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The '''Euclid–Euler theorem''' is a [[theorem]] in [[number theory]] that relates [[perfect number]]s to [[Mersenne prime]]s. It states that an even number is perfect [[if and only if]] it has the form {{math|2<sup>''p''−1</sup>(2<sup>''p''</sup> − 1)}}, where {{math|2<sup>''p''</sup> − 1}} is a [[prime number]]. The theorem is named after mathematicians [[Euclid]] and [[Leonhard Euler]], who respectively proved the "if" and "only if" aspects of the theorem.
{{Infobox Symbols|mark=∞|name=Simbol takhingga|unicode={{unichar|221E|Infinity|html=}}|different from={{unichar|267E|Permanent Paper Sign |nlink= Kertas bebas asam}}<br />{{unichar|26AD|Marriage symbol}}}}'''Simbol takhingga''' atau '''simbol ananta''' ('''{{Math|∞}}''') merupakan [[Daftar simbol matematika|simbol matematika]] yang mewakili konsep [[takhingga]]. Simbol ini disebut juga sebagai lemniskat,{{r|mind}} dinamai dari bentuk yang serupa dalam [[geometri aljabar]], yaitu kurva [[Kurva lemniskat|lemniskat]].{{r|beautiful}} Dalam [[pencitraan merek ternak]], istilah simbol ini juga disebut sebagai "angka delapan malas".{{r|zero}}
It has been conjectured that there are infinitely many Mersenne primes. Although the truth of this conjecture remains unknown, it is equivalent, by the Euclid–Euler theorem, to the conjecture that there are infinitely many even perfect numbers. However, it is also unknown whether there exists even a single odd perfect number.<ref name="stillwell" />
Simbol takhingga pertama kali dipakai dalam matematika oleh [[John Wallis]] pada abad ke-17, walaupun simbol ini memiliki sejarah yang panjang dalam pemakaian lainnya. Dalam [[matematika]], simbol takhingga seringkali diartikan sebagai proses takhingga ([[takhingga potensial]]) daripada nilai takhingga ([[takhingga aktual]]). Simbol takhingga memiliki arti teknis lain yang berkaitan dengannya, seperti pemakaian kertas yang tahan lama dalam [[Penjilidan|penjilidan buku]]<u>,</u> dan dipakai sebagai nilai simbolis takhingga dalam kesusasteraan dan mistisisme modern. Dalam [[desain grafis]], simbol takhingga umumnya dipakai sebagai elemen logo; contohnya dalam badan logo dan desain lama seperti [[bendera Métis]].
== Statement and examples ==
Simbol takhingga dan beberapa variasi lainnya tersedia di berbagai [[pengodean karakter]].
A perfect number is a [[natural number]] that equals the sum of its proper [[divisor]]s, the numbers that are less than it and divide it evenly (with [[remainder]] zero). For instance, the proper divisors of 6 are 1, 2, and 3, which sum to 6, so 6 is perfect.
A Mersenne prime is a prime number of the form {{math|1=''M''<sub>''p''</sub> = 2<sup>''p''</sup> − 1}}, one less than a [[power of two]]. For a number of this form to be prime, {{mvar|p}} itself must also be prime, but not all primes give rise to Mersenne primes in this way. For instance, {{nowrap|1=2<sup>3</sup> − 1 = 7}} is a Mersenne prime, but {{nowrap|1=2<sup>11</sup> − 1 = 2047 = 23 × 89}} is not.
== Sejarah ==
[[Berkas:First_known_usage_of_the_infinity_symbol.jpg|jmpl|Simbol takhingga dipakai pertama kali oleh [[John Wallis]] pada tahun 1655]]
{{multiple image
| total_width = 480
| image1 = John Wallis by Sir Godfrey Kneller, Bt.jpg
| alt1 = Gambar John Wallis, dilukis oleh Sir Godfrey Kneller pada tahun 1701, dari National Portrait Gallery
| caption1 = [[John Wallis]] memperkenalkan simbol takhingga {{char|<math>\infty</math>}} dalam sastra matematika.
| image2 = Infinity symbol.svg
| alt2 = Ada delapan variasi simbol takhingga
| caption2 = Simbol <math>\infty</math> dalam berbagai [[rupa huruf]].
}}
The Euclid–Euler theorem states that an even natural number is perfect if and only if it has the form {{math|2<sup>''p''−1</sup>''M''<sub>''p''</sub>}}, where {{math|''M''<sub>''p''</sub>}} is a Mersenne prime.<ref name="stillwell">{{citation|title=Mathematics and Its History|series=[[Undergraduate Texts in Mathematics]]|first=John|last=Stillwell|authorlink=John Stillwell|publisher=Springer|year=2010|isbn=978-1-4419-6052-8|page=40|url=https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA40}}.</ref> The perfect number 6 comes from {{math|1=''p'' = 2}} in this way, as {{nowrap|1=2<sup>2−1</sup>{{mvar|M}}<sub>2</sub> = 2 × 3 = 6}}, and the Mersenne prime 7 corresponds in the same way to the perfect number 28.
Sejak zaman dahulu, simbol lemniskat ini merupakan sebuah ragam hias yang umum. Contoh mengenai simbol ini umumnya dapat dilihat pada sisir [[zaman Viking]].{{r|viking}}
== History ==
[[John Wallis]], seorang matematikawan asal Inggris, diakui bahwa ia telah memperkenalkan simbol takhingga beserta pengertiannya dalam matematika pada tahun 1655, dalam karyanya ''De sectionibus conicis'',{{r|sectionibus|wallis|notations}} namun ia tidak menjelaskan simbol yang ia pilih. Simbol yang dipilih diduga berupa bentuk bilangan Romawi yang berbeda, namun bilangan yang dipilih masih belum jelas. Ada teori yang mengemukakan bahwa simbol tersebut berasal dari bilangan Romawi 100 juta, yang menyerupai simbol yang sama sangat berkaitan dengan bingkai foto.{{r|beyond}} Adapula teori yang mengemukakan bahwa simbol tersebut berasal dari notasi CIↃ yang dipakai untuk mewakili 1000.{{r|unthinkable}} Selain bilangan Romawi, simbol tersebut juga berasal dari huruf {{char|ω}} yang berbeda (dimana {{char|ω}} adalah huruf kecil dari [[omega]] dan huruf terakhir dalam [[alfabet Yunani]]).{{r|unthinkable}}
Euclid proved that {{math|2<sup>''p''−1</sup>(2<sup>''p''</sup> − 1)}} is an even perfect number whenever {{math|2<sup>''p''</sup> − 1}} is prime. This is the final result on [[number theory]] in [[Euclid's Elements|Euclid's ''Elements'']]; the later books in the ''Elements'' instead concern [[irrational number]]s, [[solid geometry]], and the [[golden ratio]]. Euclid expresses the result by stating that if a finite [[geometric series]] beginning at 1 with ratio 2 has a prime sum {{mvar|q}}, then this sum multiplied by the last term {{mvar|t}} in the series is perfect. Expressed in these terms, the sum {{mvar|q}} of the finite series is the Mersenne prime {{math|2<sup>''p''</sup> − 1}} and the last term {{mvar|t}} in the series is the power of two {{math|2<sup>''p''−1</sup>}}. Euclid proves that {{math|''qt''}} is perfect by observing that the geometric series with ratio 2 starting at {{mvar|q}}, with the same number of terms, is proportional to the original series; therefore, since the original series sums to {{math|1=''q'' = 2''t'' − 1}}, the second series sums to {{math|1=''q''(2''t'' − 1) = 2''qt'' − ''q''}}, and both series together add to {{math|2''qt''}}, two times the supposed perfect number. However, these two series are disjoint from each other and (by the primality of {{mvar|q}}) exhaust all the divisors of {{math|''qt''}}, so {{math|''qt''}} has divisors that sum to {{math|2''qt''}}, showing that it is perfect.<ref>{{citation|author=[[Euclid]]|title=The Thirteen Books of The Elements, Translated with introduction and commentary by Sir Thomas L. Heath, Vol. 2 (Books III–IX)|edition=2nd|publisher=Dover|year=1956|pages=421–426}}. See in particular Prop. IX.36.</ref>
Over a millennium after Euclid, [[Alhazen]] {{circa|1000 CE}} conjectured that {{em|every}} even perfect number is of the form {{math|2<sup>''p''−1</sup>(2<sup>''p''</sup> − 1)}} where {{math|2<sup>''p''</sup> − 1}} is prime, but he was not able to prove this result.<ref>{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham}}</ref> It was not until the 18th century, over 2000 years after Euclid,<ref>{{citation
Karena keterbatasan tipografi dalam beberapa kasus, simbol-simbol lain yang menyerupai simbol takhingga dipakai dalam arti yang sama.{{r|notations}} [[Leonhard Euler]] menggunakan simbol huruf S terbalik dan menyamping yang menyerupai simbol lemniskat.{{r|notations-incorrect}} Bahkan, simbol "O–O" dipakai menggantikan simbol takhingga itu sendiri.{{r|notations}}
| last1 = Pollack | first1 = Paul
| last2 = Shevelev | first2 = Vladimir
| doi = 10.1016/j.jnt.2012.06.008
| issue = 12
| journal = Journal of Number Theory
| mr = 2965207
| pages = 3037–3046
| title = On perfect and near-perfect numbers
| volume = 132
| year = 2012| arxiv = 1011.6160
| s2cid = 13607242
}}</ref> that [[Leonhard Euler]] proved that the formula {{math|2<sup>''p''−1</sup>(2<sup>''p''</sup> − 1)}} will yield all the even perfect numbers.<ref name="stillwell" /><ref>{{citation|first=Leonhard|last=Euler|authorlink=Leonhard Euler|chapter=De numeris amicibilibus|trans-chapter=On amicable numbers|language=Latin|contribution-url=https://scholarlycommons.pacific.edu/euler-works/798/|title=Commentationes arithmeticae|volume=2|year=1849|pages=627–636}}. Originally read to the Berlin Academy on February 23, 1747, and published posthumously. See in particular section 8, p. 88.</ref> Thus, there is a one-to-one relationship between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. After Euler's proof of the Euclid–Euler theorem, other mathematicians have published different proofs, including [[Victor-Amédée Lebesgue]], [[Robert Daniel Carmichael]], [[Leonard Eugene Dickson]], John Knopfmacher, and Wayne L. McDaniel. Dickson's proof, in particular, has been commonly used in textbooks.<ref>{{citation|last=Cohen|first=Graeme L.|date=March 1981|doi=10.2307/3617930|issue=431|journal=[[The Mathematical Gazette]]|jstor=3617930|pages=28–30|title=Even perfect numbers|volume=65}}</ref>
This theorem was included in a web listing of the "top 100 mathematical theorems", dating from 1999, which later became used by Freek Wiedijk as a [[Benchmark (computing)|benchmark]] set to test the power of different [[proof assistant]]s. {{as of|2021}}, the proof of the Euclid–Euler theorem had been formalized in 5 of the 10 proof assistants recorded by Wiedijk.<ref>{{citation|first=Freek|last=Wiedijk|url=https://www.cs.ru.nl/~freek/100/|title=Formalizing 100 Theorems|publisher=Radboud University Institute for Computing and Information Sciences|access-date=2021-07-10}}</ref>
== Pemakaian ==
=== MatematikaProof ===
Euler's proof is short<ref name="stillwell" /> and depends on the fact that the [[Divisor function|sum of divisors]] function {{mvar|σ}} is [[multiplicative function|multiplicative]]; that is, if {{mvar|a}} and {{mvar|b}} are any two [[relatively prime]] integers, then {{math|1=''σ''(''ab'') = ''σ''(''a'')''σ''(''b'')}}. For this formula to be valid, the sum of divisors of a number must include the number itself, not just the proper divisors. A number is perfect if and only if its sum of divisors is twice its value.
Dalam matematika, simbol takhingga seringkali dipakai untuk merepresentasikan [[takhingga potensial]],{{r|cosmic}} daripada kuantitas takhingga sebenarnya. Simbol takhingga dipakai diantaranya dalam [[bilangan real diperluas]], [[bilangan kardinal]], dan [[bilangan ordinal]] (notasi lain, seperti <math>\aleph_0</math> dan <math>\omega</math>, dipakai untuk bilangan takhingga). Misalnya, simbol takhingga pada bentuk [[penjumlahan]] dan [[limit]] dalam matematika seperti
=== Sufficiency ===
: <math> \sum_{n=0}^{\infty} \frac{1}{2^n} = \lim_{x\to\infty}\frac{2^x-1}{2^{x-1}} = 2</math>,
One direction of the theorem (the part already proved by Euclid) immediately follows from the multiplicative property: every Mersenne prime gives rise to an even perfect number. When {{math|1=2<sup>''p''</sup> − 1}} is prime,
<math display-block>\sigma(2^{p-1}(2^p - 1)) = \sigma(2^{p-1})\sigma(2^p - 1).</math>
The divisors of {{math|2<sup>''p''−1</sup>}} are {{math|1, 2, 4, 8, ..., 2<sup>''p''−1</sup>}}. The sum of these divisors is a [[geometric series]] whose sum is {{math|1=2<sup>''p''</sup> − 1}}. Next, since {{math|1=2<sup>''p''</sup> − 1}} is prime, its only divisors are {{math|1}} and itself, so the sum of its divisors is {{math|1=2<sup>''p''</sup>}}.
Combining these,
biasanya diartikan bahwa variabel tersebut naik membesar mendekati takhingga<u>,</u> walaupun bentuk di atas dapat diartikan sebagai nilai takhingga sebenarnya.{{r|convergence}}
<math display=block>\begin{align}
\sigma(2^{p-1}(2^p - 1)) &= \sigma(2^{p-1})\sigma(2^p - 1) \\
&= (2^p - 1)(2^p) \\
&= 2(2^{p-1})(2^p - 1).
\end{align}</math>
Therefore, {{math|2<sup>''p''−1</sup>(2<sup>''p''</sup> − 1)}} is perfect.<ref name="imsp">{{citation|title=Introduction to Mathematical Structures and Proofs|series=Undergraduate Texts in Mathematics|first=Larry|last=Gerstein|publisher=Springer|year=2012|isbn=978-1-4614-4265-3|at=Theorem 6.94, p. 339|url=https://books.google.com/books?id=qK9y768b1NQC&pg=PA339}}.</ref><ref name="pp">{{citation|title=A proof that all even perfect numbers are a power of two times a Mersenne prime|website=Prime Pages|url=https://primes.utm.edu/notes/proofs/EvenPerfect.html|access-date=2014-12-02|first=Chris K.|last=Caldwell}}.</ref><ref name="ntfagd">{{citation|title=Number Theory, Fourier Analysis and Geometric Discrepancy|volume=81|series=London Mathematical Society Student Texts|first=Giancarlo|last=Travaglini|publisher=Cambridge University Press|year=2014|isbn=978-1-107-04403-6|pages=26–27|url=https://books.google.com/books?id=mIaYAwAAQBAJ&pg=PA26}}.</ref>
=== Necessity ===
Simbol takhingga juga dapat diartikan dalam [[titik di takhingga]], termasuk ketika hanya ada satu titik yang diketahui. Simbol ini khususnya digunakan seperti titik takhingga dari [[garis proyektif]],{{r|algebraic}} dan titik yang ditambahkan ke [[ruang topologis]] untuk membentuk [[Perluasan Alexandroff|kompaktifikasi satu titik]].{{r|hitchhiker}}
In the other direction, suppose that an even perfect number has been given, and partially factor it as {{math|2<sup>''k''</sup>''x''}}, where {{mvar|x}} is odd. For {{math|2<sup>''k''</sup>''x''}} to be perfect, the sum of its divisors must be twice its value:
{{NumBlk|:|<math>2^{k+1}x = \sigma(2^k x) = (2^{k+1} - 1)\sigma(x).</math>|∗}}
The odd factor {{math|2<sup>''k''+1</sup> − 1}} on the right side of '''(∗)''' is at least 3, and it must divide {{mvar|''x''}}, the only odd factor on the left side, so {{math|1=''y'' = ''x''/(2<sup>''k''+1</sup> − 1)}} is a proper divisor of {{mvar|x}}. Dividing both sides of '''(∗)''' by the common factor {{math|1=2<sup>''k''+1</sup> − 1}} and taking into account the known divisors {{mvar|x}} and {{mvar|y}} of {{mvar|x}} gives
{{Block indent|left=1.6|<math>2^{k+1}y = \sigma(x) = x + y + {}</math>other divisors<math>{} = 2^{k+1}y + {}</math>other divisors.}}
For this equality to be true, there can be no other divisors. Therefore, {{mvar|y}} must be {{math|1}}, and {{mvar|x}} must be a prime of the form {{math|1=2<sup>''k''+1</sup> − 1}}.<ref name="imsp" /><ref name="pp" /><ref name="ntfagd" />
== References ==
=== Penggunaan teknis lainnya ===
[[Berkas:Infrarotindex_md_300_mm_IMGP1196.jpg|al=Minolta MD 4.5/300mm IF lens|jmpl|Gambar kamera lensa yang memperlihatkan ada simbol takhingga pada indikator jarak fokus lensa.|160x160px]]
Dalam cabang selain matematika, simbol takhingga dapat dipakai dalam arti yang berkaitan dengan yang lainnya. Misalnya, simbol takhingga dipakai dalam [[Penjilidan|penjilidan buku]] yang menunjukkan bahwa buku yang dicetak pada [[kertas bebas asam]] sehingga menjadi lebih awet.{{r|book}} Dalam [[kamera]] and [[lensa]]<nowiki/>nya, simbol takhingga menunjukkan bahwa [[jarak fokus]] lensa [[Pumpun ananta|diatur sebagai jarak ananta]], dan simbol takhingga "mungkin salah satu simbol takhingga paling terlama yang dipakai dalam kamera".{{r|camera}}
{{reflist}}
=== Penggunaan simbolisme dan kesusasteraan ===
[[Berkas:RWS_Tarot_08_Strength.jpg|alt=Kartu tarot kekuatan yang menggambarkan seorang wanita yang bermahkotakan simbol takhingga, sedang menutup mulut seekor singa|jmpl|Simbol takhingga muncul pada beberapa [[kartu tarot Rider–Waite]].{{r|tarot}}|289x289px]]
Dalam mistisisme modern, simbol takhingga telah diidentifikasikan sebagai variasi [[ouroboros]], gambaran kuno tentang seekor ular yang memakan ekornya sendiri juga merupakan simbol takhingga, dan ouroboros terkadang digambar dalam bentuk gambar angka delapan yang mengambarkan identifikasi simbol tersebut—daripada dalam bentuk lingkaran yang lebih sederhana.{{r|dreams}}
Dalam karya [[Vladimir Nabokov]], seperti ''[[The Gift (Novel Nabokov)|The Gift]]'' dan ''[[Pale Fire]]'', gambar berbentuk angka delapan dipakai sebagai simbol yang merujuk [[pita Möbius]] dan takhingga, seperti kasus dalam deskripsi buku tentang bentuk-bentuk jejak ban sepeda dan garis besar mengenai orang yang mengingat sebagian. Puisi Nabokov yang karya novelnya berjudulkan ''Pale Fire'' mengacu pada "keajaiban lemniskat" ({{Lang-en|miracle of the lemniscate}}).{{r|nabokov}} Ada beberapa penulis yang mengerjakan kegunaan simbol takhingga beserta pengertiannya seperti [[James Joyce]], di ''[[Ulysses (novel)|Ulysses]]'',{{r|sands}} dan [[David Foster Wallace]], di ''[[Infinite Jest]]''.{{r|wallace}}
=== Desain grafis ===
Bentuk yang terkenal dan arti simbol takhingga ini menjadikan elemen [[tipografi]] yang umum dalam [[desain grafis]]. Misalnya, [[bendera Métis]], yang dipakai oleh orang-orang [[Orang Métis Kanada|Métis]] Kanada sejak awal abad ke-19. {{r|flags}} Ada teori yang berbeda yang mengemukakan arti simbol dari bendera tersebut, di antaranya sebagai harapan untuk masa depan takhingga dalam budaya Métis, dan sebagai campuran tradisi Eropa dan [[First Nations in Canada|First Nations]],{{r|gaudry|dumont}} namun ada juga yang menimbulkan bentuk geometris dari tarian Métic,{{r|racette}}, [[buhul Celtic]],{{r|prefontaine}} ataupun [[Plains Indian Sign Language|Plains First Nations Sign Language]].{{r|barkwell}}
Simbol takhingga berwarnakan [[pelangi]] dalam [[gerakan hak autisme]] juga dipakai untuk melambangkan ada tak berhingga berbagai orang-orang dalam pergerakan dan kognitif manusia.{{r|autism}} Simbol ini dipakai dalam badan logo perusahaan [[Bakelite]] yang mengacu kepada berbagai pemakaian yang sangat luas pada material sintesis yang diproduksi.{{r|bakelite}} Ada versi dari simbol ini yang dipakai dalam merek dagang, badan logo, dan lambang lainnya seperti [[Fujitsu]],{{r|brands}} [[Cell Press]],{{r|inspires}} dan [[Piala Dunia FIFA 2022]].{{r|qatar}}
== Pengodean ==
Simbol takhingga dikodekan sebagai {{unichar|221E|infinity}} dalam Unicode,{{r|compart}} dan <code>\infty</code>: <math>\infty</math> dalam [[LaTeX]].{{r|comprehensive}} Simbol takhingga yang dilingkari dipakai sebagai lambang [[kertas bebas asam]].
{{charmap|221E|map6=[[EUC-KR]]{{r|unicode-euc-kr}} / [[Unified Hangul Code|UHC]]{{r|unicode-uhc}}|namedref2=[[Common Locale Data Repository|CLDR]] text-to-speech name{{r|cldr}}|ref1char2=\acidfree|ref1char1=\infty|namedref1=[[LaTeX]]{{r|comprehensive}}|IncludeGB=1|map8char1=A1 DB|map8=[[Big5]]{{r|encoding}}|map7char1=A2 AC|map7=[[KPS 9566|EUC-KPS-9566]]{{r|unicode-kps}}|map6char1=A1 C4|map5char1=A1 E7|267E|map5=[[EUC-JP]]{{r|unicode-euc-jp}}|map4char1=81 87|map4=[[Shift JIS]]{{r|unicode-jis}}|map3char1=A5|map3=[[Simbol (rupa huruf)#Pengodean|Pengodean fon simbol]]{{r|unicode-mac-symbol}}|map2char1=B0|map2=[[Mac OS Roman]]{{r|unicode-mac-roman}}|map1char1=EC|map1=[[Code page 437|OEM-437 (Alt Code)]]{{r|unicode-cp437}}|name2=permanent paper sign|name1=infinity|ref2char2=infinity}}
Ada kumpulan simbol Unicode yang juga diantaranya berupa bentuk simbol takhingga yang berbeda, namun simbol tersebut jarang tersedia dalam <u>[[Miscellaneous Mathematical Symbols-B]]</u>.{{r|unicode-misc}}
{{charmap|29DC|29DD|29DE|name1=incomplete infinity|name2=tie over infinity|name3=infinity negated with vertical bar|namedref1=[[LaTeX]]{{r|comprehensive}}|ref1char1=\iinfin|ref1char2=\tieinfty|ref1char3=\nvinfty}}
== Lihat pula ==
{{commons category|Simbol takhingga}}
* [[Bilangan alef]]
* [[Sejarah notasi matematika]]
== Rujukan ==
{{reflist|refs=<ref name=algebraic>{{cite book
| last = Perrin | first = Daniel
| isbn = 978-1-84800-056-8
| page = 28
| publisher = Springer
| title = Algebraic Geometry: An Introduction
| url = https://books.google.com/books?id=Vn1yR9qPvlMC&pg=PA28
| year = 2007}}</ref>
<ref name=autism>{{cite journal
| last = Gross | first = Liza
| date = September 2016
| doi = 10.1371/journal.pbio.2000958
| issue = 9
| journal = [[PLOS Biology]]
| page = e2000958
| title = In search of autism's roots
| volume = 14}}</ref>
<ref name=bakelite>{{cite journal
| last1 = Crespy | first1 = Daniel
| last2 = Bozonnet | first2 = Marianne
| last3 = Meier | first3 = Martin
| date = April 2008
| doi = 10.1002/anie.200704281
| issue = 18
| journal = Angewandte Chemie
| pages = 3322–3328
| title = 100 years of Bakelite, the material of a 1000 uses
| volume = 47}}</ref>
<ref name=barkwell>{{cite web |last1=Barkwell |first1=Lawrence J. |title=The Metis Infinity Flag |url=http://www.metismuseum.ca/resource.php/07245 |website=Virtual Museum of Métis History and Culture |publisher=Gabriel Dumont Institute |access-date=15 July 2020}}</ref>
<ref name=beautiful>{{cite book
| last = Erickson | first = Martin J.
| contribution = 1.1 Lemniscate
| contribution-url = https://books.google.com/books?id=LgeP62-ZxikC&pg=PA1
| isbn = 978-0-88385-576-8
| pages = 1–3
| publisher = [[Mathematical Association of America]]
| series = MAA Spectrum
| title = Beautiful Mathematics
| year = 2011}}</ref>
<ref name=beyond>{{cite book
| last = Maor | first = Eli | author-link = Eli Maor
| isbn = 0-691-02511-8
| mr = 1129467
| page = 7
| publisher = Princeton University Press | location = Princeton, New Jersey
| title = To Infinity and Beyond: A Cultural History of the Infinite
| url = https://books.google.com/books?id=pMY9DwAAQBAJ&pg=PA7
| year = 1991}}</ref>
<ref name=book>{{cite book
| last1 = Zboray | first1 = Ronald J.
| last2 = Zboray | first2 = Mary Saracino
| isbn = 978-0-8444-1015-9
| page = 49
| publisher = [[Center for the Book]], [[Library of Congress]]
| title = A Handbook for the Study of Book History in the United States
| url = https://archive.org/details/handbookforstudy0000zbor/page/49
| year = 2000}}</ref>
<ref name=brands>{{cite book
| last1 = Rivkin | first1 = Steve
| last2 = Sutherland | first2 = Fraser
| isbn = 978-0-19-988340-0
| page = 130
| publisher = Oxford University Press
| title = The Making of a Name: The Inside Story of the Brands We Buy
| url = https://books.google.com/books?id=WJVcgta-HToC&pg=PA130
| year = 2005}}</ref>
<ref name=camera>{{cite journal<!-- Although it would make more sense for this to be a conference, SAGE publishes it as a journal --> | last1 = Crist | first1 = Brian | last2 = Aurello | first2 = David N. | date = October 1990 | doi = 10.1177/154193129003400512 | issue = 5 | journal = Proceedings of the Human Factors Society Annual Meeting | pages = 489–493 | title = Development of camera symbols for consumers | volume = 34}}</ref>
<ref name=cldr>{{cite web |url=https://github.com/unicode-org/cldr/blob/master/common/annotations/en.xml |author=Unicode, Inc |author-link=Unicode Consortium |title=Annotations |work=[[Common Locale Data Repository]]|via=[[GitHub]]}}</ref>
<ref name=compart>{{Cite web|url=https://www.compart.com/en/unicode/U+221E|title=Unicode Character "∞" (U+221E)|work=Unicode|publisher=Compart AG|language=en|access-date=2019-11-15}}</ref>
<ref name=comprehensive>{{cite book|contribution=Table 294: stix Infinities|page=118|title=The Comprehensive LATEX Symbol List|first=Scott|last=Pakin|date=May 5, 2021|url=https://ctan.org/pkg/comprehensive|publisher=[[CTAN]]|access-date=2022-02-19}}</ref>
<ref name=convergence>{{cite journal
| last = Shipman | first = Barbara A.
| date = April 2013
| doi = 10.1080/10511970.2012.753963
| issue = 5
| journal = PRIMUS
| pages = 441–458
| title = Convergence and the Cauchy property of sequences in the setting of actual infinity
| volume = 23}}</ref>
<ref name=cosmic>{{cite book
| last = Barrow | first = John D. | author-link = John D. Barrow
| contribution = Infinity: Where God Divides by Zero
| contribution-url = https://books.google.com/books?id=uRg6iN10JCIC&pg=PA339
| isbn = 978-0-393-06177-2
| pages = 339–340
| publisher = W. W. Norton & Company
| title = Cosmic Imagery: Key Images in the History of Science
| year = 2008}}</ref>
<ref name=dreams>{{cite book|url=https://books.google.com/books?id=vhNNrX3bmo4C&pg=PA243|title=Dreams, Illusion, and Other Realities|last=O'Flaherty|first=Wendy Doniger|author-link=Wendy Doniger|publisher=University of Chicago Press|year=1986|isbn=978-0-226-61855-5|page=243}} The book also features this image on its cover.</ref>
<ref name=dumont>{{cite web |url= http://www.metisresourcecentre.mb.ca/index.php?option=com_content&view=article&id=2&Itemid=8 |title=The Métis flag |work=Gabriel Dumont Institute(Métis Culture & Heritage Resource Centre) |archive-url=https://web.archive.org/web/20130724192737/http://metisresourcecentre.mb.ca/index.php?option=com_content&view=article&id=2&Itemid=8 |archive-date=2013-07-24}}</ref>
<ref name=encoding>{{cite web |url=https://encoding.spec.whatwg.org/big5.html |title=big5 |work=Encoding Standard |publisher=[[WHATWG]] |last=van Kesteren |first=Anne |author-link=Anne van Kesteren}}</ref>
<ref name=flags>{{cite book|title=Native American Flags|first1=Donald T.|last1=Healy|first2=Peter J.|last2=Orenski|publisher=University of Oklahoma Press|year=2003|isbn=978-0-8061-3556-4|page=[https://archive.org/details/nativeamericanfl0000heal/page/284 284]|url=https://archive.org/details/nativeamericanfl0000heal/page/284}}</ref>
<ref name=gaudry>{{Cite journal|last=Gaudry |first=Adam |date=Spring 2018 |title= Communing with the Dead: The "New Métis," Métis Identity Appropriation, and the Displacement of Living Métis Culture |journal= American Indian Quarterly |volume=42 |issue=2 |pages=162–190 |jstor=10.5250/amerindiquar.42.2.0162 |doi=10.5250/amerindiquar.42.2.0162 |s2cid=165232342}}</ref>
<ref name=hitchhiker>{{cite book
| last1 = Aliprantis | first1 = Charalambos D. | author1-link = Charalambos D. Aliprantis
| last2 = Border | first2 = Kim C. | author2-link = Kim Border
| edition = 3rd
| isbn = 978-3-540-29587-7
| pages = 56–57
| publisher = Springer
| title = Infinite Dimensional Analysis: A Hitchhiker's Guide
| url = https://books.google.com/books?id=4vyXtR3vUhoC&pg=PA56
| year = 2006}}</ref>
<ref name=inspires>{{cite journal | last = Willmes | first = Claudia Gisela | date = January 2021 | doi = 10.1016/j.molmed.2020.11.001 | issue = 1 | journal = Trends in Molecular Medicine | page = 1 | pmid = 33308981 | title = Science that inspires | volume = 27}}</ref>
<ref name=mind>{{cite book
| last = Rucker | first = Rudy | author-link = Rudy Rucker
| isbn = 3-7643-3034-1
| location = Boston, Massachusetts
| mr = 658492
| page = 1
| publisher = Birkhäuser
| title = Infinity and the Mind: The science and philosophy of the infinite
| year = 1982}}</ref>
<ref name=nabokov>{{cite book|title=Nabokov: The Mystery of Literary Structures|first=Leona|last=Toker|publisher=Cornell University Press|year=1989|isbn=978-0-8014-2211-9|page=[https://archive.org/details/nabokovmysteryof00toke/page/159 159]|url=https://archive.org/details/nabokovmysteryof00toke|url-access=registration}}</ref>
<ref name=notations>{{Cite book|title-link=A History of Mathematical Notations|title=A History of Mathematical Notations, Volume II: Notations Mainly in Higher Mathematics|last=Cajori|first=Florian|author-link=Florian Cajori|publisher=Open Court |year=1929 |pages=44–48|contribution-url=https://archive.org/details/AHistoryOfMathematicalNotationVolII/page/n67|contribution=Signs for infinity and transfinite numbers}}</ref>
<ref name=notations-incorrect>{{harvtxt|Cajori|1929}} salah menampilkan simbol tersebut, ketika huruf S menyamping tanpa dibalikkan. Simbol tersebut dapat dilihat ketika [[Leonhard Euler|Euler, Leonhard]] memakainya di {{cite journal|first=Leonhard|last=Euler|author-link=Leonhard Euler|language=la|title=Variae observationes circa series infinitas|journal=Commentarii academiae scientiarum Petropolitanae|volume=9|year=1744|pages=160–188|url=http://eulerarchive.maa.org/docs/originals/E072.pdf}}, di hlm. 174</ref>
<ref name=prefontaine>{{cite journal |last1=Darren R. |first1=Préfontaine |title=Flying the Flag, Editor's note. |journal=New Breed Magazine |date=2007 |issue=Winter 2007 |page=6 |url=http://www.metismuseum.ca/resource.php/05851 |access-date=26 August 2020}}</ref>
<ref name=qatar>{{cite news|url=https://www.aljazeera.com/news/2019/09/qatar-2022-football-world-cup-logo-unveiled-190903193444377.html|publisher=Al Jazeera|title=Qatar 2022: Football World Cup logo unveiled|date=September 3, 2019}}</ref>
<ref name=racette>{{Cite book |url= http://www.metismuseum.ca/media/document.php/12153.Flags%20of%20the%20Metis.pdf |title=Flags of the Métis |last=Racette |first=Calvin |publisher=Gabriel Dumont Institute |year=1987 |isbn=0-920915-18-3}}</ref>
<ref name=sands>{{cite book | last = Bahun | first = Sanja | editor1-last = Kim | editor1-first = Rina | editor2-last = Westall | editor2-first = Claire | contribution = 'These heavy sands are language tide and wind have silted here': Tidal voicing and the poetics of home in James Joyce's Ulysses | doi = 10.1057/9781137020758_4 | pages = 57–73 | publisher = Palgrave Macmillan | title = Cross-Gendered Literary Voices: Appropriating, Resisting, Embracing | year = 2012}}</ref>
<ref name=sectionibus>{{cite book|url=https://archive.org/details/bub_gb_03M_AAAAcAAJ|title=De Sectionibus Conicis, Nova Methodo Expositis, Tractatus|last=Wallis|first=John|year=1655|pages=[https://archive.org/details/bub_gb_03M_AAAAcAAJ/page/n17 4]|language=la|chapter=Pars Prima|author-link=John Wallis}}</ref>
<ref name=tarot>{{cite journal
| last = Armson | first = Morandir
| date = June 2011
| issue = 1
| journal = Literature & Aesthetics
| pages = 196–212
| title = The transitory tarot: an examination of tarot cards, the 21st century New Age and theosophical thought
| url = https://openjournals.library.sydney.edu.au/index.php/LA/article/viewFile/5056/5761
| volume = 21}} Khususnya, lihat di hlm. 203: "Reinkarnasi dilambangkan dalam jumlah kartu tarot Waite-Smith. Simbol utama reinkarnasi yang dipakai berupa simbol takhingga atau lemniskat, roda dan lingkaran."</ref>
<ref name=unicode-cp437>{{cite web |url=https://unicode.org/Public/MAPPINGS/VENDORS/MICSFT/PC/CP437.TXT |title=cp437_DOSLatinUS to Unicode table |date=April 24, 1996 |last=Steele |first=Shawn |publisher=[[Unicode Consortium]] | access-date=2022-02-19}}</ref>
<ref name=unicode-euc-jp>{{cite web |url=https://raw.githubusercontent.com/unicode-org/icu/master/icu4c/source/data/mappings/euc-jp-2007.ucm |title=EUC-JP-2007 |publisher=[[Unicode Consortium]] |work=[[International Components for Unicode]] | access-date=2022-02-19|via=[[GitHub]]}}</ref>
<ref name=unicode-euc-kr>{{cite web |url=https://raw.githubusercontent.com/unicode-org/icu/master/icu4c/source/data/mappings/ibm-970_P110_P110-2006_U2.ucm |title=IBM-970 |publisher=[[Unicode Consortium]] |work=[[International Components for Unicode]]|date=May 9, 2007 | access-date=2022-02-19|via=[[GitHub]]}}</ref>
<ref name=unicode-jis>{{cite web |url=https://www.unicode.org/Public/MAPPINGS/OBSOLETE/EASTASIA/JIS/SHIFTJIS.TXT |title=Shift-JIS to Unicode |publisher=[[Unicode Consortium]] |date=December 2, 2015 | access-date=2022-02-19}}</ref>
<ref name=unicode-kps>{{cite web |url=https://unicode.org/Public/MAPPINGS/VENDORS/MISC/KPS9566.TXT |title=KPS 9566-2003 to Unicode |date=April 27, 2011 |publisher=[[Unicode Consortium]] | access-date=2022-02-19}}</ref>
<ref name=unicode-mac-roman>{{cite web |url=http://unicode.org/Public/MAPPINGS/VENDORS/APPLE/ROMAN.TXT |title=Map (external version) from Mac OS Roman character set to Unicode 2.1 and later |date=April 5, 2005 |publisher=[[Apple Inc.]] |via=[[Unicode Consortium]]| access-date=2022-02-19}}</ref>
<ref name=unicode-mac-symbol>{{cite web |url=http://unicode.org/Public/MAPPINGS/VENDORS/APPLE/SYMBOL.TXT |title=Map (external version) from Mac OS Symbol character set to Unicode 4.0 and later|date=April 5, 2005 |publisher=[[Apple Inc.]] |via=[[Unicode Consortium]]| access-date=2022-02-19}}</ref>
<ref name=unicode-misc>{{cite web|url=https://www.unicode.org/charts/PDF/U2980.pdf|title=Miscellaneous Mathematical Symbols-B|publisher=[[Unicode Consortium]]|archive-url=https://web.archive.org/web/20181112231107/https://www.unicode.org/charts/PDF/U2980.pdf|archive-date=2018-11-12|url-status=live| access-date=2022-02-19}}</ref>
<ref name=unicode-uhc>{{cite web |url=https://www.unicode.org/Public/MAPPINGS/VENDORS/MICSFT/WINDOWS/CP949.TXT |title=cp949 to Unicode table |last=Steele |first=Shawn |publisher=[[Unicode Consortium]] |date=January 7, 2000 | access-date=2022-02-19}}</ref>
<ref name=unthinkable>{{cite book
| last = Clegg | first = Brian | author-link = Brian Clegg (writer)
| contribution = Chapter 6: Labelling the infinite
| isbn = 978-1-84119-650-3
| publisher = Constable & Robinson Ltd
| title = A Brief History of Infinity: The Quest to Think the Unthinkable
| year = 2003}}</ref>
<ref name=viking>{{cite journal
| last = van Riel | first = Sjoerd
| journal = Lund Archaeological Review
| pages = 163–178
| title = Viking Age Combs: Local Products or Objects of Trade?
| url = https://journals.lub.lu.se/lar/article/view/21656
| volume = 23
| year = 2017}} See p. 172: "Within this type the lemniscate (∞) is a commonly used motif."</ref>
<ref name=wallace>{{cite book | last = Natalini | first = Roberto | editor1-last = Boswell | editor1-first = Marshall | editor2-last = Burn | editor2-first = Stephen J. | contribution = David Foster Wallace and the mathematics of infinity | doi = 10.1057/9781137078346_3 | pages = 43–57 | publisher = Palgrave Macmillan | series = American Literature Readings in the 21st Century | title = A Companion to David Foster Wallace Studies | year = 2013}}</ref>
<ref name=wallis>{{cite book
| last = Scott | first = Joseph Frederick
| edition = 2nd
| isbn = 0-8284-0314-7
| page = 24
| publisher = [[American Mathematical Society]]
| title = The mathematical work of John Wallis, D.D., F.R.S., (1616-1703)
| url = https://books.google.com/books?id=XX9PKytw8g8C&pg=PA24
| year = 1981}}</ref>
<ref name=zero>{{cite book|title=Zero to Lazy Eight: The Romance of Numbers|first1=Alexander|last1=Humez|first2= Nicholas D.|last2=Humez|first3= Joseph|last3= Maguire|publisher =Simon and Schuster|year=1993|isbn= 978-0-671-74281-2|page=18|url=https://books.google.com/books?id=X429EAr8g4kC&pg=PA18}}</ref>}}
| 