Numbers and computers

237 Pages·2015·1.631 MB·English

by  Ronald T. Kneusel

Ronald T. Kneusel
Numbers
and
Computers
Numbers and Computers

Ronald T. Kneusel
Numbers and Computers
123
RonaldT.Kneusel
Broomfield,CO,USA
ISBN978-3-319-17259-0 ISBN978-3-319-17260-6 (eBook)
DOI10.1007/978-3-319-17260-6
LibraryofCongressControlNumber:2015935728
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Preface
This is a book about numbers and how those numbers are represented in and
operatedonbycomputers.
Of course, numbers are fundamental to how computers operate because, in the
end,everythingacomputerworkswithisanumber.Itiscrucialthatpeoplewhode-
velopcodeunderstandthisareabecausethenumericaloperationsallowedbycom-
putersandthelimitationsofthoseoperations,especiallyintheareaoffloatingpoint
math, affect virtually everything people try to do with computers. This book aims
tohelpbyexploring,insufficient,butnotoverwhelming,detail,justwhatitisthat
computersdowithnumbers.
WhoShouldReadThisBook
Thisbookisforanyonewhodevelopssoftwareincludingsoftwareengineers,scien-
tists,computersciencestudents,andengineeringstudentsandanyonewhoprograms
forfun.
Ifyouareasoftwareengineer,youshouldalreadybefamiliarwithmanyofthe
topics in this book, especially if you have been in the field for any length of time.
Still,Iurgeyoutopresson,forperhapsyouwillfindagemortwowhicharenew
toyou.
Ifyouareoldenough,youwillrememberthedaysofFortranandmainframes.If
so,likethesoftwareengineersabove,youareprobablyalsofamiliarwiththebasics
ofhowcomputersrepresentandoperateonnumbers,but,alsolikethesoftwareen-
gineersabove,youwilllikelyfindagemortwoofyourown.Scientistsinparticular
shouldbeawareofthelimitationsandpitfallsassociatedwithusingfloating point
numberssincefewthingsinsciencearerestrictedtointegers.
Students need this book because it is essential to know what the computer is
doing under the hood. After all, if you are going to make a career of computers,
whywouldyounotwanttoknowhowthemachineworks?
vii
viii Preface
HowtoUseThisBook
Thisbookconsistsoftwomainparts.Thefirstdealswithstandardrepresentationsof
integersandfloatingpointnumbers,whiletheseconddetailsseveralothernumber
representationswhicharenicetoknowaboutandhandyfromtimetotime.Either
partisagoodplacetostart,thoughitisprobablybestifthepartsthemselvesareread
fromstarttoend.Later,afterthebookhasbeenread,youcanuseitasareference.
Thereareexercisesattheendofeachchapter.Mostoftheseareofthestraightfor-
wardpencilandpaperkind,justtotestyourunderstanding,whileothersaresmall
programming projects meant to increase your familiarity with the material. Exer-
cises that are (subjectively) more difficult will be marked with either one or two
stars(*or**)toindicatethelevelofdifficulty.
Example code is in C and/or Python version 2.7, though earlier 2.x versions
should work just as well. Intimate knowledge of these programming languages is
notnecessaryinordertounderstandtheconceptsbeingdiscussed.Ifsomethingis
not easy to see in the code, it will be described in the text. Why C? Because C is
alow-levellanguage,closetothenumberswewillbeworkingwithandbecauseC
isthegrandfather ofmostcommon programming languages incurrentuseinclud-
ing Python. In general, code will be offset from text and in a monospace font.
For readers not familiar with C and/or Python, there are a plethora of tutorials on
theWebandreferencebooksbythebookcase.Twoexamples,gearedtowardpeo-
ple less familiar with programming, are Beginning C by Ivor Horton and Python
Programming Fundamentals by Kent Lee. Both of these texts are available from
Springerinprintore-bookformat.
Attheendofeachchapterarereferencesforthematerialpresentedinthechapter.
Muchcanbelearnedbylookingatthesereferences.Almostbyinstinctwetendto
ignore sections like this as we are now programmed to ignore advertisements on
webpages.Inthisformercase,resisttemptation;inthelattercase,keepcalmand
carryon.
Acknowledgments
This book was not written in a vacuum. Here I want to acknowledge those who
helped make it a reality. First, the reviewers, who gave of their time and talent to
givemeextremelyvaluablecommentsandfriendlycriticism:RobertKneusel,M.S.,
JimPendleton,EdScott,Ph.D.,andMichaelGalloy,Ph.D.Gentlemen,thankyou.
Second, thank you to Springer, especially my editor, Courtney Clark, for moving
ahead with this book. Lastly, and most importantly, thank you to my wife, Maria,
and our children: David, Peter, Paul, Monica, Joseph, and Francis. Without your
patienceandencouragement,noneofthiswouldhavebeenwritten.
Broomfield,USA RonaldT.Kneusel
December2014 AM+DG
Contents
PartI StandardRepresentations
1 NumberSystems ............................................... 3
1.1 RepresentingNumbers ...................................... 3
1.2 TheBigThree(andOneOldGuy) ............................ 8
1.3 ConvertingBetweenNumberBases ........................... 10
1.4 ChapterSummary .......................................... 16
Exercises ...................................................... 16
References..................................................... 16
2 Integers....................................................... 19
2.1 Bits,Nibbles,Bytes,andWords .............................. 19
2.2 UnsignedIntegers .......................................... 21
2.2.1 Representation ...................................... 22
2.2.2 StorageinMemory:Endianness........................ 22
2.3 OperationsonUnsignedIntegers.............................. 25
2.3.1 BitwiseLogicalOperations............................ 25
2.3.2 Testing,Setting,Clearing,andTogglingBits ............. 30
2.3.3 ShiftsandRotates.................................... 33
2.3.4 Comparisons........................................ 37
2.3.5 Arithmetic.......................................... 41
2.3.6 SquareRoots........................................ 52
2.4 WhatAboutNegativeIntegers?............................... 53
2.4.1 Sign-Magnitude ..................................... 54
2.4.2 One’sComplement................................... 54
2.4.3 Two’sComplement .................................. 55
2.5 OperationsonSignedIntegers................................ 56
2.5.1 Comparison......................................... 56
2.5.2 Arithmetic.......................................... 57
2.6 Binary-CodedDecimal...................................... 66
2.6.1 Introduction......................................... 66
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