input and output combine variables and

403 views • 3 slides
Around canonical heights in arithmetic dynamics Shu Kawaguchi Arithmetic 2015 - Silvermania

Around canonical heights in arithmetic dynamics Shu Kawaguchi Arithmetic 2015 - Silvermania

Around canonical heights in arithmetic dynamics Shu Kawaguchi Arithmetic 2015 - Silvermania August 14, 2015 1 / 39 Plan of the talk (2005 Email? 2008 AIM) 1 Canonical heights for polarized dynamical systems 2 Canonical heights on affine space

1.23k views • 46 slides

More recommend