ffirs.qxd 9/26/2005 2:42 PM Page iii 001010110001110010101000010101011010100100100100011011000001010001101010001 THE DEFINITIVE GUIDE TO HOW COMPUTERS DO MATH Featuring The Virtual DIY Calculator Clive “MAX” Maxfield Alvin Brown 0 1 0 1 1 0 0 1 A JOHN WILEY & SONS, INC., PUBLICATION 0 001010110001 10010101000010101011010100100100100011011000001010001101010001 1 0 0 1 ffirs.qxd 9/26/2005 2:42 PM Page iv Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. 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Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. Library of Congress Cataloging-in-Publication Data is available. ISBN-13 978-0471-73278-5 ISBN-10 0-471-73278-8 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 ffirs.qxd 9/26/2005 2:42 PM Page v To all our friends who make the world such a wonderful place! ftoc.qxd 9/26/2005 2:44 PM Page vii 010101010001011101010000101010110101001001001000110110000010101101000110101 Contents Laboratories ix Do You Speak Martian? xi Chapter 0 Why This Book Is So Cool 1 This is where we discover just why this book is so cool, and also why this chapter is numbered “0.” Chapter 1 Introducing Binary and Hexadecimal Numbers 7 In this chapter we introduce the concepts of the binary and hexadecimal number systems (but in a much more interesting manner than most computer books). Chapter 2 Computers and Calculators 23 Here we rampage through the insides of a simple computer and calculator, and we also meet our virtual DIY Calculator. Chapter 3 Subroutines and Other Stuff 35 In this smorgasbord of a chapter, we first discuss logical, shift, and rotate instructions; then we plunge headfirst into the stack, subroutines, recursion, and the concept of self- modifying code. Chapter 4 Integer Arithmetic 69 This is where we learn the concepts of signed and unsigned binary arithmetic, and multibyte data representations. In the labs associated with this chapter we create some integer-based math subroutines for use in Chapter 5. Chapter 5 Creating an Integer Calculator 123 This is where things really start to get interesting because we use the subroutines we developed in Chapter 4 to implement a simple four-function integer calculator. vviiii ftoc.qxd 9/26/2005 2:44 PM Page viii viii CONTENTS Chapter 6 More Functions and Experiments 135 This may be the last chapter, but turn that frown upside down into a smile because this is where we discover additional projects you can undertake, and also how you can compete with other readers to create better math routines for your DIY Calculator. Interactive Laboratories 155–412 This is where you will find the interactive laboratories associated with Chapters 2 through 5 (see more details on page ix). Appendix A Installing Your DIY Calculator 413 In this appendix, we learn how to install the virtual DIY Calculator presented on the CD-ROM accompanying this book. Appendix B Addressing Modes 417 Here we learn about the various addressing modes supported by the virtual microprocessor powering the DIY Calculator. Appendix C Instruction Set Summary 429 The tables in this appendix are useful for quickly determining which instructions are available with which addressing modes. Appendix D Additional Resources 433 There is a wealth of additional resources for individual readers and educators on the CD-ROM accompanying the book and from the DIY Calculator’s website. Also, there are lots of other really interesting books to read. This appendix explains it all. About the Authors 441 Acknowledgments 443 Index 445 flast.qxd 9/26/2005 2:47 PM Page ix 010101010001011101010000101010110101001001001000110110000010101101000110101 Laboratories LABS FOR CHAPTER 2: CREATING AND RUNNING PROGRAMS 155 Lab 2a: Creating a Simple Program 157 Lab 2b: Constant Labels and .EQU Directives 169 Lab 2c: Driving the Calculator’s Main Display 173 Lab 2d: Reading from the Calculator’s Keypad 185 Lab 2e: Writing to the Calculator’s Six LEDs 193 Lab 2f: Using the Memory Walker and Other 201 Diagnostic Displays LABS FOR CHAPTER 3: SUBROUTINES AND OTHER STUFF 219 Lab 3a: Using Logical Instructions, Shifts, and Rotates 221 Lab 3b: Understanding the Program Counter (PC) 237 Lab 3c: Using the Index Register (X) 245 Lab 3d: Using the Stack and Stack Pointer (SP) 255 Lab 3e: Using Subroutines 261 Lab 3f: Using Recursion 267 LABS FOR CHAPTER 4: INTEGER ARITHMETIC 275 Lab 4a: Creating a Testbench Program 277 Lab 4b: Creating a 16-bit ADD Subroutine 299 Lab 4c: Creating a 16-bit SUBTRACT Subroutine 311 Lab 4d: Creating a 16-bit NEGATE Subroutine 317 Lab 4e: Checking for Overflow in the ADD and 323 SUBTRACT Routines Lab 4f: Creating a 16-bit MULTIPLY Subroutine 333 Lab 4g: Creating a 16-bit DIVIDE Subroutine 351 LABS FOR CHAPTER 5: CREATING AN INTEGER CALCULATOR 373 Lab 5a: Creating the Calculator Framework 375 Lab 5b: Adding Some Low-Level Utility Routines 383 iixx flast.qxd 9/26/2005 2:47 PM Page x x LABORATORIES Lab 5c: Creating a Decimal GETNUM (“get number”) 389 Subroutine Lab 5d: Creating a Decimal DISPNUM (“display number”) 395 Subroutine Lab 5e: Implementing a Four-Function Integer Calculator 405 flast.qxd 9/26/2005 2:47 PM Page xi 010101010001011101010000101010110101001001001000110110000010101101000110101 Do You Speak Martian? There are an abundance of books on computer architectures, computer logic, and computer mathematics, and most of these works discuss vari- ous techniques for representing and manipulating numbers inside com- puters. Sadly, however, it appears that the majority of these tomes are written by visitors from the planet Mars, whose keen understanding of higher mathematics is somewhat offset by their limited grasp of the English language. “Say it’s not so!” you cry, but the proof is irrefutable. When was the last time you waded through a book on computer mathematics without your brain overheating? Much like reading Being and Nothingness by the famous French philosopher Jean Paul Sartre, one could mull over many of these cryptic masterpieces until the end of time without gaining so much as the faintest clue as to what was in their authors’ minds. This is why we have been moved to write this modest attempt at in- troducing the basics of computer arithmetic in words we can all under- stand. An impossible task, some may say, but we dare to fly in the face of conventional wisdom. Now, read on. ... iixx c00.qxd 7/20/2005 8:21 AM Page 1 The Definitive Guide to How Computers Do Math: Featuring the Virtual DIY Calculator. Clive “MAX” Maxfield and Alvin Brown. Copyright  2005 John Wiley & Sons, Inc. 100001001010111111001010100001010101101010010010010001101100000101000110101 CHAPTER 0 WHY THIS BOOK IS SO COOL “ The simplest schoolboy is now familiar with facts for which Archimedes would have ” sacrificed his life. ERNESTRENAN(1823–1892) in Souvenirs D’enfance et de Jeunesse(1887) 100001001010111111001010100001010101101010010010010001101100000101000110101 10100010101010001110010101000010101011010100100 1010I1n0 t0hi0s 1ch1a0p1te1r w00e w00ill0 le1a0rn100011010110000101010100011 0 1001010100001010101101010010010010001101100000 0 ●● The sock color of choice for the discerning 101100011010V1ik1in0g0 w0a0rr1io0r101010001110010101000010101 01010101001001001000110110000010100011010110000 ●● Why this book is so cool 1 1010101000111001010100001010101101010010010010 ●● That there are jobs awaiting time-travelers 0 0011011000001010001101011000010101010001110010 0 ●● Some “stuff” about calculators 1011000010●●10W1h0y1 th1i0s c1h0ap1t0er0 i1s n0u0m1b0e0re1d0 “00”0110110000010100 0110101100001010101000111001010100001010101101 0100100100100011011000001010001101011000010101 0100011100101010000101010110101001001001000110 1100000101000110101100001010101000111001010100 0010101011010100100100100011011000001010001101 0110000101010100011100101010000101010110101001 0010010001101100000101000110101100001010101000 1110010101000010101011010100100100100011011000 0010100011010110000101010100011100101010000101 0101101010010010010001101100000101000110101000 0011011000001010001101011000010101010001110010 1010000101010110101001001001000110110000010100 01010101010101010010100 0110101100001010101000111001010100001010101101 0100100100100011011000001010001101011000010101 0100011100101010000101010110101001001001000110 1 1100000101000110101100001010101000111001010100 0110000101010100011100101010000101010110101001