9

This article is about the number. For the years, see 9 BC and AD 9. For other uses, see 9 (disambiguation) and Number nine (disambiguation).

← 8 9 10 →
−1 0 1 2 3 4 5 6 7 8 9 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
Cardinal	nine
Ordinal	9th (ninth)
Numeral system	nonary
Factorization	32
Divisors	1,3,9
Greek numeral	Θ´
Roman numeral	IX, ix
Greek prefix	ennea-
Latin prefix	nona-
Binary	10012
Ternary	1003
Senary	136
Octal	118
Duodecimal	912
Hexadecimal	916
Amharic	፱
Arabic, Kurdish, Persian, Sindhi, Urdu	٩
Armenian numeral	Թ
Bengali	৯
Chinese numeral	九, 玖
Devanāgarī	९
Greek numeral	θ´
Hebrew numeral	ט
Tamil numerals	௯
Khmer	៩
Telugu numeral	౯
Thai numeral	๙
Malayalam	൯
Babylonian numeral	𒐝
Egyptian hieroglyph	𓐂
Morse code	____.

9 (nine) is the natural number following 8 and preceding 10.

Evolution of the Hindu–Arabic digit

See also: Hindu–Arabic numeral system

Circa 300 BC, as part of the Brahmi numerals, various Indians wrote a digit 9 similar in shape to the modern closing question mark without the bottom dot. The Kshatrapa, Andhra and Gupta started curving the bottom vertical line coming up with a 3-look-alike. ^[1] How the numbers got to their Gupta form is open to considerable debate. The Nagari continued the bottom stroke to make a circle and enclose the 3-look-alike, in much the same way that the sign @ encircles a lowercase a. As time went on, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3-look-alike became smaller. Soon, all that was left of the 3-look-alike was a squiggle. The Arabs simply connected that squiggle to the downward stroke at the middle and subsequent European change was purely cosmetic.

While the shape of the glyph for the digit 9 has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .

The modern digit resembles an inverted number 6 (six.) To disambiguate the two on objects and labels that can be inverted, they are often underlined. It is sometimes handwritten with two strokes and a straight stem, resembling a raised lower-case letter q, which distinguishes it from the 6. Similarly, in seven-segment display, the number 9 can be constructed either with a hook at the end of its stem or without one. Most LCD calculators use the former, but some VFD models use the latter.

Mathematics

Nine is the fourth composite number, and the first composite number that is odd. Nine is the third square number (3²), and the second non-unitary square prime of the form p², and, the first that is odd, with all subsequent squares of this form odd as well. Nine has the even aliquot sum of 4, and with a composite number sequence of two (9, 4, 3, 1, 0) within the 3-aliquot tree. It is the first member of the first cluster of two semiprimes (9, 10), preceding (14, 15). ^[2] Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers in decimal, a method known as long ago as the 12th century. ^[3]

By Mihăilescu's theorem, 9 is the only positive perfect power that is one more than another positive perfect power, since the square of 3 is one more than the cube of 2. ^{[4] [5]}

Non-intersecting chords between four points on a circle

9 is the sum of the cubes of the first two non-zero positive integers ${\displaystyle 1^{3}+2^{3}}$ which makes it the first cube-sum number greater than one. ^[6]

It is also the sum of the first three nonzero factorials ${\displaystyle 1!+2!+3!}$, and equal to the third exponential factorial, since ${\displaystyle 9=3^{2^{1}}.}$ ^[7]

Nine is the number of derangements of 4, or the number of permutations of four elements with no fixed points. ^[8]

9 is the fourth refactorable number, as it has exactly three positive divisors, and 3 is one of them. ^[9]

A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes. ^[10]

If an odd perfect number exists, it will have at least nine distinct prime factors. ^[11]

9 is a Motzkin number, for the number of ways of drawing non-intersecting chords between four points on a circle. ^[12]

Four concentric magic circles with 9 in the center (by Yang Hui), where numbers on each circle and diameter around the center generate a magic sum of 138.

The first non-trivial magic square is a ${\displaystyle 3}$ x ${\displaystyle 3}$ magic square made of nine cells, with a magic constant of 15. ^[13] Meanwhile, a ${\displaystyle 9}$ x ${\displaystyle 9}$ magic square has a magic constant of 369. ^[14]

There are nine Heegner numbers, or square-free positive integers ${\displaystyle n}$ that yield an imaginary quadratic field ${\displaystyle \mathbb {Q} \left[{\sqrt {-n}}\right]}$ whose ring of integers has a unique factorization, or class number of 1. ^[15]

Geometry

Polygons and tilings

A polygon with nine sides is called a nonagon. ^[16] Since 9 can be written in the form ${\displaystyle 2^{m}3^{n}p}$, for any nonnegative natural integers ${\displaystyle m}$ and ${\displaystyle n}$ with ${\displaystyle p}$ a product of Pierpont primes, a regular nonagon is constructed with a regular compass, straightedge, and angle trisector. ^[17] Also an enneagon, a regular nonagon is able to fill a plane-vertex alongside an equilateral triangle and a regular 18-sided octadecagon (3.9.18), and as such, it is one of only nine polygons that are able to fill a plane-vertex without uniformly tiling the plane. ^[18] In total, there are a maximum of nine semiregular Archimedean tilings by convex regular polygons, when including chiral forms of the snub hexagonal tiling. More specifically, there are nine distinct uniform colorings to both the triangular tiling and the square tiling (the simplest regular tilings) while the hexagonal tiling, on the other hand, has three distinct uniform colorings.

The fewest number of squares needed for a perfect tiling of a rectangle is nine. ^[19]

Polyhedra

There are nine uniform edge-transitive convex polyhedra in three dimensions:

• the five regular Platonic solids: the tetrahedron, octahedron, cube, dodecahedron and icosahedron;
• the two quasiregular Archimedean solids: the cuboctahedron and the icosidodecahedron; and
• two Catalan solids: the rhombic dodecahedron and the rhombic triacontahedron, which are duals to the only two quasiregular polyhedra.

Nine distinct stellation's by Miller's rules are produced by the truncated tetrahedron. ^[20] It is the simplest Archimedean solid, with a total of four equilateral triangular and four hexagonal faces.

Collectively, there are nine regular polyhedra in the third dimension, when extending the convex Platonic solids to include the concave regular star polyhedra known as the Kepler-Poinsot polyhedra. ^{[21] [22]}

Higher dimensions

In four-dimensional space, there are nine paracompact hyperbolic honeycomb Coxeter groups, as well as nine regular compact hyperbolic honeycombs from regular convex and star polychora . ^[23] There are also nine uniform demitesseractic (${\displaystyle \mathrm {D} _{4}}$) Euclidean honeycombs in the fourth dimension.

There are only three types of Coxeter groups of uniform figures in dimensions nine and thereafter, aside from the many families of prisms and proprisms: the ${\displaystyle \mathrm {A} _{n}}$ simplex groups, the ${\displaystyle \mathrm {B} _{n}}$ hypercube groups, and the ${\displaystyle \mathrm {D} _{n}}$ demihypercube groups. The ninth dimension is also the final dimension that contains Coxeter-Dynkin diagrams as uniform solutions in hyperbolic space. Inclusive of compact hyperbolic solutions, there are a total of 238 compact and paracompact Coxeter-Dynkin diagrams between dimensions two and nine, or equivalently between ranks three and ten. The most important of the last ${\displaystyle {\tilde {E}}_{9}}$ paracompact groups is the group ${\displaystyle {\tilde {T}}_{9}}$ with 1023 total honeycombs, the simplest of which is 6₂₁ whose vertex figure is the 5₂₁ honeycomb: the vertex arrangement of the densest-possible packing of spheres in 8 dimensions which forms the ${\displaystyle \mathbb {E} _{8}}$ lattice. The 6₂₁ honeycomb is made of 9-simplexes and 9-orthoplexes, with 1023 total polytope elements making up each 9-simplex. It is the final honeycomb figure with infinite facets and vertex figures in the k₂₁ family of semiregular polytopes, first defined by Thorold Gosset in 1900.

List of basic calculations

Multiplication	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	20	25	50	100	1000
9 × x	9	18	27	36	45	54	63	72	81	90	99	108	117	126	135	144	180	225	450	900	9000

Division	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
9 ÷ x	9	4.5	3	2.25	1.8	1.5	1.285714	1.125	1	0.9	0.81	0.75	0.692307	0.6428571	0.6
x ÷ 9	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	1	1.1	1.2	1.3	1.4	1.5	1.6

Exponentiation	1	2	3	4	5	6	7	8	9	10
9x	9	81	729	6561	59049	531441	4782969	43046721	387420489	3486784401
x9	1	512	19683	262144	1953125	10077696	40353607	134217728	387420489	1000000000

Radix	1	5	10	15	20	25	30	40	50	60	70	80	90	100
	110	120	130	140	150	200	250	500	1000	10000	100000	1000000
x9	1	5	119	169	229	279	339	449	559	669	779	889	1109	1219
	1329	1439	1549	1659	1769	2429	3079	6159	13319	146419	1621519	17836619

In base 10

9 is the highest single-digit number in the decimal system.

Divisibility

A positive number is divisible by nine if and only if its digital root is nine:

• 9 × 2 = 18 (1 + 8 = 9)
• 9 × 3 = 27 (2 + 7 = 9)
• 9 × 9 = 81 (8 + 1 = 9)
• 9 × 121 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9)
• 9 × 234 = 2106 (2 + 1 + 0 + 6 = 9)
• 9 × 578329 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = 27; 2 + 7 = 9)
• 9 × 482729235601 = 4344563120409 (4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45; 4 + 5 = 9)

That is, if any natural number is multiplied by 9, and the digits of the answer are repeatedly added until it is just one digit, the sum will be nine. ^[24]

In base-${\displaystyle N}$, the divisors of ${\displaystyle N-1}$ have this property.

Multiples of 9

There are other interesting patterns involving multiples of nine:

• 9 × 12345679 = 111111111
• 18 × 12345679 = 222222222
• 81 × 12345679 = 999999999

The difference between a base-10 positive integer and the sum of its digits is a whole multiple of nine. Examples:

• The sum of the digits of 41 is 5, and 41 − 5 = 36. The digital root of 36 is 3 + 6 = 9.
• The sum of the digits of 35967930 is 3 + 5 + 9 + 6 + 7 + 9 + 3 + 0 = 42, and 35967930 − 42 = 35967888. The digital root of 35967888 is 3 + 5 + 9 + 6 + 7 + 8 + 8 + 8 = 54, 5 + 4 = 9.

If dividing a number by the amount of 9s corresponding to its number of digits, the number is turned into a repeating decimal. (e.g. 274/999 = 0.274274274274...)

Another consequence of 9 being 10 − 1 is that it is a Kaprekar number, preceding the ninth and tenth triangle numbers, 45 and 55 (where all 9, 99, 999, 9999, ... are Keprekar numbers). ^[25]

Six recurring nines appear in the decimal places 762 through 767 of π. (See six nines in pi).

Alphabets and codes

• In the NATO phonetic alphabet, the digit 9 is called "Niner".
• Five-digit produce PLU codes that begin with 9 indicate organic foods.

Culture and mythology

Indian culture

Nine is a number that appears often in Indian culture and mythology. ^[26] Some instances are enumerated below.

• Nine influencers are attested in Indian astrology.
• In the Vaisheshika branch of Hindu philosophy, there are nine universal substances or elements: Earth, Water, Air, Fire, Ether, Time, Space, Soul, and Mind. ^[27]
• Navaratri is a nine-day festival dedicated to the nine forms of Durga. ^{[28] [29]}
• Navaratna, meaning "nine jewels" may also refer to Navaratnas – accomplished courtiers, Navratan – a kind of dish, or a form of architecture.
• In Indian aesthetics, there are nine kinds of Rasa.

Chinese culture

• Nine (九; pinyin :jiǔ) is considered a good number in Chinese culture because it sounds the same as the word "long-lasting" (久; pinyin :jiǔ). ^[30]
• Nine is strongly associated with the Chinese dragon, a symbol of magic and power. There are nine forms of the dragon, it is described in terms of nine attributes, and it has nine children. It has 117 scales – 81 yang (masculine, heavenly) and 36 yin (feminine, earthly). All three numbers are multiples of 9 (9 × 13 = 117, 9 × 9 = 81, 9 × 4 = 36) ^[31] as well as having the same digital root of 9.
• The dragon often symbolizes the Emperor, and the number nine can be found in many ornaments in the Forbidden City.
• The circular altar platform (Earthly Mount) of the Temple of Heaven has one circular marble plate in the center, surrounded by a ring of nine plates, then by a ring of 18 plates, and so on, for a total of nine rings, with the outermost having 81 = 9 × 9 plates.
• The name of the area called Kowloon in Hong Kong literally means: nine dragons .
• The nine-dotted line (Chinese :南海九段线; pinyin :nánhǎi jiǔduàn xiàn; lit.'Nine-segment line of the South China Sea') delimits certain island claims by China in the South China Sea.
• The nine-rank system was a civil service nomination system used during certain Chinese dynasties.
• 9 Points of the Heart (Heal) / Heart Master (Immortality) Channels in Traditional Chinese Medicine.

Ancient Egypt

• The nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt. ^[32]
• The Ennead is a group of nine Egyptian deities, who, in some versions of the Osiris myth, judged whether Horus or Set should inherit Egypt.

European culture

• In Norse mythology, the universe is divided into nine worlds which are all connected by the world tree Yggdrasil. ^[33]
• In Norse mythology as well, the number nine is associated with Odin, as that is how many days he hung from the world tree Yggdrasil before attaining knowledge of the runes.

Greek mythology

• The nine Muses in Greek mythology are Calliope (epic poetry), Clio (history), Erato (erotic poetry), Euterpe (lyric poetry), Melpomene (tragedy), Polyhymnia (song), Terpsichore (dance), Thalia (comedy), and Urania (astronomy).
• It takes nine days (for an anvil) to fall from heaven to earth, and nine more to fall from earth to Tartarus.
• Leto labored for nine days and nine nights for Apollo, according to the Homeric Hymn to Delian Apollo.

Mesoamerican mythology

• The Lords of the Night, is a group of nine deities who each ruled over every ninth night forming a calendrical cycle.

Aztec mythology

• Mictlan the underworld in Aztec mythology, consists of nine levels.

Mayan mythology

• The Mayan underworld Xibalba consists of nine levels.
• El Castillo, the Mayan step-pyramid in Chichén Itzá, consists of nine steps. It is said that this was done to represent the nine levels of Xibalba.

Australian culture

The Pintupi Nine, a group of 9 Aboriginal Australian women who remained unaware of European colonisation of Australia and lived a traditional desert-dwelling life in Australia's Gibson Desert until 1984.

Anthropology

Idioms

• "to go the whole nine yards-"
• "A cat-o'-nine-tails suggests perfect punishment and atonement." – Robert Ripley.
• "A cat has nine lives"
• "to be on cloud nine"
• " A stitch in time saves nine"
• "found true 9 out of 10 times"
• "possession is nine tenths of the law"
• The word "K-9" pronounces the same as canine and is used in many US police departments to denote the police dog unit. Despite not sounding like the translation of the word canine in other languages, many police and military units around the world use the same designation.
• Someone dressed "to the nines" is dressed up as much as they can be.
• In North American urban culture, "nine" is a slang word for a 9mm pistol or homicide, the latter from the Illinois Criminal Code for homicide.

Technique

International maritime signal flag for 9

Playing cards showing the 9 of all four suits

• Stanines, a method of scaling test scores, range from 1 to 9.
• There are 9 square feet in a square yard.

Literature

• There are nine circles of Hell in Dante's Divine Comedy .
• The Nine Bright Shiners, characters in Garth Nix's Old Kingdom trilogy. The Nine Bright Shiners was a 1930s book of poems by Anne Ridler ^[34] and a 1988 fiction book by Anthea Fraser; ^[35] the name derives from "a very curious old semi-pagan, semi-Christian" song. ^[36]
• The Nine Tailors is a 1934 mystery novel by British writer Dorothy L. Sayers, her ninth featuring sleuth Lord Peter Wimsey.
• Nine Unknown Men are, in occult legend, the custodians of the sciences of the world since ancient times.
• In J. R. R. Tolkien's Middle-earth, there are nine rings of power given to men, and consequently, nine ringwraiths. Additionally, The Fellowship of the Ring consists of nine companions.
• In Lorien Legacies there are nine Garde sent to Earth.
• Number Nine is a character in Lorien Legacies .
• In the series A Song of Ice and Fire , there are nine regions of Westeros (the Crownlands, the North, the Riverlands, the Westerlands, the Reach, the Stormlands, the Vale of Arryn, the Iron Islands and Dorne). Additionally, there is a group of nine city-states in western Essos known collectively as the Free Cities (Braavos, Lorath, Lys, Myr, Norvos, Pentos, Qohor, Tyrosh and Volantis).
• In The Wheel of Time series, Daughter of the Nine Moons is the title given to the heir to the throne of Seanchan, and the Court of the Nine Moons serves as the throne room of the Seanchan rulers themselves. Additionally, the nation of Illian is partially governed by a body known as the Council of Nine, and the flag of Illian displays nine golden bees on it. Furthermore, in the Age of Legends, the Nine Rods of Dominion were nine regional governors who administered individual areas of the world under the ruling world government.

Organizations

• Divine Nine – The National Pan-Hellenic Council (NPHC) is a collaborative organization of nine historically African American, international Greek-lettered fraternities and sororities.

Places and thoroughfares

• List of highways numbered 9
• Ninth Avenue is a major avenue in Manhattan.
• South Africa has 9 provinces
• Negeri Sembilan, a Malaysian state located in Peninsular Malaysia, is named as such as it was historically a confederation of nine (Malay : sembilan) settlements ( nagari ) of the Minangkabau migrated from West Sumatra.

Religion and philosophy

Christianity

• In Christianity there are nine Fruit of the Holy Spirit which followers are expected to have: love, joy, peace, patience, kindness, goodness, faithfulness, gentleness, and self-control. ^[37]
• The Bible recorded that Christ died at the 9th hour of the day (3 pm). ^[38]

Islam

There are three verses that refer to nine in the Quran.

We surely gave Moses nine clear signs.¹ ˹You, O Prophet, can˺ ask the Children of Israel. When Moses came to them, Pharaoh said to him, “I really think that you, O Moses, are bewitched.”

—  Surah Al-Isra (The Night Journey/Banī Isrāʾīl):101 ^[39]

Note 1: The nine signs of Moses are: the staff, the hand (both mentioned in Surah Ta-Ha 20:17-22), famine, shortage of crops, floods, locusts, lice, frogs, and blood (all mentioned in Surah Al-A'raf 7:130-133). These signs came as proofs for Pharaoh and the Egyptians. Otherwise, Moses had some other signs such as water gushing out of the rock after he hit it with his staff, and splitting the sea.

Now put your hand through ˹the opening of˺ your collar, it will come out ˹shining˺ white, unblemished.² ˹These are two˺ of nine signs for Pharaoh and his people. They have truly been a rebellious people.”

— Surah Al-Naml (The Ant):12 ^[40]

Note 2: Moses, who was dark-skinned, was asked to put his hand under his armpit. When he took it out it was shining white, but not out of a skin condition like melanoma.

And there were in the city nine ˹elite˺ men who spread corruption in the land, never doing what is right.

— Surah Al-Naml (The Ant):48 ^[41]

• Ramadan, the month of fasting and prayer, is the ninth month of the Islamic calendar.

A nine-pointed star

• Nine, as the highest single-digit number (in base ten), symbolizes completeness in the Baháʼí Faith. In addition, the word Baháʼ in the Abjad notation has a value of 9, and a 9-pointed star is used to symbolize the religion.
• The number 9 is revered in Hinduism and considered a complete, perfected and divine number because it represents the end of a cycle in the decimal system, which originated from the Indian subcontinent as early as 3000 BC.
• In Buddhism, Gautama Buddha was believed to have nine virtues, which he was (1) Accomplished, (2) Perfectly Enlightened, (3) Endowed with knowledge and Conduct or Practice, (4) Well-gone or Well-spoken, (5) the Knower of worlds, (6) the Guide Unsurpassed of men to be tamed, (7) the Teacher of gods and men, (8) Enlightened, and (9) Blessed.
• Important Buddhist rituals usually involve nine monks.
• The first nine days of the Hebrew month of Av are collectively known as "The Nine Days" (Tisha HaYamim), and are a period of semi-mourning leading up to Tisha B'Av, the ninth day of Av on which both Temples in Jerusalem were destroyed.
• Nine is a significant number in Norse Mythology. Odin hung himself on an ash tree for nine days to learn the runes.
• The Fourth Way Enneagram is one system of knowledge which shows the correspondence between the 9 integers and the circle.
• Tian's Trigram Number, of Feng Shui, in Taoism.

Science

Astronomy

• Before 2006 (when Pluto was officially designated as a non-planet), there were nine planets in the Solar System.
• Messier object M9 is a magnitude 9.0 globular cluster in the constellation Ophiuchus.
• The New General Catalogue object NGC 9, a spiral galaxy in the constellation Pegasus.

Chemistry

• The purity of chemicals (see Nine (purity)).
• Nine is the atomic number of fluorine.

Physiology

A human pregnancy normally lasts nine months, the basis of Naegele's rule.

Psychology

Common terminal digit in psychological pricing.

Sports

Billiards: A Nine-ball rack with the no. 9 ball at the center

• Nine-ball is the standard professional pocket billiards variant played in the United States.
• In association football (soccer), the centre-forward/striker traditionally (since at least the fifties) wears the number 9 shirt.
• In baseball:
  • There are nine players on the field including the pitcher.
  • There are nine innings in a standard game.
  • 9 represents the right fielder's position.
  • NINE: A Journal of Baseball History and Culture, published by the University of Nebraska Press ^[42]
• In rugby league, the jersey number assigned to the hooker in most competitions. (An exception is the Super League, which uses static squad numbering.)
• In rugby union, the number worn by the starting scrum-half.

Technology

• ISO 9 is the ISO's standard for the transliteration of Cyrillic characters into Latin characters
• In the Rich Text Format specification, 9 is the language code for the English language. All codes for regional variants of English are congruent to 9 mod 256.
• The9 Limited (owner of the9.com) is a company in the video-game industry, including former ties to the extremely popular MMORPG World of Warcraft .

Music

• "Revolution 9", a sound collage which appears on The Beatles' eponymous 1968 album The Beatles (aka The White Album), prominently features a loop of a man's voice repeating the phrase "Number nine". ^[43]
• There are 9 semitones in a Major 6th interval in music. ^[44]
• There was a superstition among some notable classical music composers that they would die after completing their ninth symphony. Some composers who died after composing their ninth symphony include Ludwig van Beethoven, Anton Bruckner, Antonin Dvorak and Gustav Mahler. ^[45]
• Beethoven's Symphony No. 9 is regarded as a masterpiece, and one of the most frequently performed symphonies in the world.
• A staff of a score contains nine notes who comprise five lines and four spaces.

See also

• Mathematics portal

• 9 (disambiguation)
• 0.999...
• Cloud Nine
• List of highways numbered 9

References

1. Lippman, David (12 July 2021). "6.0.2: The Hindu-Arabic Number System". Mathematics LibreTexts. Retrieved 31 March 2024.
2. Sloane, N. J. A. (ed.). "SequenceA001358(Semiprimes (or biprimes): products of two primes.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 27 February 2024.
3. Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN   0-8218-2102-4. p.91
4. Mihăilescu, Preda (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. 572. Berlin: De Gruyter: 167–195. doi:10.1515/crll.2004.048. MR   2076124. S2CID   121389998.
5. Metsänkylä, Tauno (2004). "Catalan's conjecture: another old Diophantine problem solved" (PDF). Bulletin of the American Mathematical Society . 41 (1). Providence, R.I.: American Mathematical Society: 43–57. doi: 10.1090/S0273-0979-03-00993-5 . MR   2015449. S2CID   17998831. Zbl   1081.11021.
6. Sloane, N. J. A. (ed.). "SequenceA000537(Sum of first n cubes; or n-th triangular number squared.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 19 June 2023.
7. "Sloane's A049384 : a(0)=1, a(n+1) = (n+1)^a(n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 1 June 2016.
8. Sloane, N. J. A. (ed.). "SequenceA000166(Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 10 December 2022.
9. Sloane, N. J. A. (ed.). "SequenceA033950(Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 19 June 2023.
10. Davenport, H. (1939), "On Waring's problem for cubes", Acta Mathematica , 71, Somerville, MA: International Press of Boston: 123–143, doi: 10.1007/BF02547752 , MR   0000026, S2CID   120792546, Zbl   0021.10601
11. Pace P., Nielsen (2007). "Odd perfect numbers have at least nine distinct prime factors". Mathematics of Computation . 76 (260). Providence, R.I.: American Mathematical Society: 2109–2126. arXiv: math/0602485 . Bibcode:2007MaCom..76.2109N. doi: 10.1090/S0025-5718-07-01990-4 . MR   2336286. S2CID   2767519. Zbl   1142.11086.
12. "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 1 June 2016.
13. William H. Richardson. "Magic Squares of Order 3". Wichita State University Dept. of Mathematics. Retrieved 6 November 2022.
14. Sloane, N. J. A. (ed.). "SequenceA006003(Also the sequence M(n) of magic constants for n X n magic squares)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 8 December 2022.
15. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
16. Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
17. Gleason, Andrew M. (1988). "Angle trisection, the heptagon, and the triskaidecagon". American Mathematical Monthly . 95 (3). Taylor & Francis, Ltd: 191–194. doi:10.2307/2323624. JSTOR   2323624. MR   0935432. S2CID   119831032.
18. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine . 50 (5). Taylor & Francis, Ltd.: 228–234. doi:10.2307/2689529. JSTOR   2689529. S2CID   123776612. Zbl   0385.51006.
19. Sloane, N. J. A. (ed.). "SequenceA219766(Number of nonsquare simple perfect squared rectangles of order n up to symmetry)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
20. Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 26 November 2022. Retrieved 15 December 2022.
21. Weisstein, Eric W. "Regular Polyhedron". Mathworld -- A WolframAlpha Resource. Retrieved 27 February 2024.
22. Coxeter, H. S. M. (1948). Regular Polytopes (1st ed.). London: Methuen & Co., Ltd. p. 93. ISBN   0-486-61480-8. MR   0027148. OCLC   798003.
23. Coxeter, H. S. M. (1956), "Regular honeycombs in hyperbolic space", Proceedings of the International Congress of Mathematicians, vol. III, Amsterdam: North-Holland Publishing Co., pp. 167–169, MR   0087114
24. Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155
25. Sloane, N. J. A. (ed.). "SequenceA006886(Kaprekar numbers.)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 27 February 2024.
26. DHAMIJA, ANSHUL (16 May 2018). "The Auspiciousness Of Number 9". Forbes India. Retrieved 1 April 2024.
27. "Vaisheshika | Atomism, Realism, Dualism | Britannica". www.britannica.com. Retrieved 13 April 2024.
28. "Navratri | Description, Importance, Goddess, & Facts | Britannica". www.britannica.com. 11 April 2024. Retrieved 13 April 2024.
29. Lochtefeld, James G. (2002). The illustrated encyclopedia of hinduism. New York: the Rosen publ. group. ISBN   978-0-8239-2287-1.
30. "Lucky Number Nine, Meaning of Number 9 in Chinese Culture". www.travelchinaguide.com. Retrieved 15 January 2021.
31. Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN   1-4179-6429-4.
32. "The Global Egyptian Museum | Nine Bows". www.globalegyptianmuseum.org. Retrieved 16 November 2023.
33. Mark, Joshua J. "Nine Realms of Norse Cosmology". World History Encyclopedia. Retrieved 16 November 2023.
34. Jane Dowson (1996). Women's Poetry of the 1930s: A Critical Anthology. Routledge. ISBN   0-415-13095-6.
35. Anthea Fraser (1988). The Nine Bright Shiners. Doubleday. ISBN   0-385-24323-5.
36. Charles Herbert Malden (1905). Recollections of an Eton Colleger, 1898–1902. Spottiswoode. p.  182. nine-bright-shiners.
37. Galatians 5:22–23
38. "Meaning of Numbers in the Bible The Number 9". Bible Study. Archived from the original on 17 November 2007.
39. "Surah Al-Isra - 101". Quran.com. Retrieved 17 August 2023.
40. "Surah An-Naml - 12". Quran.com. Retrieved 17 August 2023.
41. "Surah An-Naml - 48". Quran.com. Retrieved 17 August 2023.
42. "Web site for NINE: A Journal of Baseball History & Culture". Archived from the original on 4 November 2009. Retrieved 20 February 2013.
43. Glover, Diane (9 October 2019). "#9 Dream: John Lennon and numerology". www.beatlesstory.com. Beatles Story. Retrieved 6 November 2022. Perhaps the most significant use of the number 9 in John's music was the White Album's 'Revolution 9', an experimental sound collage influenced by the avant-garde style of Yoko Ono and composers such as Edgard Varèse and Karlheinz Stockhausen. It featured a series of tape loops including one with a recurring 'Number Nine' announcement. John said of 'Revolution 9': 'It's an unconscious picture of what I actually think will happen when it happens; just like a drawing of a revolution. One thing was an engineer's testing voice saying, 'This is EMI test series number nine.' I just cut up whatever he said and I'd number nine it. Nine turned out to be my birthday and my lucky number and everything. I didn't realise it: it was just so funny the voice saying, 'number nine'; it was like a joke, bringing number nine into it all the time, that's all it was.'
44. Truax, Barry (2001). Handbook for Acoustic Ecology (Interval). Burnaby: Simon Fraser University. ISBN   1-56750-537-6..
45. "The Curse of the Ninth Haunted These Composers | WQXR Editorial". WQXR. 17 October 2016. Retrieved 16 January 2022.

Further reading

• Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ISBN   3-7913-1933-7, ISBN   978-3-7913-1933-9