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పంక్తి 39:
==భారతీయ సంస్కృతి==
* తొమ్మిది సంఖ్యకు భారతదేశంలో ముఖ్యంగా ఆంధ్రప్రదేశ్ లో అధిక ప్రాధాన్యత నిస్తారు. ఎందుకంటే వివిధ డిజిట్ సంఖ్యలలో పెద్ద సంఖ్యగా ఉదాహరణకు ఒక అంకె సంఖ్యలలో 9 పెద్ద సంఖ్య, రెండు అంకెల స్థానంలో 99 పెద్ద సంఖ్య, అంతేకాక ఆధ్యాత్మిక పరంగా కూడా కొన్ని కారణాలు ఉన్నాయి. ఉదాహరణకు నవగ్రహాలు, నవరాత్రులు మొదలగునవి.
==గణితం==
Nine is a [[composite number]], its proper [[divisor]]s being [[1 (number)|1]] and [[3 (number)|3]]. It is 3 times 3 and hence the third [[square number]]. Nine is a [[Motzkin number]]. It is the first composite [[lucky number]], along with the first composite odd number.
Nine is the highest single-digit number in the [[decimal|decimal system]]. It is the second non-unitary square [[prime number|prime]] of the form (p<sup>2</sup>) and the first that is odd. All subsequent squares of this form are odd. It has a unique [[aliquot sum]] [[4 (number)|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 number]]s.<ref>Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 93</ref>
Since 9 = 3<sup>2<sup>1</sup></sup>, 9 is an [[exponential factorial]].
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.<ref>Robert Dixon, ''Mathographics''. New York: Courier Dover Publications: 24</ref> A group of nine of anything is called an ennead.
In [[decimal|base 10]] a positive number is divisible by nine [[if and only if]] its [[digital root]] is 9.<ref>[[Martin Gardner]], ''A Gardner's Workout: Training the Mind and Entertaining the Spirit''. New York: A. K. Peters (2001): 155</ref> That is, if you multiply nine by any [[natural number]], and repeatedly add the digits of the answer until it is just one digit, you will end up with nine:
* 2 × 9 = 18 (1 + 8 = 9)
* 3 × 9 = 27 (2 + 7 = 9)
* 9 × 9 = 81 (8 + 1 = 9)
* 121 × 9 = 1089 (1 + 0 + 8 + 9 = 18; 1 + 8 = 9)
* 234 × 9 = 2106 (2 + 1 + 0 + 6 = 9)
* 578329 × 9 = 5204961 (5 + 2 + 0 + 4 + 9 + 6 + 1 = 27; 2 + 7 = 9)
* 482729235601 × 9 = 4344563120409 (4 + 3 + 4 + 4 + 5 + 6 + 3 + 1 + 2 + 0 + 4 + 0 + 9 = 45; 4 + 5 = 9)
There are other interesting patterns involving multiples of nine:
* 12345679 x 9 = 111111111
* 12345679 x 18 = 222222222
* 12345679 x 81 = 999999999
This works for all the multiples of 9.
''n'' = [[3 (number)|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 [[positional notation|base N]], the [[divisor]]s 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:
* The sum of the digits of 41 is 5, and 41-5 = 36. The digital root of 36 is 3+6 = 9, which, as explained above, demonstrates that it is divisible by nine.
* The sum of the digits of 35967930 is 3+5+9+6+7+9+3+0 = 42, and 35967930-42 = 35967888. The digital root of 35967888 is 3+5+9+6+7+8+8+8 = 54, 5+4 = 9.
Subtracting two base-10 positive integers that are transpositions of each other yields a number that is a whole multiple of nine. Examples:
* 41 - 14 = 27 (2 + 7 = 9)
* 36957930 - 35967930 = 990000, a multiple of nine.
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:
* 99765330 - 35967930 = 63797400; 6+3+7+9+7+4+0+0 = 36; 3+6 = 9.
* 99765330 - 03356799 = 96408531; 9+6+4+0+8+5+3+1 = 36; 3+6 = 9.
* 35967930 - 03356799 = 32611131; 3+2+6+1+1+1+3+1 = 18; 1+8 = 9.
[[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.<ref>[[Cajori, Florian]] (1991, 5e) ''A History of Mathematics'', AMS. ISBN 0-8218-2102-4. p.91</ref>
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]].
If an odd [[perfect number]] is of the form 36''k'' + 9, it has at least nine distinct prime factors.<ref>Eyob Delele Yirdaw, "[http://arxiv.org/abs/0804.0152v1 Proving Touchard's Theorem from Euler's Form]" ArXiv preprint.</ref>
If you divide 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...)
Nine is the binary complement of number [[6 (number)|six]]:
<pre>
9 = 1001
6 = 0110</pre>
===List of basic calculations===
{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Multiplication]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
! style="width:5px;"|
!11
!12
!13
!14
!15
!16
!17
!18
!19
!20
! style="width:5px;"|
!21
!22
!23
!24
!25
! style="width:5px;"|
!50
!100
!1000
|-
|<math>9 \times x</math>
|'''9'''
|[[18 (number)|18]]
|[[27 (number)|27]]
|[[36 (number)|36]]
|[[45 (number)|45]]
|[[54 (number)|54]]
|[[63 (number)|63]]
|[[72 (number)|72]]
|[[81 (number)|81]]
|[[90 (number)|90]]
!
|[[99 (number)|99]]
|[[108 (number)|108]]
|[[117 (number)|117]]
|[[126 (number)|126]]
|[[135 (number)|135]]
|[[144 (number)|144]]
|[[153 (number)|153]]
|[[162 (number)|162]]
|[[171 (number)|171]]
|[[180 (number)|180]]
!
|[[189 (number)|189]]
|[[198 (number)|198]]
|[[207 (number)|207]]
|[[216 (number)|216]]
|[[225 (number)|225]]
!
|450
|[[900 (number)|900]]
|[[9000 (number)|9000]]
|}
{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Division (mathematics)|Division]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
! style="width:5px;"|
!11
!12
!13
!14
!15
|-
|<math>9 \div x</math>
|'''9'''
|4.5
|3
|2.25
|1.8
|1.5
|1.{{overline|285714}}
|1.125
|1
|0.9
!
|0.{{overline|81}}
|0.75
|0.{{overline|692307}}
|0.6{{overline|428571}}
|0.6
|-
|<math>x \div 9</math>
|0.{{overline|1}}
|0.{{overline|2}}
|0.{{overline|3}}
|0.{{overline|4}}
|0.{{overline|5}}
|0.{{overline|6}}
|0.{{overline|7}}
|0.{{overline|8}}
|1
|1.{{overline|1}}
!
|1.{{overline|2}}
|1.{{overline|3}}
|1.{{overline|4}}
|1.{{overline|5}}
|1.{{overline|6}}
|}
{|class="wikitable" style="text-align: center; background: white"
|-
! style="width:105px;"|[[Exponentiation]]
!1
!2
!3
!4
!5
!6
!7
!8
!9
!10
! style="width:5px;"|
!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 (number)|512]]
|19683
|262144
|1953125
|10077696
|40353607
|134217728
|387420489
|[[1000000000 (number)|1000000000]]
!
|2357947691
|5159780352
|10604499373
|}
{|class="wikitable" style="text-align: center; background: white"
|-
! rowspan="2" style="width:105px;"|[[Radix]]
!1
!5
!10
!15
!20
!25
!30
<!--
!35
-->
!40
<!--
!45
-->
!50
!60
!70
!80
!90
!100
|-
!110
!120
!130
!140
!150
<!--
!160
!170
!180
!190
-->
!200
!250
!500
!1000
!10000
!100000
!1000000
|
|
|-
|rowspan="2"|<math>x_{9} \ </math>
|1
|5
|11<sub>9</sub>
|16<sub>9</sub>
|22<sub>9</sub>
|27<sub>9</sub>
|33<sub>9</sub>
|44<sub>9</sub>
|55<sub>9</sub>
|66<sub>9</sub>
|77<sub>9</sub>
|88<sub>9</sub>
|110<sub>9</sub>
|121<sub>9</sub>
|-
|132<sub>9</sub>
|143<sub>9</sub>
|154<sub>9</sub>
|165<sub>9</sub>
|176<sub>9</sub>
|242<sub>9</sub>
|307<sub>9</sub>
|615<sub>9</sub>
|1331<sub>9</sub>
|14641<sub>9</sub>
|162151<sub>9</sub>
|1783661<sub>9</sub>
|}
===Numeral systems===
{| class=wikitable
|-
! [[Radix|Base]] !! [[Numeral system]]
|-
| 2 || [[Binary numeral system|binary]] || 1001
|-
| 3 || [[Ternary numeral system|ternary]] || 100
|-
| 4 || [[quaternary numeral system|quaternary]] || 21
|-
| 5 || [[quinary]] || 14
|-
| 6 || [[senary]] || 13
|-
| 7 || [[septenary]] || 12
|-
| 8 || [[octal]] || 11
|-
| 9 || [[novenary]] || 10
|-
| colspan=2 | 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 (number)|10]] [[logarithm]] of the probability of the event's [[Probability axioms|complement]].
For example, an event that is 99% likely to occur has an unlikelihood of 1% or 0.01, which amounts to −log<sub>10</sub> 0.01 = 2 nines of probability.
[[0 (number)|Zero]] probability gives zero nines (−log<sub>10</sub> 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, [[5 nines|"five nines" (99.999%)]] availability implies a total [[downtime]] of no more than five minutes per year - typically a very high degree of [[:wikt:reliability|reliability]]; but never 100%.
==ఇవి కూడా చూడండి==
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