9 (nine) is the natural number following 8 and preceding 10.
Evolution of the Hindu–Arabic digit
editThis article needs additional citations for verification. (March 2024) |
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 form of the number nine (9) could possibly derived from the Arabic letter waw, in which its isolated form (و) resembles the number 9.
The modern digit resembles an inverted 6. 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
editNine is the fourth composite number, and the first composite number that is odd. Nine is the third square number (32).[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]
9 is the sum of the cubes of the first two non-zero positive integers which makes it the first cube-sum number greater than one.[6]
9 is the fourth refactorable number, as it has exactly three positive divisors, and 3 is one of them.[7]
A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes.[8]
If an odd perfect number exists, it will have at least nine distinct prime factors.[9]
The first non-trivial magic square is a x magic square made of nine cells, with a magic constant of 15.[10]
There are nine Heegner numbers, or square-free positive integers that yield an imaginary quadratic field whose ring of integers has a unique factorization, or class number of 1.[11]
Geometry
editPolygons and tilings
editThe regular hexagon, which is one of only three polygons to tile the plane with copies of itself laid edge-to-edge, contains a total of nine diagonals.
A polygon with nine sides is called a nonagon.[12] Since 9 can be written in the form , for any nonnegative natural integers and with a product of Pierpont primes, a regular nonagon is constructed with a regular compass, straightedge, and angle trisector.[13] 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.[14] 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.[15]
Polyhedra
editThere 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.[16] 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.[17][18]
Higher dimensions
editIn 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.[19] There are also nine uniform demitesseractic ( ) 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 simplex groups, the hypercube groups, and the 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 paracompact groups is the group 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 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
editMultiplication | 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 |
Base 10
9 is the largest single-digit number in the decimal system.
Alphabets and codes
edit- 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
editThis article needs additional citations for verification. (November 2023) |
Indian culture
editNine is a number that appears often in Indian culture and mythology.[20] 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.[21]
- Navaratri is a nine-day festival dedicated to the nine forms of Durga.[22][23]
- 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
edit- 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).[24] - 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)[25] 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
edit- The nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt.[26]
- 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
edit- In Norse mythology, the universe is divided into nine worlds which are all connected by the world tree Yggdrasil.[27]
- 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
edit- 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
edit- The Lords of the Night, is a group of nine deities who each ruled over every ninth night forming a calendrical cycle.
Aztec mythology
edit- Mictlan the underworld in Aztec mythology, consists of nine levels.
Mayan mythology
edit- 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
editThe 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
editIdioms
edit- "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
edit- Stanines, a method of scaling test scores, range from 1 to 9.
- There are 9 square feet in a square yard.
Religion and philosophy
editChristianity
edit- 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.[28]
- The Bible recorded that Christ died at the 9th hour of the day (3 pm).[29]
Islam
editThere 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.”
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.”
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[32]
- Ramadan, the month of fasting and prayer, is the ninth month of the Islamic calendar.
Other
edit- Nine, as the largest 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
editAstronomy
edit- 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
edit- The purity of chemicals (see Nine (purity)).
- Nine is the atomic number of fluorine.
Physiology
editA human pregnancy normally lasts nine months, the basis of Naegele's rule.
Psychology
editCommon terminal digit in psychological pricing.
See also
editReferences
edit- ^ Lippman, David (12 July 2021). "6.0.2: The Hindu-Arabic Number System". Mathematics LibreTexts. Retrieved 31 March 2024.
- ^ Sloane, N. J. A. (ed.). "Sequence A001358 (Semiprimes (or biprimes): products of two primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 27 February 2024.
- ^ Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
- ^ 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.
- ^ 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.
- ^ Sloane, N. J. A. (ed.). "Sequence A000537 (Sum of first n cubes; or n-th triangular number squared.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 19 June 2023.
- ^ Sloane, N. J. A. (ed.). "Sequence A033950 (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.
- ^ 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
- ^ 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.
- ^ William H. Richardson. "Magic Squares of Order 3". Wichita State University Dept. of Mathematics. Retrieved 6 November 2022.
- ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
- ^ Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
- ^ 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.
- ^ 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.
- ^ Sloane, N. J. A. (ed.). "Sequence A219766 (Number of nonsquare simple perfect squared rectangles of order n up to symmetry)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 26 November 2022. Retrieved 15 December 2022.
- ^ Weisstein, Eric W. "Regular Polyhedron". Mathworld -- A WolframAlpha Resource. Retrieved 27 February 2024.
- ^ Coxeter, H. S. M. (1948). Regular Polytopes (1st ed.). London: Methuen & Co., Ltd. p. 93. ISBN 0-486-61480-8. MR 0027148. OCLC 798003.
- ^ 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
- ^ DHAMIJA, ANSHUL (16 May 2018). "The Auspiciousness Of Number 9". Forbes India. Retrieved 1 April 2024.
- ^ "Vaisheshika | Atomism, Realism, Dualism | Britannica". www.britannica.com. Retrieved 13 April 2024.
- ^ "Navratri | Description, Importance, Goddess, & Facts | Britannica". www.britannica.com. 11 April 2024. Retrieved 13 April 2024.
- ^ Lochtefeld, James G. (2002). The illustrated encyclopedia of hinduism. New York: the Rosen publ. group. ISBN 978-0-8239-2287-1.
- ^ "Lucky Number Nine, Meaning of Number 9 in Chinese Culture". www.travelchinaguide.com. Retrieved 15 January 2021.
- ^ Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN 1-4179-6429-4.
- ^ "The Global Egyptian Museum | Nine Bows". www.globalegyptianmuseum.org. Retrieved 16 November 2023.
- ^ Mark, Joshua J. "Nine Realms of Norse Cosmology". World History Encyclopedia. Retrieved 16 November 2023.
- ^ Galatians 5:22–23
- ^ "Meaning of Numbers in the Bible The Number 9". Bible Study. Archived from the original on 17 November 2007.
- ^ "Surah Al-Isra - 101". Quran.com. Retrieved 17 August 2023.
- ^ "Surah An-Naml - 12". Quran.com. Retrieved 17 August 2023.
- ^ "Surah An-Naml - 48". Quran.com. Retrieved 17 August 2023.
Further reading
edit- Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ISBN 3-7913-1933-7, ISBN 978-3-7913-1933-9