9

{{See also|Hindu–Arabic numeral system}}{{Verification section|date=September 2024}}File:Evo9glyph.svg

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 {{num|3}}-look-alike.{{Cite web |last=Lippman |first=David |date=2021-07-12 |title=6.0.2: The Hindu-Arabic Number System |url=https://math.libretexts.org/Courses/Cosumnes_River_College/Math_300%3A_Mathematical_Ideas_Textbook_(Muranaka)/06%3A_Miscellaneous_Extra_Topics/6.00%3A_Historical_Counting_Systems/6.0.02%3A_The_Hindu-Arabic_Number_System |access-date=2024-03-31 |website=Mathematics LibreTexts |language=en}} 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 45px.

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.

File:Seven-segment 9.svg

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.{{Cite OEIS|A033950|Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.|access-date=2023-06-19}}

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.Cajori, Florian (1991, 5e) A History of Mathematics, AMS. {{ISBN|0-8218-2102-4}}. p.91

9 is the only square number that is the sum of two consecutive, positive cubes: $3^2\; =\; 9\; =\; 1^3\; +\; 2^3${{cite book |last1=Wells |first1=David |title=The Penguin Dictionary of Curious and Interesting Numbers |date=1997 |publisher=Penguin |isbn=0-14-026149-4 |edition=2nd |language=English}}

If an odd perfect number exists, it will have at least nine distinct prime factors.{{Cite journal |last=Pace P. |first=Nielsen |year=2007 |title=Odd perfect numbers have at least nine distinct prime factors |url=https://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-01990-4/ |journal=Mathematics of Computation |location=Providence, R.I. |publisher=American Mathematical Society |volume=76 |issue=260 |pages=2109–2126 |arxiv=math/0602485 |bibcode=2007MaCom..76.2109N |doi=10.1090/S0025-5718-07-01990-4 |mr=2336286 |s2cid=2767519 |zbl=1142.11086 |doi-access=free}}

File:MotzkinChords4.svg between four points on a circle]]

9 is the sum of the cubes of the first two non-zero positive integers $1^\{3\}\; +\; 2^\{3\}$ which makes it the first cube-sum number greater than one.{{Cite OEIS |A000537 |Sum of first n cubes; or n-th triangular number squared. |access-date=2023-06-19 }} A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes.{{citation |last=Davenport|first=H. |author-link=Harold Davenport |title=On Waring's problem for cubes |journal=Acta Mathematica |publisher=International Press of Boston |location=Somerville, MA |volume=71 |year=1939 |pages=123–143 |doi=10.1007/BF02547752 |doi-access=free |mr=0000026 |zbl=0021.10601 |s2cid=120792546 }}

File:Yang Hui magic circle.svgs 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 $n$ that yield an imaginary quadratic field $\backslash Q\backslash left[\backslash sqrt\{-n\}\backslash right]$ whose ring of integers has a unique factorization, or class number of 1.Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93

A polygon with nine sides is called a nonagon.Robert Dixon, Mathographics. New York: Courier Dover Publications: 24 A regular nonagon can be constructed with a regular compass, straightedge, and angle trisector.{{Cite journal|last=Gleason |first=Andrew M. |author-link=Andrew M. Gleason |url=https://www.tandfonline.com/doi/abs/10.1080/00029890.1988.11971989?journalCode=uamm20 |title=Angle trisection, the heptagon, and the triskaidecagon |journal=American Mathematical Monthly |volume= 95 |issue=3 |year= 1988 |publisher=Taylor & Francis, Ltd |pages=191–194 |doi=10.2307/2323624 |mr=935432 |jstor=2323624 |s2cid=119831032 }}

The lowest number of squares needed for a perfect tiling of a rectangle is 9.{{cite OEIS|A219766|Number of nonsquare simple perfect squared rectangles of order n up to symmetry}}

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

{{More citations needed|date=November 2023}}

Nine is a number that appears often in Indian culture and mythology.{{Cite web |last=DHAMIJA |first=ANSHUL |date=May 16, 2018 |title=The Auspiciousness Of Number 9 |url=https://www.forbesindia.com/article/9th-anniversary-special/the-auspiciousness-of-number-9/50205/1 |access-date=2024-04-01 |website=Forbes India |language=en}} For example, there are nine influencers attested to 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.{{Cite web |title=Vaisheshika {{!}} Atomism, Realism, Dualism {{!}} Britannica |url=https://www.britannica.com/topic/Vaisheshika |access-date=2024-04-13 |website=www.britannica.com |language=en}} And Navaratri is a nine-day festival dedicated to the nine forms of Durga.{{Cite web |date=2024-04-11 |title=Navratri {{!}} Description, Importance, Goddess, & Facts {{!}} Britannica |url=https://www.britannica.com/topic/Navratri |access-date=2024-04-13 |website=www.britannica.com |language=en}}{{Cite book |last=Lochtefeld |first=James G. |title=The illustrated encyclopedia of Hinduism |date=2002 |publisher=the Rosen publ. group |isbn=978-0-8239-2287-1 |location=New York}}

• Nine ({{lang|zh|九}}; {{Lang-zh|p=jiǔ}}) is considered a good number in Chinese culture because it sounds the same as the word "long-lasting" ({{lang|zh|久}}; {{Lang-zh|p=jiǔ}}).{{Cite web |title=Lucky Number Nine, Meaning of Number 9 in Chinese Culture |url=https://www.travelchinaguide.com/intro/lucky-number9.htm |access-date=15 January 2021 |website=www.travelchinaguide.com}}
• 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 ({{nowrap|1=9 × 13 = 117}}, {{nowrap|1=9 × 9 = 81}}, {{nowrap|1=9 × 4 = 36}}).{{cite book |title=Myths of China And Japan |author=Donald Alexander Mackenzie |url=https://books.google.com/books?id=vzbeLy4TBa4C&q=chinese+dragon+scales+yin+36&pg=PA46 |publisher=Kessinger |year=2005 |isbn=1-4179-6429-4}}

• "To go the whole nine yards"
• "A cat has nine lives"
• "To be on cloud nine"
• 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.

File:ICS Niner.svg for 9]]

File:Bahai star.svg

• 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 Norse mythology, 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.
• Nine is the number associated with Satan in LaVeyan Satanism. Anton LaVey wrote in The Satanic Rituals that this is because nine is the number of the ego since it "always returns to itself" even after being multiplied by any number.

The purity of chemicals (see Nine (purity)).

The centrioles of many living organisms, including humans, contain a 9-fold symmetry.

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

Common terminal digit in psychological pricing.

{{Wiktionary|nine}}

{{Portal|Mathematics}}

• 9 (disambiguation)
• 0.999...
• Cloud Nine

{{Reflist|35em}}

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

{{Integers|zero}}

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Category:Integers

Category:9 (number)

Category:Superstitions about numbers