9 (number)

This article is about the number. For other uses, see 9 (disambiguation).
8 9 10
−1 0 1 2 3 4 5 6 7 8 9
Cardinal nine
Ordinal 9th
(ninth)
Numeral system nonary
Factorization 32
Roman numeral IX
Unicode symbol(s) Ⅸ, ⅸ
Greek prefix ennea-
Latin prefix nona-
Binary 10012
Ternary 1003
Quaternary 214
Quinary 145
Senary 136
Octal 118
Duodecimal 912
Hexadecimal 916
Vigesimal 920
Base 36 936
Amharic
Arabicl & Kurdish ٩
Urdu ۹
Armenian numeral Թ
Bengali
Chinese/Japanese
/Korean numeral
九 (jiu)
玖 (formal writing)
Devanāgarī (nau)
Greek numeral θ´
Hebrew numeral ט (Tet)
Tamil numerals
Khmer
Telugu numeral
Thai numeral
Look up nine in Wiktionary, the free dictionary.

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

Alphabets and codes

Commerce

Culture and mythology

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

Evolution of the glyph

According to Georges Ifrah, the origin of the 9 integers can be attributed to the ancient Indian civilization, and was adopted by subsequent civilizations in conjunction with the 0.[2]

In the beginning, various Indians wrote 9 similar 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 @ character 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 9 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .

This numeral resembles an inverted 6. To disambiguate the two on objects and documents that can be inverted, the 9 is often underlined, as is done for the 6. Another distinction from the 6 is that it is sometimes handwritten with a straight stem, resembling a raised lower-case letter q.

Idioms and popular phrases

Internet

Literature

Mathematics

Nine is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and hence the third square number. Nine is a Motzkin number.[6] It is the first composite lucky number, along with the first composite odd number.

9 is the only positive perfect power that is one more than another positive perfect power, by Mihăilescu's Theorem.

Nine is the highest single-digit number in the decimal system. It is the second non-unitary square prime of the form (p2) and the first that is odd. All subsequent squares of this form are odd. It has a unique aliquot sum 4 which is itself a square prime. Nine is; and can be, the only square prime with an aliquot sum of the same form. The aliquot sequence of nine has 5 members (9,4,3,1,0) this number being the second composite member of the 3-aliquot tree. It is the aliquot sum of only one number the discrete semiprime 15.

There are nine Heegner numbers.[7]

Since 9 = 321, 9 is an exponential factorial.[8]

8 and 9 form a Ruth-Aaron pair under the second definition that counts repeated prime factors as often as they occur.

In bases 12, 18 and 24, nine is a 1-automorphic number and in base 6 a 2-automorphic number (displayed as '13').

A polygon with nine sides is called a nonagon or enneagon.[9] A group of nine of anything is called an ennead.

In base 10 a positive number is divisible by nine if and only if its digital root is 9.[10] That is, if any natural number is multiplied by nine, and repeatedly add the digits of the answer until it is just one digit, the sum will be nine:

There are other interesting patterns involving multiples of nine:

This works for all the multiples of 9. n = 3 is the only other n > 1 such that a number is divisible by n if and only if its digital root is n. In base N, the divisors of N  1 have this property. Another consequence of 9 being 10  1, is that it is also a Kaprekar number.

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

Subtracting two base-10 positive integers that are transpositions of each other yields a number that is a whole multiple of nine. Examples:

This works regardless of the number of digits that are transposed. For example, the largest transposition of 35967930 is 99765330 (all digits in descending order) and its smallest transposition is 03356799 (all digits in ascending order); subtracting pairs of these numbers produces:

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.[11]

Every prime in a Cunningham chain of the first kind with a length of 4 or greater is congruent to 9 mod 10 (the only exception being the chain 2, 5, 11, 23, 47).

Six recurring nines appear in the decimal places 762 through 767 of π, see Six nines in pi.

If an odd perfect number is of the form 36k + 9, it has at least nine distinct prime factors.[12]

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...)

It appears in the Padovan sequence, preceded by the terms 4, 5, 7 (it is the sum of the first two of these).[13]

Nine is the binary complement of number six:

9 = 1001
6 = 0110

List of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000
9 × x 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225 450 900 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 11 12 13
9x 9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 31381059609 282429536481 2541865828329
x9 1 512 19683 262144 1953125 10077696 40353607 134217728 387420489 1000000000 2357947691 5159780352 10604499373
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

Numeral systems

Base Numeral system
2 binary 1001
3 ternary 100
4 quaternary 21
5 quinary 14
6 senary 13
7 septenary 12
8 octal 11
9 novenary 10
over 9 (decimal, hexadecimal) 9

Probability

In probability, the nine is a logarithmic measure of probability of an event, defined as the negative of the base-10 logarithm of the probability of the event's complement. For example, an event that is 99% likely to occur has an unlikelihood of 1% or 0.01, which amounts to −log10 0.01 = 2 nines of probability. Zero probability gives zero nines (−log10 1 = 0). A 100% probability is considered to be impossible in most circumstances: that results in infinite improbability. The effectivity of processes and the availability of systems can be expressed (as a rule of thumb, not explicitly) as a series of "nines". For example, "five nines" (99.999%) availability implies a total downtime of no more than five minutes per year − typically a very high degree of reliability; but never 100%.

Organizations

Places and thoroughfares

Religion and philosophy

Science

Astronomy

Chemistry

Physiology

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

Sports

Billiards: A Nine-ball rack with the no. 9 ball at the center

Auto racing

Baseball

Billiards

Rugby

Soccer

All sports

The jersey number 9 has been retired by several North American sports teams in honor of past playing greats (or in one case, an owner):

Technology

Other fields

Playing cards showing the 9 of all four suits

See also

References

  1. Donald Alexander Mackenzie (2005). Myths of China And Japan. Kessinger. ISBN 1-4179-6429-4.
  2. Georges Ifrah (1985). From One to Zero: A Universal History of Numbers. Viking. ISBN 0-670-37395-8.
  3. Jane Dowson (1996). Women's Poetry of the 1930s: A Critical Anthology. Routledge. ISBN 0-415-13095-6.
  4. Anthea Fraser (1988). The Nine Bright Shiners. Doubleday. ISBN 0-385-24323-5.
  5. Charles Herbert Malden (1905). Recollections of an Eton Colleger, 1898-1902. Spottiswoode.
  6. "Sloane's A001006 : Motzkin numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  7. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
  8. "Sloane's A049384 : a(0)=1, a(n+1) = (n+1)^a(n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  9. Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
  10. Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155
  11. Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
  12. Eyob Delele Yirdaw, "Proving Touchard's Theorem from Euler's Form" ArXiv preprint.
  13. "Sloane's A000931 : Padovan sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-11.
  14. "Web site for NINE: A Journal of Baseball History & Culture". Retrieved 20 February 2013.

Further reading

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