9 (original) (raw)

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Integer number 9

Natural number

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

Evolution of the Hindu–Arabic digit

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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.

9 is the fourth composite number, and the first odd composite number. 9 is also a refactorable number.[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]

If an odd perfect number exists, it will have at least nine distinct prime factors.[4]

Non-intersecting chords between four points on a circle

9 is the sum of the cubes of the first two non-zero positive integers 1 3 + 2 3 {\displaystyle 1^{3}+2^{3}} which makes it the first cube-sum number greater than one.[5] A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes.[6]

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.

There are nine Heegner numbers, or square-free positive integers n {\displaystyle n} that yield an imaginary quadratic field Q [ − n ] {\displaystyle \mathbb {Q} \left[{\sqrt {-n}}\right]} whose ring of integers has a unique factorization, or class number of 1.[7]

A polygon with nine sides is called a nonagon.[8] A regular nonagon can be constructed with a regular compass, straightedge, and angle trisector.[9]

The lowest number of squares needed for a perfect tiling of a rectangle is 9.[10]

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

List of basic calculations

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Alphabets and codes

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• 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

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Nine is a number that appears often in Indian culture and mythology.[11] 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.[12]
• Navaratri is a nine-day festival dedicated to the nine forms of Durga.[13][14]
• 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.
• 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).[15]
• 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)[16] 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.
• The nine bows is a term used in Ancient Egypt to represent the traditional enemies of Egypt.[17]
• 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.
• In Norse mythology, the universe is divided into nine worlds which are all connected by the world tree Yggdrasil.[18]
• 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.
• 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

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• The Lords of the Night, is a group of nine deities who each ruled over every ninth night forming a calendrical cycle.
• Mictlan the underworld in Aztec mythology, consists of nine levels.
• 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.

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.

• "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.

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.

Religion and philosophy

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• 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.[19]
• The Bible recorded that Christ died at the 9th hour of the day (3 pm).[20]

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[21]

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[23]

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

A nine-pointed star

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

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

Common terminal digit in psychological pricing.

Look up nine in Wiktionary, the free dictionary.

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

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.). "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.
3. ^ Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
4. ^ 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.
5. ^ 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.
6. ^ 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
7. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
8. ^ Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
9. ^ 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.
10. ^ 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.
11. ^ DHAMIJA, ANSHUL (16 May 2018). "The Auspiciousness Of Number 9". Forbes India. Retrieved 1 April 2024.
12. ^ "Vaisheshika | Atomism, Realism, Dualism | Britannica". www.britannica.com. Retrieved 13 April 2024.
13. ^ "Navratri | Description, Importance, Goddess, & Facts | Britannica". www.britannica.com. 11 April 2024. Retrieved 13 April 2024.
14. ^ Lochtefeld, James G. (2002). The illustrated encyclopedia of hinduism. New York: the Rosen publ. group. ISBN 978-0-8239-2287-1.
15. ^ "Lucky Number Nine, Meaning of Number 9 in Chinese Culture". www.travelchinaguide.com. Retrieved 15 January 2021.
16. ^ Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN 1-4179-6429-4.
17. ^ "The Global Egyptian Museum | Nine Bows". www.globalegyptianmuseum.org. Retrieved 16 November 2023.
18. ^ Mark, Joshua J. "Nine Realms of Norse Cosmology". World History Encyclopedia. Retrieved 16 November 2023.
19. ^ Galatians 5:22–23
20. ^ "Meaning of Numbers in the Bible The Number 9". Bible Study. Archived from the original on 17 November 2007.
21. ^ "Surah Al-Isra - 101". Quran.com. Retrieved 17 August 2023.
22. ^ "Surah An-Naml - 12". Quran.com. Retrieved 17 August 2023.
23. ^ "Surah An-Naml - 48". Quran.com. Retrieved 17 August 2023.

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