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Revision as of 09:36, 4 August 2024 edit100.36.106.199 (talk) All egregious WP:SYNTH violations, randomly combining properties of limited significance to make new, super-properties of literally no significance at all (because the process of running a number through a three-deep chain of random OEIS connections can be done with any number, it produces numerology instead of meaning). Please actually read the relevant policies!Tag: Undo← Previous edit Revision as of 09:42, 4 August 2024 edit undoRadlrb (talk | contribs)Extended confirmed users9,249 edits →In mathematics: improve layout, and start adding algebraic propertiesTags: Mobile edit Mobile web editNext edit →
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==In mathematics== ==In mathematics==
* Eighteen is the tenth ], its ]s being 1, 2, 3, 6 and 9. Three of these divisors (3, 6 and 9) add up to 18, hence 18 is a ].<ref>{{Cite OEIS|A005835|Pseudoperfect (or semiperfect) numbers|access-date=2016-05-31}}</ref> 18 is the first inverted square-prime of the form ''p''·''q''<sup>2</sup>. Eighteen is the tenth ], its ]s being 1, 2, 3, 6 and 9. It is the first inverted square-prime of the form ''p''·''q''<sup>2</sup>.

⚫ * It is an ], as the sum of its proper divisors is greater than itself (1 + 2 + 3 + 6 + 9 = ]). It is known to be a ], despite not being ] to this sum.
===Integer properties===
⚫ * In ], it is a ], since it is divisible by the ], 1 + 8 = ]. It is the only number in decimal where the sum of its digits is half of itself.
⚫ 18 has three of its divisors (3, 6 and 9) that add up to itself, hence 18 is a ].<ref>{{Cite OEIS|A005835|Pseudoperfect (or semiperfect) numbers|access-date=2016-05-31}}</ref> It is also an ], as the sum of its proper divisors is greater than itself (1 + 2 + 3 + 6 + 9 = ]). It is known to be a ], despite not being ] to this sum.
⚫ * There are 18 discriminants of imaginary ]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 ]s).<ref>{{Cite OEIS |A014602 |Discriminants of imaginary quadratic fields with class number 1 (negated). |access-date=2024-08-03 }}</ref>
* 18 is the ] of ] (the only number to have this sum, aside from the prime number ]),<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 ] (], ], ], 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 ], ] = ]<sup>3</sup> (with thrice 6 being 18). }}
* It is the number of one-sided ]es.
*Only four numbers have an ] of 18, they are ], ], ] and ]<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).
⚫ * It is a Fine number.<ref>{{cite OEIS|A000957|name=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|access-date=2022-06-01}}</ref>
*As ]s, only five numbers have divisors whose average is 18, they are 30, ], ], ] and ].<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>
*18 is the number of ] of the smallest ] that is the product of the first three prime numbers, ] (equal to partitions into distinct ] 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>

⚫ In ], it is a ], since it is divisible by the ], 1 + 8 = ]. It is the only number in decimal where the sum of its digits is half of itself.

⚫ There are 18 discriminants of imaginary ]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 ]s).<ref>{{Cite OEIS |A014602 |Discriminants of imaginary quadratic fields with class number 1 (negated). |access-date=2024-08-03 }}</ref>

⚫ 18 is the number of one-sided ]es, and it a ''fine number''.<ref>{{cite OEIS|A000957|name=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|access-date=2022-06-01}}</ref>

=== In algebra ===
There are 18 classes of infinite families of ] that classify as ]: 16 are ]s, while the remaining 2 ]s and ]s.


==In science== ==In science==

Revision as of 09:42, 4 August 2024

"XVIII" redirects here. For the Eighteen Visions album, see XVIII (album). Natural number
← 17 18 19 →
← 10 11 12 13 14 15 16 17 18 19 → ← 0 10 20 30 40 50 60 70 80 90 →
Cardinaleighteen
Ordinal18th
(eighteenth)
Numeral systemoctodecimal
Factorization2 × 3
Divisors1, 2, 3, 6, 9, 18
Greek numeralΙΗ´
Roman numeralXVIII, xviii
Binary100102
Ternary2003
Senary306
Octal228
Duodecimal1612
Hexadecimal1216
Hebrew numeralי"ח
Babylonian numeral𒌋𒐜

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·q.

Integer properties

18 has three of its divisors (3, 6 and 9) that add up to itself, hence 18 is a semiperfect number. 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), 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.
  • Only four numbers have an Euler totient of 18, they are 19, 27, 38 and 54 (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.
  • 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).

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); this is twice the number of such discriminants of class number 1 (the Heegner numbers).

18 is the number of one-sided pentominoes, and it a fine number.

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

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.
  • 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.
  • 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'.
  • 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.

Age 18

See also: Age of majority, Voting age, Drinking age, Smoking age, Marriageable age, and Age of consent

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.
    • Minimum age to purchase firearms in thirty-eight states with the exception of handguns (21 under federal law).
    • 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

See also

Notes

  1. Sloane, N. J. A. (ed.). "Sequence A005835 (Pseudoperfect (or semiperfect) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-31.
  2. 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.
  3. Sloane, N. J. A. (ed.). "Sequence A002808 (The composite numbers.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-08-04.
  4. Sloane, N. J. A. (ed.). "Sequence A000010 (Euler totient function.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-08-04.
  5. 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.
  6. 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.
  7. 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.
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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
  13. 言必有「中」：「十八」「實發」 人見人愛. 文匯報. Accessed 2015-01-12. Archived 2015-01-15.
  14. "Chai, Its Meaning and Significance | Shiva, Jewish Mourning". www.shiva.com. Retrieved 2020-11-18.
  15. "STATUTORY RAPE: A GUIDE TO STATE LAWS AND REPORTING REQUIREMENTS. SEXUAL INTERCOURSE WITH MINORS". aspe.hhs.gov.
  16. "Minimum Age". Giffords.
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