తొమ్మిది: కూర్పుల మధ్య తేడాలు

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పంక్తి 380: పంక్తి 380:
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>&nbsp;0.01&nbsp;=&nbsp;2 nines of probability.
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>&nbsp;0.01&nbsp;=&nbsp;2 nines of probability.
[[0 (number)|Zero]] probability gives zero nines (−log<sub>10</sub>&nbsp;1&nbsp;=&nbsp;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%.
[[0 (number)|Zero]] probability gives zero nines (−log<sub>10</sub>&nbsp;1&nbsp;=&nbsp;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%.

==మూలాలు==
{{మూలాలజాబితా}}


==ఇవి కూడా చూడండి==
==ఇవి కూడా చూడండి==

13:31, 14 జూన్ 2013 నాటి కూర్పు

8 9 10
−1 0 1 2 3 4 5 6 7 8 9
Cardinalnine
Ordinal9th
(ninth)
Factorization32
Roman numeralIX
Unicode symbol(s)Ⅸ, ⅸ
Greek prefixennea-
Latin prefixnona-
Binary10012
Ternary1003
Quaternary214
Quinary145
Octal118
Duodecimal912
Hexadecimal916
Vigesimal920
Base 36936
Amharic
Arabic-Indic numeral٩
Armenian numeralԹ
Bengali
Chinese/Japanese
/Korean numeral
九 (jiu)
玖 (formal writing)
Devanāgarī (Nao)
Greek numeralθ´
Hebrew numeralט (Tet)
Tamil numeral
Khmer
Telugu numeral
Thai numeral

తొమ్మిది (9) ఒక సహజ సంఖ్య, దీనికి ముందు గల సంఖ్య ఎనిమిది (8), మరియు తరువాత వచ్చే సంఖ్య పది (10).

వాడుకలో తొమ్మిదవ, తొమ్మిదో అనే పదాలను ఉపయోగిస్తారు. కొన్ని చోట్ల తొమ్మిదికి బదులు నవ ఉపయోగిస్తారు. ఉదాహరణకు తొమ్మిది గ్రహాలను నవగ్రహాలు అంటారు.

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 divisors being 1 and 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 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.[1]

Since 9 = 321, 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.[2] 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.[3] 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 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:

  • 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.[4]

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 36k + 9, it has at least nine distinct prime factors.[5]

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 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 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 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
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
9 81 729 6561 59049 531441 4782969 43046721 387420489 3486784401 31381059609 282429536481 2541865828329
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
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%.

మూలాలు

  1. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 93
  2. Robert Dixon, Mathographics. New York: Courier Dover Publications: 24
  3. Martin Gardner, A Gardner's Workout: Training the Mind and Entertaining the Spirit. New York: A. K. Peters (2001): 155
  4. Cajori, Florian (1991, 5e) A History of Mathematics, AMS. ISBN 0-8218-2102-4. p.91
  5. Eyob Delele Yirdaw, "Proving Touchard's Theorem from Euler's Form" ArXiv preprint.

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