9


List of numbers — Integers 

Cardinal  9 nine 
Ordinal  9th ninth 
Numeral system  novenary 
Factorization  <math> 3^2 </math> 
Divisors  1, 3, 9 
Amharic  ፱ 
Roman numeral  IX 
Roman numeral (Unicode)  Ⅸ, ⅸ 
prefixes  ennea (from Greek)
nona (from Latin) 
Binary  1001 
Octal  11 
Duodecimal  9 
Hexadecimal  9 
ArabicIndic numeral  ٩ 
Armenian numeral  Թ 
Bengali  ৯ 
Chinese/Japanese numeral  九 玖 (formal writing) 
Devanāgarī  ९ 
Greek numeral  θ´ 
Hebrew numeral  ט (Tet) 
Tamil numeral  ௯ 
Khmer  ៩ 
Thai numeral  ๙ 
9 (nine) is a number, numeral, and glyph that represents the number. It is the natural number^{[1]} that follows 8 and precedes 10. It is an integer and a cardinal number, that is, a number that is used for counting.^{[2]} In addition, it is classified as a real number,^{[3]} distinguishing it from imaginary numbers. Nine is the highest singledigit number in the decimal system. A group of nine is called an ennead.
Contents 
According to Georges Ifrah,^{[4]} the origin of the 9 integers can be traced to the ancient Indian civilization. These numbers were adopted by subsequent civilizations in conjunction with the 0.
At first, various Indians wrote 9 similar to the modern closing question mark without the bottom dot. The Kshtrapa, Andhra, and Gupta started curving the bottom vertical line coming up with a 3 lookalike. The Nagari continued the bottom stroke to make a circle and enclose the 3 lookalike, in much the same way that the @ character encircles a lowercase a. Over time, the enclosing circle became bigger and its line continued beyond the circle downwards, as the 3 lookalike became smaller. Soon, all that was left of the 3 lookalike 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, for example, in .
This numeral resembles an inverted 6. To disambiguate the two on objects and documents that could be inverted, the 9 has often been underlined (as is done for the 6). Another distinction from the 6 is that 9 is often handwritten with a straight stem.
Nine is a composite number, its proper divisors being 1 and 3. It is 3 times 3 and hence the third square number. 9 is a Motzkin number. It is the first composite lucky number.
9 is the second nonunitary square prime (3^{2}). It has a unique aliquot sum 4 which is itself a square prime. 9 is the only square prime with an aliquot sum of the same form. The aliquot sequence of 9 has 5 members (9,4,3,1,0) this number being the second composite member of the 3aliquot tree.
There are nine Heegner numbers.
Since <math>9 = 3^{2^1}</math>, 9 is an exponential factorial.
Eight and 9 form a RuthAaron pair under the second definition that counts repeated prime factors as often as they occur.
A polygon with nine sides is called an enneagon (technically) or nonagon (in common usage).^{[5]} A group of nine of anything is called an ennead.
In base 10, a number is evenly divisible by nine if and only if its digital root is 9.^{[6]} That is, if you multiply nine by any whole number (except zero), and repeatedly add the digits of the answer until it's just one digit, you will end up with nine:
The only other number with this property is three. 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 base10 positive integer and the sum of its digits is a whole multiple of nine. Examples:
Subtracting two base10 positive integers that are transpositions of each other yields a number that is a whole multiple of nine. Some 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 twelfth century.^{[7]}
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 pi. This is known as the Feynman point.^{[8]}
If an odd perfect number is of the form 36k + 9, it has at least nine distinct prime factors.^{[9]}
Nine is the binary complement of number six:
9 = 1001 6 = 0110
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 
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  

<math>9 \times x</math>  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  

<math>9 \div x</math>  9  4.5  3  2.25  1.6  1.5  <math>1.\overline{285714}</math>  1.125  1  0.9  <math>0.\overline{81}</math>  0.75  <math>0.\overline{692307}</math>  <math>0.6\overline{428571}</math>  0.6  
<math>x \div 9</math>  <math>0.\overline{1}</math>  <math>0.\overline{2}</math>  <math>0.\overline{3}</math>  <math>0.\overline{4}</math>  <math>0.\overline{5}</math>  <math>0.\overline{6}</math>  <math>0.\overline{7}</math>  <math>0.\overline{8}</math>  1  <math>1.\overline{1}</math>  <math>1.\overline{2}</math>  <math>1.\overline{3}</math>  <math>1.\overline{4}</math>  <math>1.\overline{5}</math>  <math>1.\overline{6}</math> 
Exponentiation  1  2  3  4  5  6  7  8  9  10  11  12  13  

<math>9 ^ x\,</math>  9  81  729  6561  59049  531441  4782969  43046721  387420489  3486784401  31381059609  282429536481  2541865828329  
<math>x ^ 9\,</math>  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  
<math>x_{9} \ </math>  1  5  <math>11_{9} \ </math>  <math>16_{9} \ </math>  <math>22_{9} \ </math>  <math>27_{9} \ </math>  <math>33_{9} \ </math>  <math>44_{9} \ </math>  <math>55_{9} \ </math>  <math>66_{9} \ </math>  <math>77_{9} \ </math>  <math>88_{9} \ </math>  <math>110_{9} \ </math>  <math>121_{9} \ </math> 
<math>132_{9} \ </math>  <math>143_{9} \ </math>  <math>154_{9} \ </math>  <math>165_{9} \ </math>  <math>176_{9} \ </math>  <math>242_{9} \ </math>  <math>307_{9} \ </math>  <math>615_{9} \ </math>  <math>1331_{9} \ </math>  <math>14641_{9} \ </math>  <math>162151_{9} \ </math>  <math>1783661_{9} \ </math> 
Nine (九 pinyin jiǔ) is considered a good number in Chinese culture because it sounds the same as the word "longlasting" (久 pinyin jiǔ). The Japanese consider 9 to be unlucky, however, because it sounds similar to the Japanese word for "pain" or "distress" (苦 kunrei ku). Fear of the number nine is called enneaphobia.
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 9×13 scales, 9×9 being yang (masculine, or bad influence) and 9×4 being yin (feminine, or good influence).^{[12]}
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.
All links retrieved January 26, 2016.
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