← 8 9 10 →
−1 0 1 2 3 4 5 6 7 8 9
Cardinalnine
Ordinal9th
(ninth)
Numeral systemnonary
Factorization32
Divisors1,3,9
Greek numeralΘ´
Roman numeralIX, ix
Greek prefixennea-
Latin prefixnona-
Binary10012
Ternary1003
Senary136
Octal118
Duodecimal912
Hexadecimal916
Amharic
Arabic, Kurdish, Persian, Sindhi, Urdu٩
Armenian numeralԹ
Bengali
Chinese numeral九, 玖
Devanāgarī
Greek numeralθ´
Hebrew numeralט
Tamil numerals
Khmer
Telugu numeral
Thai numeral
Malayalam

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 BCE, 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. 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 6. To disambiguate the two on objects and documents that can be inverted, they are often underlined. Another distinction from the 6 is that it is sometimes handwritten with two strokes and a straight stem, resembling a raised lower-case letter q. 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. 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.[1] 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.[2][3]

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.[4] It is also the sum of the first three nonzero factorials and equal to the third exponential factorial, since [5]

Non-intersecting chords between four points on a circle

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

9 is a Motzkin number, for the number of ways of drawing non-intersecting chords between four points on a circle.[10]

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 x magic square made of nine cells, with a magic constant of 15.[11] Meanwhile, a x magic square has a magic constant of 369.[12]

A polygon with nine sides is called a nonagon.[13] Since 9 can be written in the form , for any nonnegative natural integers and with a product of Pierpont primes, a regular nonagon can be constructed with a regular compass, straightedge, and angle trisector.[14]

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.[15]

There are nine distinct uniform colorings of the triangular tiling and the square tiling, which are the two simplest regular tilings; the hexagonal tiling, on the other hand, has three distinct uniform colorings.

There are a maximum of nine semiregular Archimedean tilings by convex regular polygons, when including chiral forms of the snub hexagonal tiling.

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

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.

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.[17] 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.

In decimal

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

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

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.[18]

In base-, the divisors of have such a property, which makes 3 the only other number aside from 9 in decimal that shares this property. Another consequence of 9 being 10 − 1 is that it is a Kaprekar number.

There are other interesting patterns involving multiples of nine:

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

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...)

Casting out nines is a quick way of testing the calculations of sums, differences, products, and quotients of integers known as long ago as the 12th century.[19]

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

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

Alphabets and codes

Culture and mythology

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Indian culture

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

Chinese culture

Ancient Egypt

European culture

Greek mythology

Mesoamerican mythology

Aztec mythology

Mayan mythology

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

Technique

International maritime signal flag for 9
Playing cards showing the 9 of all four suits

Literature

Organizations

Places and thoroughfares

Religion and philosophy

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[27]

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[28]

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[29]
A nine-pointed star
A nine-pointed star

Science

Astronomy

Chemistry

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

Technology

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Music

See also

References

  1. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
  2. ^ Mihăilescu, Preda (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. Berlin: De Gruyter. 572: 167–195. doi:10.1515/crll.2004.048. MR 2076124. S2CID 121389998.
  3. ^ Metsänkylä, Tauno (2004). "Catalan's conjecture: another old Diophantine problem solved" (PDF). Bulletin of the American Mathematical Society. Providence, R.I.: American Mathematical Society. 41 (1): 43–57. doi:10.1090/S0273-0979-03-00993-5. MR 2015449. S2CID 17998831. Zbl 1081.11021.
  4. ^ 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.
  5. ^ "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.
  6. ^ Sloane, N. J. A. (ed.). "Sequence A000166 (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.
  7. ^ 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.
  8. ^ Davenport, H. (1939), "On Waring's problem for cubes", Acta Mathematica, Somerville, MA: International Press of Boston, 71: 123–143, doi:10.1007/BF02547752, MR 0000026, S2CID 120792546, Zbl 0021.10601
  9. ^ Pace P., Nielsen (2007). "Odd perfect numbers have at least nine distinct prime factors". Mathematics of Computation. Providence, R.I.: American Mathematical Society. 76 (260): 2109–2126. arXiv:math/0602485. Bibcode:2007MaCom..76.2109N. doi:10.1090/S0025-5718-07-01990-4. MR 2336286. S2CID 2767519. Zbl 1142.11086.
  10. ^ "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 1 June 2016.
  11. ^ William H. Richardson. "Magic Squares of Order 3". Wichita State University Dept. of Mathematics. Retrieved 6 November 2022.
  12. ^ Sloane, N. J. A. (ed.). "Sequence A006003 (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.
  13. ^ Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
  14. ^ Gleason, Andrew M. (1988). "Angle trisection, the heptagon, and the triskaidecagon". American Mathematical Monthly. Taylor & Francis, Ltd. 95 (3): 191–194. doi:10.2307/2323624. JSTOR 2323624. MR 0935432. S2CID 119831032.
  15. ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. Taylor & Francis, Ltd. 50 (5): 228-234. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
  16. ^ Webb, Robert. "Enumeration of Stellations". www.software3d.com. Archived from the original on 25 November 2022. Retrieved 15 December 2022.
  17. ^ 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
  18. ^ Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155
  19. ^ Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
  20. ^ "Lucky Number Nine, Meaning of Number 9 in Chinese Culture". www.travelchinaguide.com. Retrieved 15 January 2021.
  21. ^ Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN 1-4179-6429-4.
  22. ^ "The Global Egyptian Museum | Nine Bows". www.globalegyptianmuseum.org. Retrieved 16 November 2023.
  23. ^ Mark, Joshua J. "Nine Realms of Norse Cosmology". World History Encyclopedia. Retrieved 16 November 2023.
  24. ^ Jane Dowson (1996). Women's Poetry of the 1930s: A Critical Anthology. Routledge. ISBN 0-415-13095-6.
  25. ^ Anthea Fraser (1988). The Nine Bright Shiners. Doubleday. ISBN 0-385-24323-5.
  26. ^ Charles Herbert Malden (1905). Recollections of an Eton Colleger, 1898–1902. Spottiswoode. p. 182. nine-bright-shiners.
  27. ^ "Surah Al-Isra - 101". Quran.com. Retrieved 17 August 2023.
  28. ^ "Surah An-Naml - 12". Quran.com. Retrieved 17 August 2023.
  29. ^ "Surah An-Naml - 48". Quran.com. Retrieved 17 August 2023.
  30. ^ "Meaning of Numbers in the Bible The Number 9". Bible Study. Archived from the original on 17 November 2007.
  31. ^ "Web site for NINE: A Journal of Baseball History & Culture". Archived from the original on 4 November 2009. Retrieved 20 February 2013.
  32. ^ 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.'
  33. ^ Truax, Barry (2001). Handbook for Acoustic Ecology (Interval). Burnaby: Simon Fraser University. ISBN 1-56750-537-6..
  34. ^ "The Curse of the Ninth Haunted These Composers | WQXR Editorial". WQXR. 17 October 2016. Retrieved 16 January 2022.

Further reading