9

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8 9 10
−1 0 1 2 3 4 5 6 7 8 9
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
Arabic, Kurdish, Persian, Sindhi, Urdu ٩
Armenian numeral Թ
Bengali
Chinese numeral 九, 玖
Devanāgarī
Santali
Greek numeral θ´
Hebrew numeral ט
Tamil numerals
Khmer
Telugu script
Thai numeral
Malayalam
Babylonian numeral 𒐝
Egyptian hieroglyph 𓐂
Morse code ____.
NATO phonetic alphabet NINER

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

Contents

Evolution of the Hindu–Arabic digit

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 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 TextFigs196.png .

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.

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.

Mathematics

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]

9 is the only square number that is the sum of two consecutive, positive cubes: [4]

If an odd perfect number exists, it will have at least nine distinct prime factors. [5]

Non-intersecting chords between four points on a circle MotzkinChords4.svg
Non-intersecting chords between four points on a circle

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] A number that is 4 or 5 modulo 9 cannot be represented as the sum of three cubes. [7]

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. Yang Hui magic circle.svg
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 that yield an imaginary quadratic field whose ring of integers has a unique factorization, or class number of 1. [8]

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

The "nine dots" puzzle. The puzzle asks to link all nine dots using four straight lines or fewer, without lifting the pen. 9dots.svg
The "nine dots" puzzle. The puzzle asks to link all nine dots using four straight lines or fewer, without lifting the pen.

The nine dots puzzle is a mathematical puzzle whose task is to connect nine squarely arranged points with a pen by four (or fewer) straight lines without lifting the pen or retracing any lines.

Geometry

A polygon with nine sides is called a nonagon. [9] A regular nonagon can be constructed with a regular compass, straightedge, and angle trisector. [10]

The lowest number of squares needed for a perfect tiling of a rectangle is 9. [11]

List of basic calculations

Multiplication 123456789101112131415162025501001000
9 × x9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 180 225 450 900 9000
Division 123456789101112131415
9 ÷ x94.532.251.81.51.2857141.12510.90.810.750.6923070.64285710.6
x ÷ 90.10.20.30.40.50.60.70.811.11.21.31.41.51.6
Exponentiation 12345678910
9x9817296561590495314414782969430467213874204893486784401
x91 512 1968326214419531251007769640353607134217728387420489 1000000000

Culture and mythology

Indian culture

Nine is a number that appears often in Indian culture and mythology. [12] 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. [13] Navaratri is a nine-day festival dedicated to the nine forms of Durga. [14] [15]

Chinese culture

Anthropology

Idioms

International maritime signal flag for 9 ICS Niner.svg
International maritime signal flag for 9

Religion and philosophy

A nine-pointed star Bahai star.svg
A nine-pointed star

Science

Chemistry

This gold has four nines, as it is 99.99% pure Four nines fine gold.jpeg
This gold has four nines, as it is 99.99% pure

Nines are a notation for expressing the purity of a chemical.

In chemical nomenclature, the prefix "non-" stands for the number nine, for example in molecular chains with nine atoms such as "nonane" for an alkane with nine carbon atoms. The prefixes play a role in the naming of molecules, especially in organic chemistry to indicate the number of similar atoms or groups in the molecule (e.g. "nonan" for 9 carbon atoms, "nonavalent" for nine-valent. Nonapeptides are peptides that consist of nine amino acids.

Physiology

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

Psychology

Common terminal digit in psychological pricing.

See also

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.). "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.
  3. Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN   0-8218-2102-4. p.91
  4. Wells, David (1997). The Penguin Dictionary of Curious and Interesting Numbers (2nd ed.). Penguin. ISBN   0-14-026149-4.
  5. 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.
  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. 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
  8. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
  9. Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
  10. 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.
  11. 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.
  12. DHAMIJA, ANSHUL (16 May 2018). "The Auspiciousness Of Number 9". Forbes India. Retrieved 1 April 2024.
  13. "Vaisheshika | Atomism, Realism, Dualism | Britannica". www.britannica.com. Retrieved 13 April 2024.
  14. "Navratri | Description, Importance, Goddess, & Facts | Britannica". www.britannica.com. 11 April 2024. Retrieved 13 April 2024.
  15. Lochtefeld, James G. (2002). The illustrated encyclopedia of Hinduism. New York: the Rosen publ. group. ISBN   978-0-8239-2287-1.
  16. "Lucky Number Nine, Meaning of Number 9 in Chinese Culture". www.travelchinaguide.com. Retrieved 15 January 2021.
  17. Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN   1-4179-6429-4.

Further reading