Physical and Philosophical Limi s in he Rep esen a ion o
I a ional Numbe s:
F om The modynamics o he Con inuum – A Pedagogical No e
Rica do Adonis Ca accioli Ab ego1∗Michael Joel Spilsbu y Fuen es2†
Ma co An onio Reyes Pagoada1‡
1Depa men o Elec ical Enginee ing, Uni e sidad Nacional Au ónoma de Hondu as (UNAH),
Campus Co és, Hondu as
2Depa men o Physics, Uni e sidad Nacional Au ónoma de Hondu as (UNAH), Hondu as
No embe 24, 2025
Abs ac
This a icle is a pedagogical and documen a y no e. We e iew, in an exposi o y way,
how he comple e physical ep esen a ion o i a ional numbe s, such as π, is cons ained
by he modynamics, in o ma ion heo y, and he s uc u e o space ime. **We a gue ha
he collec i e o ce o hese physical cons ain s p o ides conc e e suppo o philosophi-
cal pe spec i es ha ques ion ac ual in ini ies in physics.** Classical esul s such as Lan-
daue ’s p inciple, he Bekens ein bound, and he holog aphic p inciple (which yields an
uppe in o ma ion limi o ∼10122 bi s o he obse able uni e se), oge he wi h quan-
um limi s including B eme mann’s and he Ma golus–Le i in bound, imply ha no ini e
physical sys em—no e en he obse able uni e se—can ma e ialize in ini ely many digi s
o an i a ional numbe . We in oduce a quali a i e hie a chy o i a ionali y acco ding o
compu a ional and ene ge ic cos , and discuss al e na i e philosophical iewpoin s ( ini ism,
ul a ini ism, cons uc i ism) and disc e e models in physics. The aim o his no e is o
assemble well-known ideas in o a cohe en na a i e ha cla i ies wha i can mean, in
p ac ice, o “physically ep esen ” a numbe , and o con as he ope a ional success o he
ma hema ical con inuum wi h i s impossible ull ealiza ion in he physical uni e se.
1 In oduc ion
I a ional numbe s, such as πo √2, eme ge na u ally om geome y and analysis, ye hei
decimal expansions a e in ini e and non epea ing. In pu e ma hema ics his poses no di icul y:
he eal line is pos ula ed as a comple e, uncoun able con inuum, and i a ional numbe s a e
de ined ia limi s, Cauchy sequences, Dedekind cu s, o equi alen cons uc ions.
In physics and compu a ion, howe e , ep esen ing a numbe always in ol es ma e , ene gy,
and ime. Regis e s, memo ies, and measu ing de ices a e physical sys ems subjec o he mo-
dynamic and quan um cons ain s. This aises a simple bu p o ound ques ion: in wha sense
can an i a ional numbe be physically ep esen ed?
The pu pose o his a icle is delibe a ely modes and pedagogical. We do no p opose
new physical bounds o ma hema ical heo ems. Ins ead, we syn hesize known esul s om he
he modynamics o in o ma ion, quan um limi s on compu a ion, and cosmology o a gue ha :
∗Co esponding au ho . E-mail: [email p o ec ed]. ORCID: 0009-0006-3522-5818.
†E-mail: [email p o ec ed].
‡E-mail: [email p o ec ed].
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• he e exis s ic uppe bounds on he numbe o digi s o any numbe ha can be s o ed
o gene a ed in any ini e physical sys em;
• hese bounds a e especially s iking when applied o i a ional numbe s;
• he ma hema ical con inuum is he e o e bes unde s ood, om a physical poin o iew,
as an ex emely success ul idealiza ion a he han as a li e ally ealizable s uc u e.
We also ske ch a quali a i e hie a chy o i a ionali y in e ms o compu a ional and ene ge ic
cos , and b ie ly connec hese ideas wi h philosophical posi ions such as ini ism and wi h
disc e e app oaches o undamen al physics.
2 The modynamic and In o ma ion-Theo e ic Bounds
Landaue ’s p inciple s a es ha e asing one bi o in o ma ion in a sys em coupled o a he mal
ba h a empe a u e Tdissipa es a leas an ene gy
Ebi ≥kBTln 2,(1)
whe e kBis Bol zmann’s cons an [1]. Rep esen ing a decimal digi equi es
log2(10) ≈3.32
bi s. The e o e, a lowe bound on he ene gy cos pe decimal digi is
Edec ≥kBTln 10.(2)
I Ndecimal digi s a e s o ed in a memo y a empe a u e T, he o al ene ge ic cos obeys
E(N)≥N kBTln 10.(3)
Beyond Landaue ’s p inciple, he Bekens ein bound [2] and he holog aphic p inciple [3]
cons ain he maximum amoun o in o ma ion ha can be con ained wi hin a ini e egion
o space ime. Roughly speaking, hese bounds s a e ha he en opy (and hus in o ma ion
capaci y) o a egion scales wi h i s su ace a ea a he han i s olume. Fo he obse able
uni e se, he Bekens ein–Hawking en opy o he cosmological ho izon is o en es ima ed o be
o o de
Smax ∼10122kB,(4)
which co esponds o ∼10122 bi s, o abou 3×10121 decimal digi s o in o ma ion in any
encoding. This p o ides an independen , concep ually di e en uppe bound on he o al in o -
ma ion con en o he uni e se, complemen a y o he ene ge ic a gumen based on Landaue ’s
p inciple.
3 Quan um Limi s on Compu a ion
Quan um mechanics cons ains no only s o age, bu also he a e a which in o ma ion can be
p ocessed.
B eme mann’s limi [4] s a es ha a sys em o mass-ene gy Mcanno p ocess in o ma ion
a a a e exceeding
RB≤2Mc2
hbi s pe second, (5)
whe e cis he speed o ligh and his Planck’s cons an .
2
The Ma golus–Le i in bound [5] asse s ha he maximum numbe o dis inc elemen a y
ope a ions pe second ha can be pe o med by a sys em wi h a e age ene gy E(abo e i s
g ound s a e) is bounded by
RML ≤2E
πℏ,(6)
wi h ℏ=h/2π.
Taken oge he , hese quan um limi s show ha he gene a ion, manipula ion, and eading o
digi s o a numbe a e subjec o undamen al speed limi s. E en wi h ideal ha dwa e, a bi a ily
long compu a ions canno be pe o med wi hin ini e ime and ini e ene gy.
4 A C ude Ene ge ic Bound o he Obse able Uni e se
Le Euni deno e he o al ene gy o he obse able uni e se. Following s anda d cosmological
es ima es, a ough o de o magni ude is [6]
Euni ∼1070 J.(7)
Taking he empe a u e o he cosmic mic owa e backg ound as T≈2.73 K, Landaue ’s bound
implies ha he maximum numbe o decimal digi s ha could e e be physically ep esen ed
(s o ed o i e e sibly p ocessed) in he obse able uni e se is
Nmax ≈Euni
kBTln 10 ≈1.15 ×1090.(8)
This is an ex emely gene ous uppe bound: i assumes ha all he ene gy in he uni e se
is a ailable o compu a ion and ha all o i is used op imally o s o ing decimal digi s a
empe a u e 2.73 K. In eali y, g a i a ional, s uc u al, and p ac ical cons ain s, as well as
he Bekens ein and holog aphic bounds discussed abo e, would signi ican ly educe his maxi-
mum. **In pa icula , he Bekens ein-Hawking bound (which se s he limi a ∼10121 decimal
digi s) is concep ually mo e undamen al, as i depends only on he a ea o he ho izon and no
on empe a u e o he local e iciency o compu a ion. The ac ha wo en i ely independen
physical p inciples ( he modynamics and g a i y) con e ge o a ini e and eno mous, ye di -
e en , limi clea ly illus a es ha he o al in o ma ion capaci y o he uni e se is ini e, and
he e o e canno accommoda e he ull in ini e expansion o any i a ional numbe .**
5 Hie a chy o I a ionali y and Compu a ional Complexi y
No all i a ional numbe s a e equally di icul o gene a e o app oxima e. F om he s andpoin
o algo i hmic in o ma ion heo y and compu a ional complexi y [7, 8], one can quali a i ely
classi y di e en classes o numbe s acco ding o he esou ces needed o compu e hei digi s.
Le Ndeno e he numbe o digi s (in some ixed base) ha we wish o gene a e. A ough
quali a i e hie a chy is shown in Table 1.
The p ecise complexi y depends on he model o compu a ion and he chosen algo i hms,
bu he gene al idea is ha a ional and algeb aic numbe s ypically admi e icien schemes,
while anscenden al compu able numbe s usually equi e e o p opo ional o N o gene a e
Ndigi s. Non-compu able eals, such as Chai in’s Ω, do no admi any algo i hmic gene a ion
o hei digi s a all: no ini e p og am can ou pu hei ull expansion.
F om a physical pe spec i e, his hie a chy ansla es in o di e en ene ge ic and empo al
cos s o gene a ing app oxima ions o each class. The exis ence o non-compu able eals u he
unde sco es he gap be ween he ma hema ical con inuum and wha can be ealized o e en
app oxima ed algo i hmically.
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Table 1: Quali a i e compu a ional and ene ge ic complexi y o di e en classes o numbe s,
as a unc ion o he numbe o digi s N. The exponen s ka e ixed cons an s depending on he
algo i hm and ep esen a ion.
Type Example Typical complexi y
Ra ional 1/7O(log N)
Algeb aic i a ional √2O(log N)k
Compu able anscenden al π,e ON(log N)k
Non-compu able Chai in’s ΩIn ini e (no algo i hm)
5.1 Geome ic s. Algo i hmic Rep esen a ion
One migh hope o “chea ” he p oblem o digi s by ep esen ing an i a ional as a geome ic
magni ude. Fo ins ance, √2a ises as he leng h o he diagonal o a uni squa e. In a Euclidean
idealiza ion, his ep esen a ion appea s o bypass decimal expansions en i ely.
Howe e , any physical measu emen o leng h is subjec o unce ain y and e o . The
Heisenbe g unce ain y p inciple imposes quan um limi s on he p ecision wi h which posi ions
and momen a can be simul aneously known, while noise, ini e esolu ion o ins umen s, and he
possible exis ence o a minimal leng h scale (e.g., ela ed o he Planck leng h) u he cons ain
measu emen accu acy.
Thus, encoding √2 ia he diagonal o a physical squa e does no e ade physical limi a ions:
we s ill canno ead o i s alue wi h a bi a y p ecision. The geome ic encoding eplaces digi s
wi h an analog magni ude, bu ul ima e p ecision emains limi ed by he laws o physics.
6 Re e sible and Quan um Compu a ion
Landaue ’s bound applies o i e e sible ope a ions, such as bi e asu es. Re e sible compu-
a ion and quan um compu a ion ha e been p oposed, in pa , o educe ene gy dissipa ion in
in o ma ion p ocessing [7, 9]. In an ideal e e sible compu e , logical ope a ions a e in e ible
and, in p inciple, can app oach a bi a ily low ene gy dissipa ion pe s ep.
Ne e heless, se e al undamen al limi a ions emain:
•Ini ializa ion, e o co ec ion, and coupling o measu emen appa a us ypically in ol e
i e e sibili y and hus nonze o Landaue cos .
•Decohe ence and noise necessi a e o e heads ha g ow wi h sys em size and equi ed
ideli y.
•Quan um compu e s a e s ill subjec o B eme mann- ype and Ma golus–Le i in- ype
bounds on he o al a e o ope a ions.
Consequen ly, e en wi h e e sible o quan um compu a ion, in ini e sequences o digi s
canno be gene a ed o s o ed wi h ini e physical esou ces. These models can imp o e he
e iciency o compu a ion bu canno elimina e he unde lying physical cons ain s.
7 Wha Does I Mean o Physically Rep esen a Numbe ?
In scien i ic p ac ice, i a ional numbe s usually appea ei he :
•symbolically, as in o mulas C= 2π ,E=ℏω, o
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• ia ini e app oxima ions, such as π≈3.14159.
These ini e app oxima ions a e su icien o expe imen al p edic ions and enginee ing applica-
ions, because all measu emen s ha e ini e p ecision.
To speak o a physical ep esen a ion o a numbe , one migh equi e:
1. a ini e physical sys em whose s a e encodes (in some scheme) he numbe , and
2. an ope a ional p ocedu e o ex ac he encoded alue o a desi ed p ecision.
Unde his ope a ional iewpoin , no physical sys em can ep esen in ini ely many digi s o
any numbe . Ins ead, sys ems ep esen ini e unca ions o app oxima ions, whose achie able
p ecision is bounded by ene gy, ime, and noise.
This c ea es a concep ual gap be ween he ideal ma hema ical no ion o an i a ional, which
has an in ini e expansion, and any physically ealizable encoding, which can only app oxima e
ha expansion up o a ini e ( hough possibly e y high) p ecision.
8 The Ope a ional Success o he Con inuum
Despi e hese physical limi a ions, he ma hema ical con inuum has been ex ao dina ily suc-
cess ul in modeling na u e. Classical mechanics, gene al ela i i y, and quan um ield heo y a e
o mula ed using di e en iable mani olds, con inuous ields, and eal- alued unc ions.
F om an ope a ional poin o iew, howe e , all physically measu able quan i ies a e a ional
(o a mos compu able) numbe s unca ed o ini e p ecision. De ec o s and ins umen s
ou pu ini e s ings o digi s, no ac ual eal numbe s. Expe imen al es s o con inuous heo ies
always in ol e ini e samplings a ini e esolu ion.
This sugges s ha he con inuum should be ega ded, in physics, as a powe ul and ex emely
accu a e idealiza ion: a ic ion ha cap u es he beha io o la ge and complex sys ems in a
compac and ma hema ically elegan way, e en hough no indi idual eal numbe can be ully
ins an ia ed in he ma e ial uni e se.
9 Al e na i es o he Con inuum in Fundamen al Physics
Mo i a ed in pa by hese conside a ions, se e al app oaches o quan um g a i y and unda-
men al physics a emp o desc ibe space ime and ields in pu ely disc e e e ms. Examples
include:
•causal se heo y [11], whe e space ime is modeled as a locally ini e pa ially o de ed se ;
•loop quan um g a i y [10], whe e geome ic ope a o s such as a ea and olume ha e dis-
c e e spec a.
These amewo ks aim o g ound physical eali y on coun able s uc u es, po en ially elimi-
na ing he need o uncoun able in ini ies in he undamen al desc ip ion. Whe he such disc e e
models can ully ep oduce he success ul p edic ions o con inuum-based heo ies emains an
ac i e a ea o esea ch.
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10 Philosophical Pe spec i es
The ma hema ical s a us o he con inuum and o ac ual in ini y has long been deba ed in
he philosophy o ma hema ics. Fini ism, ul a ini ism, and cons uc i ism, in a ious o ms,
ejec he exis ence o comple ed in ini e o ali ies and emphasize ma hema ics g ounded in
cons uc i e o physically ealizable p ocedu es [13].
The physical a gumen s e iewed in his no e lend suppo o such pe spec i es, a leas
a he le el o ep esen a ion: e en i in ini e se s and eal numbe s a e indispensable ools in
ma hema ical heo y, hey canno be ully ma e ialized as physical objec s. In his sense, in ini e
ma hema ical en i ies may bes be seen as use ul ic ions a he han as on ologically obus
cons i uen s o he physical uni e se. **This physical limi a ion di ec ly suppo s an i- ealis
iews on he exis ence o he ma hema ical con inuum in he na u al wo ld, con as ing wi h
s ong Pla onis iews whe e ma hema ical objec s exis independen ly o human cons uc ion
o physical cons ain s.**
Pen ose and o he s ha e also explo ed he in e play be ween mind, compu a ion, and physical
law, ques ioning whe he human ma hema ical insigh can be ully cap u ed by o mal sys ems
and by physically ealizable compu a ions [12]. The impossibili y o physically ealizing in ini e
ma hema ical s uc u es is one ace o his b oade discussion.
11 Conclusion and Ou look
We ha e su eyed, in a compac and in en ionally pedagogical way, a collec ion o classical
esul s om he he modynamics o compu a ion, quan um limi s, and cosmology, and applied
hem o he ques ion o how i a ional numbe s can be ep esen ed in he physical uni e se. The
main poin s can be summa ized as ollows:
•Landaue ’s p inciple, he Bekens ein bound, and he holog aphic p inciple impose s ic
limi s on in o ma ion s o age in ini e sys ems.
•Quan um bounds such as hose o B eme mann and Ma golus–Le i in cons ain he a e
a which in o ma ion can be p ocessed.
•**When applied o he obse able uni e se as a whole, hese independen bounds (yielding
limi s o o de 1090 and 10121 decimal digi s, espec i ely) con i m a ini e uppe limi on
he o al amoun o dis inc nume ical in o ma ion ha can be physically ins an ia ed.**
•No physical sys em, he e o e, can ealize in ini ely many digi s o any i a ional numbe .
A bes , we can app oxima e such numbe s o ini e p ecision.
•The ma hema ical con inuum emains an ex ao dina ily success ul idealiza ion in physics,
bu i s ull s uc u e canno be embedded in a ini e, esou ce-limi ed uni e se.
This a icle is in ended as a didac ic and documen a y syn hesis, no as a sou ce o new
bounds o heo ems. Se e al di ec ions o u he wo k emain, including:
•mo e de ailed quan i a i e compa isons be ween di e en in o ma ion- heo e ic bounds in
conc e e cosmological scena ios;
•analyses o how hese limi s cons ain high-p ecision nume ical simula ions in cosmology
and pa icle physics;
•explo a ion o disc e e o ini e-in o ma ion o mula ions o physical heo ies, and hei
implica ions o he ole o he con inuum in science.
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Acknowledgmen s
The au ho s hank colleagues o discussions on quan um compu a ion, philosophy o ma he-
ma ics, and disc e e al e na i es o he con inuum. Any emaining e o s o o e simpli ica ions
a e en i ely ou own.
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