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==In mathematics== |
==In mathematics== |
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Eighteen is the tenth [[composite number]], its [[divisor]]s being 1, 2, 3, 6 and 9. It is the first inverted square-prime of the form ''p''·''q''<sup>2</sup>. |
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===Integer properties=== |
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| ⚫ | 18 has three of its divisors (3, 6 and 9) that add up to itself, hence 18 is a [[semiperfect number]].<ref>{{Cite OEIS|A005835|Pseudoperfect (or semiperfect) numbers|access-date=2016-05-31}}</ref> It is also an [[abundant number]], as the sum of its proper divisors is greater than itself (1 + 2 + 3 + 6 + 9 = [[21 (number)|21]]). It is known to be a [[friendly number|solitary number]], despite not being [[coprime]] to this sum. |
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* 18 is the [[Divisor function|sum-of-divisors]] of [[10]] (the only number to have this sum, aside from the prime number [[17 (number)|17]]),<ref>{{Cite OEIS |A000203 |the sum of the divisors of n. |access-date=2024-08-04 }}</ref> where 18 is in equivalence with the sum of the first even and odd primes and [[Composite number |composites]] ([[2]], [[3]], [[4]], 9), all less than 10, which is also the composite index of 18.<ref>{{Cite OEIS |A002808 |The composite numbers. |access-date=2024-08-04 }}</ref>{{efn|1=Meanwhile, the product of these four digits is a [[Cubic number|cube]], [[216 (number)|216]] = [[6]]<sup>3</sup> (with thrice 6 being 18). }} |
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* It is the number of one-sided [[pentomino]]es. |
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*Only four numbers have an [[Euler totient]] of 18, they are [[19 (number)|19]], [[27 (number)|27]], [[38 (number)|38]] and [[54 (number)|54]]<ref>{{Cite OEIS |A000010 |Euler totient function. |access-date=2024-08-04 }}</ref> (the latter is the sum of eight integers, from 2 through 9). |
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*As [[arithmetic number]]s, only five numbers have divisors whose average is 18, they are 30, [[33 (number)|33]], [[36 (number)|36]], [[66 (number)|66]] and [[70 (number)|70]].<ref>{{Cite OEIS |A003601 |Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n). |access-date=2024-08-04 }}</ref><ref>{{Cite OEIS |A102187 |Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer). |access-date=2024-08-04 }}</ref> |
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*18 is the number of [[Integer partition|self-conjugate partitions]] of the smallest [[sphenic number]] that is the product of the first three prime numbers, [[30 (number)|30]] (equal to partitions into distinct [[Parity (mathematics)|odd]] parts).<ref>{{Cite OEIS |A000700 |number of partitions of n into distinct odd parts; number of self-conjugate partitions; number of symmetric Ferrers graphs with n nodes. |access-date=2024-08-04 }}</ref> |
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| ⚫ | There are 18 discriminants of imaginary [[quadratic field]]s with class number 2 (negated);<ref>{{Cite OEIS |A014603 |Discriminants of imaginary quadratic fields with class number 2 (negated). |access-date=2024-08-03 }}</ref> this is twice the number of such discriminants of class number 1 (the [[Heegner number]]s).<ref>{{Cite OEIS |A014602 |Discriminants of imaginary quadratic fields with class number 1 (negated). |access-date=2024-08-03 }}</ref> |
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=== In algebra === |
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There are 18 classes of infinite families of [[Group theory|groups]] that classify as [[Finite simple group|finite and simple]]: 16 are [[Lie group]]s, while the remaining 2 [[cyclic group]]s and [[alternating group]]s. |
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==In science== |
==In science== |
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Revision as of 09:42, 4 August 2024
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| Cardinal | eighteen | |||
| Ordinal | 18th (eighteenth) | |||
| Numeral system | octodecimal | |||
| Factorization | 2 × 32 | |||
| Divisors | 1, 2, 3, 6, 9, 18 | |||
| Greek numeral | ΙΗ´ | |||
| Roman numeral | XVIII | |||
| Binary | 100102 | |||
| Ternary | 2003 | |||
| Senary | 306 | |||
| Octal | 228 | |||
| Duodecimal | 1612 | |||
| Hexadecimal | 1216 | |||
| Hebrew numeral | י"ח | |||
| Babylonian numeral | 𒌋𒐜 | |||
Look up eighteen in Wiktionary, the free dictionary.
18 (eighteen) is the natural number following 17 and preceding 19.
In mathematics
Eighteen is the tenth composite number, its divisors being 1, 2, 3, 6 and 9. It is the first inverted square-prime of the form p·q2.
Integer properties
18 has three of its divisors (3, 6 and 9) that add up to itself, hence 18 is a semiperfect number.[1] It is also an abundant number, as the sum of its proper divisors is greater than itself (1 + 2 + 3 + 6 + 9 = 21). It is known to be a solitary number, despite not being coprime to this sum.
- 18 is the sum-of-divisors of 10 (the only number to have this sum, aside from the prime number 17),[2] where 18 is in equivalence with the sum of the first even and odd primes and composites (2, 3, 4, 9), all less than 10, which is also the composite index of 18.[3][a]
- Only four numbers have an Euler totient of 18, they are 19, 27, 38 and 54[4] (the latter is the sum of eight integers, from 2 through 9).
- As arithmetic numbers, only five numbers have divisors whose average is 18, they are 30, 33, 36, 66 and 70.[5][6]
- 18 is the number of self-conjugate partitions of the smallest sphenic number that is the product of the first three prime numbers, 30 (equal to partitions into distinct odd parts).[7]
In base ten, it is a Harshad number, since it is divisible by the sum of its digits, 1 + 8 = 9. It is the only number in decimal where the sum of its digits is half of itself.
There are 18 discriminants of imaginary quadratic fields with class number 2 (negated);[8] this is twice the number of such discriminants of class number 1 (the Heegner numbers).[9]
18 is the number of one-sided pentominoes, and it a fine number.[10]
In algebra
There are 18 classes of infinite families of groups that classify as finite and simple: 16 are Lie groups, while the remaining 2 cyclic groups and alternating groups.
In science
Chemistry
- Eighteen is the atomic number of argon.
- Group 18 of the periodic table is called the noble gases.
- The 18-electron rule is a rule of thumb in transition metal chemistry for characterising and predicting the stability of metal complexes.
- 18 is the number of ways to represent the word 'Floccinaucinihilipilification' using chemical element symbols.
In religion and literature
- The Hebrew word for "life" is חי (chai), which has a numerical value of 18. Consequently, the custom has arisen in Jewish circles to give donations and monetary gifts in multiples of 18 as an expression of blessing for long life.[11]
- In Judaism, in the Talmud; Pirkei Avot (5:25), Rabbi Yehudah ben Teime gives the age of 18 as the appropriate age to get married ("Ben shmonah esra lechupah", at eighteen years old to the Chupah (marriage canopy)). (See Coming of age, Age of majority).
- Shemoneh Esrei (sh'MOH-nuh ES-ray) is a prayer that is the center of any Jewish religious service. Its name means "eighteen". The prayer is also known as the Amidah.
- In Ancient Roman custom the number 18 can symbolise a blood relative.
- Joseph Heller's novel Catch-22 was originally named Catch-18 because of the Hebrew meaning of the number, but was amended to the published title to avoid confusion with another war novel, Mila 18.[12]
- There are 18 chapters in the Bhagavad Gita, which is contained in the Mahabharata, which has 18 books. The Kurukshetra War which the epic depicts, is between 18 armies (11 on the Kuru side, 7 on the Pandava). The war itself lasts for 18 days. In the other Hindu epic, the Ramayana, the war between Rama and the demons also lasted 18 days.
- In Babism the first 18 disciples of the Báb were known as the Letters of the Living.
As lucky or unlucky number
- In Chinese tradition, 18 is pronounced 十八; shí bā and is considered a lucky number due to similarity with 實發; shì fā 'definitely get rich', 'to get rich for sure'.[13]
- According to applications of numerology in Judaism, the letters of the word chai ("living") add up to 18. Thus, 18 is considered a lucky number and many gifts for B'nai Mitzvot and weddings are in $18 increments.[14]
Age 18
In most countries, 18 is the age of majority, in which a minor becomes a legal adult. It is also the voting age, marriageable age, drinking age and smoking age in most countries, though sometimes these ages are different than the age of majority. Many websites restrict adult content to visitors who claim to be aged over 18.
- In the United States, 18 is the:
- Age for sexual consent in eleven states and under federal law.[15]
- Minimum age to purchase firearms in thirty-eight states with the exception of handguns (21 under federal law).[16]
- Marriageable age without parental consent except for Nebraska (19), Mississippi and Puerto Rico (21).
- The minimum age at which one can purchase, rent, or buy tickets to NC-17-rated films or buy video games with an Adults Only rating.
- In the UK, 18 is the legal age to purchase a BBFC "18" rated film.
- In Japan, 18 is the minimum age at which one can purchase, rent, or buy tickets to R18+ rated movies or buy video games with a Z rating.
In sports
- In association football (soccer), "the 18" is a slang term for the penalty area.
- In Australian rules football (except in AFL Women's), each team has 18 players on the field during play.
- There are 18 holes on a regulation golf course.
- In Nippon Professional Baseball (Japanese baseball), No.18 is known as Ace number.
- 18 is the record number of most NBA championships in NBA history, which the Boston Celtics achieved.
See also
Notes
- ^ Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
- ^ Sloane, N. J. A. (ed.). "Sequence A000203 (the sum of the divisors of n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-08-04.
- ^ Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-08-04.
- ^ Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-08-04.
- ^ Sloane, N. J. A. (ed.). "Sequence A003601 (Numbers n such that the average of the divisors of n is an integer: sigma_0(n) divides sigma_1(n).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-08-04.
- ^ Sloane, N. J. A. (ed.). "Sequence A102187 (Arithmetic means of divisors of arithmetic numbers (arithmetic numbers, A003601, are those for which the average of the divisors is an integer).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-08-04.
- ^ Sloane, N. J. A. (ed.). "Sequence A000700 (number of partitions of n into distinct odd parts; number of self-conjugate partitions; number of symmetric Ferrers graphs with n nodes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-08-04.
- ^ Sloane, N. J. A. (ed.). "Sequence A014603 (Discriminants of imaginary quadratic fields with class number 2 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-08-03.
- ^ Sloane, N. J. A. (ed.). "Sequence A014602 (Discriminants of imaginary quadratic fields with class number 1 (negated).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-08-03.
- ^ Sloane, N. J. A. (ed.). "Sequence A000957 (Fine's sequence (or Fine numbers): number of relations of valence >= 1 on an n-set; also number of ordered rooted trees with n edges having root of even degree)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-06-01.
- ^ Benjaminson, Chani. "What is the reason for the custom to give money gifts in multiples of 18 -- 18, 36, 54, etc.?". Chabad.org. Archived from the original on 2007-12-27. Retrieved January 19, 2022.
- ^ N James. The Early Composition History of Catch-22. In Biographies of Books: The Compositional Histories of Notable American Writings, J Barbour, T Quirk (edi.) pp. 262-90. Columbia: University of Missouri Press, 1996
- ^ 言必有「中」：「十八」「實發」 人見人愛. 文匯報. Accessed 2015-01-12. Archived 2015-01-15.
- ^ "Chai, Its Meaning and Significance | Shiva, Jewish Mourning". www.shiva.com. Retrieved 2020-11-18.
- ^ "STATUTORY RAPE: A GUIDE TO STATE LAWS AND REPORTING REQUIREMENTS. SEXUAL INTERCOURSE WITH MINORS". aspe.hhs.gov.
- ^ "Minimum Age". Giffords.
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