Math4200,HonorsPaper
RyanWallentine,SpringSemester2013
“Godmadenaturalnumbers;allelseistheworkofman.”
‐‐LeopoldKronecker
Theabovequotestatesthatthenaturalnumbersarewhatweweregiven,andallother
numbers,suchasintegersandrationalnumbers,werecreatedfromthem.Thisimpliesthatall
othersetsofnumberscanbeformedoutofthenaturalnumbers.Theobjectofthispaperistoshow
theconnectionbetweeneachsetofnumbersbyshowinghowwecanformeachsetfromthe
naturalnumbers.
Wewillbeginbydefiningallourterms.LetJrepresentthesetofallnaturalnumbers,W
representthesetofallwholenumbers,Zrepresentthesetofallintegers,Qrepresentthesetofall
rationalnumbers,andRrepresentthesetofallrealnumbers.WewillusePeano’sAxiomstodefine
thenaturalnumbersinordertogiveusastartingpoint.Peano’sAxiomsareasfollows:
Axiom1:1isanaturalnumber.Thatis,oursetisnotempty;itcontainsanobjectcalled
“one.”
Axiom2:Foreachx,thereexistsexactlyonenaturalnumber,calledthesuccessorofx,
whichwillbedenotedbyx’. thinkx’ x 1 Axiom3:Wealwayshavex’ 1.Thatis,thereisnonumberwhosesuccessoris1.
Axiom4:Ifx’ y’thenx y.Thatis,foranygivennumber,thereexistseithernonumberor
exactlyonenumberwhosesuccessoristhegivennumber.
Axiom5 AxiomofInduction :LettherebegivenasetJofnaturalnumbers,withthe
followingproperties:
I.
II.
J
1belongstoJ.
IfxbelongstoJ,thensodoesx’.
ThenJcontainsallthenaturalnumbers.Thus,if1’ 2,2’ 3,3’ 4,…,then
1,2,3,4,… .
Fromthisdefinitionofthenaturalnumbers,wecaneasilycreatethesetofallwhole
numbersbyincludingzero.WewillletWhavethesamepropertiesasJ,exceptforwewilllet0
belongtoWsuchthat0’ 1and0isnotthesuccessorofanynumber.Thus,W 0,1,2,3,… .We
willnowdefineadditionandmultiplicationasfollows:
Addition
:Forallx,x 1 x’.Forallxandy,x y’ x y ’.
Noticethat2 1’,sowehavethefollowing:4 2 4 1’ 4 1 ’ 4’ ’ 6.
Multiplication * :Forallx,x*1 x.Forallxandy,x*y’ x*y x.
Forexample,ify 1andx 4,thenx*y’ 4*1’ 4*1 4 4 4 8.
Wewillnowdefine0asfollows:0 x xandx*0 0forallx.Fromherewecaneasilycreate
theintegersbyletting–xdenotetheadditiveinverseofx.Thismeansthatx ‐x 0forallx.
Theintegers,willthenbedefinedasfollows:
Z x:x∈J ∪ 0 ∪ ‐x:x∈J ThusZ …,‐2,‐1,0,1,2,… . Toseemoreaboutbuildingtheintegers,see
http://en.wikipedia.org/wiki/Integer#Construction .
Wewillnowconstructtherationalnumbersfromtheintegersbyconstructingequivalence
classesusingtheintegers.Theseequivalenceclasseswillbepairsofintegersputtogetheras
follows: a,b :a,b∈Z,b 0 .Also, a,b isequivalentto c,d ifandonlyifad bc think a,b and
c,d as and respectively. Forexample,let 3,7 denotetheequivalencedefinedinthepair
3,7 .Then, 3,7 a,b ∈S: a,b 3,7 .Forexample, 6,14 3,7 .Wecanrepresent
3,7 bythepairs 3,7 or 6,14 ,ormoreformerlyas or .ThesetofallrationalnumbersQis
builtfromtheseequivalenceclasses. Furtherinformationaboutconstructingtherationals:
http://en.wikipedia.org/wiki/Rational_number#Formal_construction .Additionand
multiplicationfortherationalnumbersareasfollows:
Addition: a,b
c,d ad bc,bd Multiplication: a,b * c,d ac,bd Nowwearereadytoconstructtherealnumbers.LetT rn :rnisrationaland rn isa
Cauchysequence. Wewilllettwosequences rn and sn inTbeequivalenttoeachotherifand
onlyiflim →
0.Fromhere,itiseasytoshowthatthisrelationisreflexive,transitive,
andsymmetric.ThesetTispartitionedintodisjointequivalenceclasseswhichwewillcallreal
numbers.Forexample,letan 2foralln.Theequivalenceclass an istherealnumber2.The
realnumber2isalsorepresentedby 2 .Sincelim
lim
→
2
2
2
2
→
2
2andlim
→
2
2,then
0sincethelimitofthesumisthesumofthelimits.
Considertherealnumbere.Wecanrepresentthisnumberbythelimitofthebinomial
expansion:lim
→
1
1
.Considerthebinomialexpansionforn 1,000,000.Wethenhave
2.71828046…whichweseeisveryclosetothevalueofe.Anotherwayto
approximatethenumbereisthroughtheTaylorseriesexpansionfor
givesus∑
!
!
!
!
!
∑
!
forx 1.This
⋯Byaddingthetermsofthisseriesthroughn 15,weget
2.71828182…whichisalsoratherclose.Thesecontinuetogetcloserandclosertoeasn
approachesinfinity.
Nowconsidertherealnumber .Justlikee,therearevariouswaysthatwecan
approximate aswell.Forexample,wehavetheGregory‐Leibnizseriesasfollows:
4∑
4
⋯ .Thisapproximation,however,convergestooslowlyforit
tobeapracticalapproximation.AnotherapproximationwasgivenbyNewton:
!
2∑
2 1
!
1
1
1
⋯
.
Wewillnowlookataseriestorepresent√2.Letrnbethesequenceoflargestrational
numberswithdenominatorlessthanorequaltonsuchthat rn 2 2.Thisgivesusthefollowing
sequence: 1,1, , , ,… whichisaCauchysequencesuchthatlim
andsin x ∑
Thus,√2 !
!
showthat√2 ∑
,thenwegetthat√2 2sin
!
2∑
√2.Giventhatsin
!
√
∑
.
!
⋯.Inalikemanner,byusingcos x inplaceofsin x ,wecan
!
!
→
2
!
!
!
!
⋯.Thusweseethatwecanfind
variousrepresentationsforvariousrealnumbersincludingtheirrationalnumberslike ,e,and√2.
WewillnowtakealookatthenumberΦ phi whichrepresentsthegoldenratio.This
numberisinterestingbecausemanyartistsandarchitectshaveproportionedtheirworksto
approximatetheratiobecausetheproportionisaestheticallypleasing.Twoquantitiesaandbare
saidtobeinthegoldenratioΦif
Φ.ThisresultsinΦ
√
moreinformationcanbe
foundathttp://en.wikipedia.org/wiki/Golden_ratio .Themostinterestingthingaboutthis
numbercomesfromonemethodofitsconstruction.FromtheFibonaccisequencedefinedas
fn fn‐1 fn‐2wheref0 0andf1 1,wecancreateasequencethatconvergestoΦ.Consider
lim
.Oneclosedformulaforfn →
√
√
.Itiseasytoshowfromthisclosedformulathat
Φ.Thus,thesequenceoftheratiosofsuccessivepairsofnumberstakenfromthe
FibonaccisequencewillconvergetoΦ.
√