Large Numbers and
Addin Delia Nur Azf
LARGE NUMBERS AND INFINITY Addin Delia Nur Azf Muhajir Muhajir $"%&!"'"!&() Mathemati*s +du*ation , !"% Abstract
Infnity is dicult concept to understand, especially in mathematics. Not all students and teachers are able to understand it. In this paper, writer tries to explain infnity by starting it with large numbers; names o large numbers and special large numbers. Discussing infnity is not completed without mentioning its history; when the frst discovery o infnity is and how people used to understand it, also the popular eno!s paradox. "hen, we will continue with the cardinality o infnite sets to #now what and how big is infnity. $t the end o this paper, writer reviews a %ournal about a research which had been done in &yprus to #now '( elementary school teachers! perception o infnity. $nother %ournal which we will be going to review shows the level o ') students! understanding understanding o infnity in *inland.
-hat -hat is the largest largest number number ever ever kno.n kno.n in this this univer universe/ se/ -ell0 ell0 the ans.ers may vary0 based on the one .hom .e asked to1 ,hildren .ho are still in 2indergarten .ill likely to ans.er a million or a billion1 +lementary students students ans.er .ith a trillion0 trillion0 a million million billion0 billion0 or a billion trillion1 trillion1 3igh s*hoolers may ans.er even larger numbers su*h as0 a 4uadrillion0 a billion trillion0 or a thousand 4uadrillion1 And0 some o5 us .ill ans.er the 4uestion .ith infnity 1 -e all kno. that that infnity infnity is0 .ell0 infnite1 6ut0 .hi*h one o5 these are the largest/ Is infnity the largest number/ Is infnity larger than any number 7ossibilities/ 8he ans.ers to both o5 the 4uestion are9 no1 :irstly0 infnity is not a number1 Se*ondly0 there is no su*h largest number1 6e*ause there is a bigger number to every number kno.n0 like the 7o7ular 7hrase9 ;there is sky above a sky<1 It iss endless0 the number .ill kee7 going higher and higher0 larger and larger0 and nobody kno.s the end1
Yogyaka Yogyakarta rta State University University
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Large Numbers and
Addin Delia Nur Azf
A. Large Numbers -hat are the re4uirements 5or a number to be *alled large
number/ A**ording to -iki7edia0 large number starts at a million or 6
10
1 Some s7esif* large numbers have their o.n names0 5or 6
e=am7le
10
9
or a million0
12
10
or a billion0
10
so on1 8hese named large numbers based on multi7ly o5 & and
≥6
10
10
to 7o.er o5 a 15
10
1 It goes u7 to a 4uadrillion $
18
4uintillion $
or a trillion0 and
)0 a
21
)0 a se=tillion $
10
)0 et*1 8he numbers< names
ending .ith >illion are all derived by adding 7ref=es $bi?0 tri?0 4uad?0 303
10
et*10 derived 5rom Latin) to the stem >illion1 ,entillion $
) is the
highest name ending in ?@illion1 Billion used in5ormally to mean large but ins7esif* large number $-iki7edia0 !"#)1 A number one .ith a hundred zeros a5ter it has its o.n name1 100
8he number is *alled gogool $ 2anser
and
Ne.man
10
"C'!
)1 8he .ord gogool0 a**ording to book0
;Mathemati*s
and
8he
Imagination<0 .as invented by 2anser
basi*ally a number " .ith a gogool zeros a5ter it $
10
100
)1 It .as frst
suggested that a gogoolplex should be "0 5ollo.ed by .riting zeros until you got tired $note that this *on*e7t .as suggested by a C?years? old boy) $Dubis*h0 "C%)1 I am not going to .rite googolplex in
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Large Numbers and
Addin Delia Nur Azf
number 5orm be*ause there is not enough matter in this universe to .rite googolplex do.n1 Large numbers are large0 but still fnite1 -e *ould eventually get there $even the 7ro*ess is really long)1 No. try to imagine a number larger than googolplex 1 3o. about googolplex to the power o 100
googolplex
( 10 ) 10
100
10
10
/ -ell0 it is e=tremely large0 but it is not even a
hal5 .ay to infnity1 Another 4uestion 7o7s out0 ho. large is infnity/ Is it really big/ Eemember that even an enormous number is not even *lose to it1 B. Infnity Let us start this *ha7ter .ith more 4uestions1 -hat a*tually is
infnity/ Is infnity a number/ Fr is it just an e=7ression o5 something enormous/ -ho is the frst man to *ome u7 .ith an idea o5 infnity/ :irst o5 all0 infnity is not huge1 It is not tremendously large1 It is not even e=tremely humongous enormous $L161 Gates0 !"#)1 8hen ho. big is infnity/ Is it as big as universe/ Fr maybe infnity is even bigger than it/ Infnity $symbol9 H) is an abstra*t *on*e7t des*ribing something without any bound or larger than any number $-iki7edia0 !"#)1 Infnite0 a**ording to F=5ord di*tionary0 is limitless or endless in
s7a*e0 e=tent0 or size im7ossible to measure or *al*ulate1 As .ritten in a mathemati*al .eb 7age0 athsIs*un0 infnity is an idea o5 something that has no end0 or endless1 So that is it1 Infnity is endess1 As mentioned in the introdu*tion0 infnity is not a number0 not a
real number1 So0 it *annot be measured1 Any measurement .ill not be long enough to measure it1 +ven the entire universe *annot *om7ete infnity1 8here .as no eviden*e that infnite *olle*tions o5 7hysi*al obje*ts e=isted $6urton0 !"")1 "1 8he +arly 3istory o5 Infnity At frst0 there is no a*tual re*ord o5 earlier *ivilizations regarding0 *on*e7tualizing0 or dis*ussing infnity0 but .e .ill take a look the beginning o5 infnity 5rom the an*ient Greeks1 8here is a .ord0 apeiron0 .hi*h is meant unbounded0 infnite0 indefnite0 or
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Large Numbers and
Addin Delia Nur Azf
undefned $Allen0 "CCC)1 8o the Greeks0 the *on*e7t o5 infnity .as 5or*ed u7on them 5rom the 7hysi*al .orld by three traditional observations9 ? 8ime seems .ithout end1 ? S7a*e and time *an be unendingly subdivided1 ? S7a*e is .ithout bound1 8his *on*e7t is 4uiet a**e7table to human
infnity
.ill
not
be
*om7leted
.ithout
mentioning the 7o7ular 7arado= by Beno o5 +lea $*1 '%! 6,)0 the tortoise and A*hilles $A*hilles .as the great Greek hero o5 3omer
'
Large Numbers and
Addin Delia Nur Azf
some f=ed time to *ross hal5 the distan*e to the other side o5 the room0 say se*onds1 3o. long .ill it take to *ross hal5 the remaining distan*e/ 3al5 as longonly " se*ond1 ,overing hal5 o5 the remaining distan*e $an eighth o5 the total) .ill take only hal5 a se*ond1 And so on1 And on*e I have *overed all the infnitely many sub?distan*es and added u7 all the time it took to traverse them/ Fnly ' se*onds0 and here I am0 on the other side o5 the room a5ter all1 And 7oor old A*hilles .ould have .on his ra*e1 &1 ,ardinality o5 Infnite Sets A +uro7ean mathemati*ian0 Georg ,antor $"'%?"C")0 defne infnite set $using the notion o5 one?to?one *orres7onden*e) as9 a set that *an be 7ut into a one?to?one *orres7onden*e .ith a 7ro7er subset o5 itsel51 :or e=am7le9 "
&
'
%
#
" ' C "# % &# -e *an make a one?to?one *orres7onden*e o5 set o5 natural numbers and set o5 s4uare numbers1 It is leading to a *on*lusion that both set has the same *ardinality even some o5 us may say that set o5 s4uare numbers is bigger than natural numbers1 -ell0 it is *lear that natural numbers is infnite be*ause .e *an make a one?to?one *orres7onden*e .ith its subset set o5 s4uare numbers1 8his does not .ork only 5or natural numbers1 No. take a look at this e=am7le9 "
&
'
%
#
1
1
1
1
1
1
1
2
3
4
5
6
Interesting0 isn
Yogyakarta State University
%
Large Numbers and
Addin Delia Nur Azf
-e kno. the *ardinality o5 some fnite sets like9 set o5 5a*tors o5 "0 set o5 natural numbers belo. (0 set o5 dates in a month0 et*1 3o. about the *ardinality o5 infnite sets/ 8he *ardinality o5 natural numbers is ale7h?null1 Ale7h numbers are a se4uen*e numbers used to re7resent the *ardinality o5 infnite sets1 Ale7h $ ) is the frst letter in 3ebre. al7habet $-iki7edia0 !"#)1
Set o5 natural numbers is the frst
infnite sets0 also set o5 even numbers0 set o5 s4uare numbers0 set o5 rational numbers0 et*1 OGo .at*h 3o. to ,ount Jast Infnity by Mi*hael Stevens $htt7s9PP ...1youtube1*omP.at*h/vQSrUCYDoR+) 5or a sim7le0 amazing0 and *lear e=7lanation about infnity and infnite sets !. Tea"#ers$ and Students$ Understanding ab%ut Infnity -e all kno. the *on*e7t o5 infnity is *om7li*ated and hard to
understand by the tea*hers and students themselves1 In their journal0 2attou Maria and her 5riends sho. i5 they did a resear*h to kno. tea*hers< 7er*e7tion about infnity as a 7ro*ess or an obje*t1 8he resear*h had been done to '& elementary s*hool tea*hers in ,y7rus1 8he data .ere *olle*ted through a sel5?re7ort 4uestionnaire .hi*h took ! minutes to *om7lete1 8he 4uestionnaire .as *om7rised o5 5our tasks that aimed to identi5y 7er*e7tions related to the *on*e7t o5 infnity1 8he majority o5 tea*hers *om7rehend infnity as an unlimited 7ro*ess1 8hey mainly *on*eive infnity as a mathemati*al idea .ith limited a77li*ations to daily li5e1 8he 5a*t that tea*hers 4uoted e=am7les 5rom various felds o5 mathemati*s su*h as geometry0 trigonometry0 sets0 et* indi*ates that the *on*e7t o5 infnity is 7resented
throughout
the
mathemati*s
*urri*ulum
in
,y7rus1
Although0 some em7iri*al e=am7les .ere 7rovided0 these in*luded large fnite numbers1 Due to human
o5 sand or *om7uting the number
10
0 the 7erson *orrelates them
.ith the *on*e7t o5 infnity $2attou0 !!C)1 Markku S1 3annula and her 5riends also did a resear*h to kno. the level o5 students< understanding on infnity1 8he obje*ts o5 this
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Large Numbers and
Addin Delia Nur Azf
resear*h .ere '! students in :inland in grade % and during the a*ademi* years !!"?!!&1 8hey gave out "C mathemati*al tasks0 & o5 them .ere about infnity1 8hey also did an intervie. to the students1 In the f5th grade most students have no *lue o5 infnity0 and the situation is not mu*h better in the seventh grade1 Yet0 there is an obvious develo7ment 5rom grade % to ( in student
REFEREN!ES
Allen0
G1
Donald1
No
year1
8he
-istory
o
Infnity0
$htt79PP...1math1tamu1eduPdallenP historyPinfnity17d50 a**essed on "C A7ril !"#)1 6agni0 Giorgio 81 !!%1 Infnite eries/ rom -istory to athematics 0ducation in ,MI8 International Vournal 5or Mathemati*s 8ea*hing and Yogyakarta State University
(
Large Numbers and
Learning0
Addin Delia Nur Azf
$htt79PP...1*imt1
7lymouth1a*1ukPjournalPbagni17d50
a**essed on "C A7ril !"#)1 6urton0 David M1 !""1 "he -istory o athematics/ $n Introduction1 Ne. York9 M*Gra.?3ill ,om7any1 Dubis*h0 Eoy1 "C%1 "he Nature o Number 1 Ne. York9 8he Eonald Jress ,om7any1 Dunham0 -illiam1 "CC"1 1ourney through +enius1 London9 Jenguin 6ooks1 Gates0 Les 6ill1 !"#1 2hat is Infnity3, an arti*le in Math Is :un an enjoyable
and
easy?to?learn
mathemati*s
.eb
site0
$htt7s9PP...1mathsis5un1*omPnumbersPinfnity1 html0 a**essed on & Vune !"#)1 Gohar0 Dina1 !!(1 &antor!s 4uest to 5nderstand the Infnite0 an arti*le in 8he -orld o5 8rotter Math a mathemati*s .eb site 5or tea*hers< sour*es0 $htt79PP...1trotter math1netP7ersonalPinfnity1html0 a**essed on & Vune !"#)1 Markku S1 3annula0 +rkki Jehkonen0 3anna Maijala0 Eiitta Soro1 !!#1 6evels o tudents! 5nderstanding on Infnity in 8ea*hing Mathemati*s
and
,om7uter
S*ien*e
Vournal
University
o5
Debre*en0
$htt79PPtm*s1math1klte1huP,ontentsP!!#?Wol?IW?Issue?IIP "!1%'%X8M,S1!!#1!"C17d50 a**essed on A7ril !"#)1 Maria 2attou0 Mi*hael 8hanasia0 21 2aterina0 ,1 ,onstantinos0 Jhili77ou George1 !!C1 "eachers! 7erception about Infnity/ a 7rocess or an 8b%ect3 in Vournal o5 De7artment o5 +du*ation University o5 ,y7rus0
$htt79PPi5e1enslyon15rP7ubli*ationsP
edition?ele*troni4ueP*erme#P.g"!?
!C?kattou17d50 a**essed on A7ril !"#)1 Jlatoni* Eealms $Intera*tive Mathemati*s +n*y*lo7edia)1 No year1 eno!s 7aradox
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the
"ortoise
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$chilles0
$htt79PP7latoni*realms1*omPen*y*lo7ediaPzenos?7arado=?o5?the? tortoise?and?a*hilles0 a**essed on " Vune !"#)1
Yogyakarta State University
Large Numbers and
Addin Delia Nur Azf
-iki7edia $the 5ree en*y*lo7edia)1 Last modifed in !"#1 $leph number 0 $htt7s9PPen1 .iki7edia1orgP.ikiPAle7hXnumber0 a**essed on "C A7ril !"#)1 -iki7edia $the 5ree en*y*lo7edia)1 Last modifed in !"#1 Infnity 0 $htt7s9PPen1.iki7edia1 orgP.ikiPInfnity0 a**essed on "C A7ril !"#)1 -iki7edia $the 5ree en*y*lo7edia)1 Last modifed in !"#1 Names o large numbers0
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$htt7s9PPen1
.iki7edia1orgP.ikiPBeno(sX7arado=es0
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