formulasearchengine - User contributions [en]

Kite (geometry)

2014-02-02T10:28:09Z

112.207.1.119: /* Area */

In [[physics]], the '''reduced mass''' is the "effective" [[inertial mass]] appearing in the [[two-body problem]] of [[Newtonian mechanics]]. It is a quantity which allows the two-body problem to be solved as if it were a [[one-body problem]]. Note, however, that the mass determining the [[gravitational force]] is ''not'' reduced. In the computation one mass ''can'' be replaced by the reduced mass, if this is compensated by replacing the other mass by the sum of both masses. The reduced mass is frequently denoted by <math>\scriptstyle \mu </math> ([[Greek alphabet|Greek]] lower case [[Mu (letter)|mu]]), although the [[standard gravitational parameter]] is also denoted by <math>\scriptstyle \mu </math> (and so are [[Mu (letter)#Physics and engineering|a number of other physical quantities]] as well). It has the [[dimensional analysis|dimensions]] of mass, and [[SI unit]] kg.

==Equation==

Given two bodies, one with mass ''m''1 and the other with mass ''m''2, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass <ref>Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) 0-89573-752-3</ref><ref>Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8</ref>

:<math>m_\text{red} = \mu = \cfrac{1}{\cfrac{1}{m_1}+\cfrac{1}{m_2}} = \cfrac{m_1 m_2}{m_1 + m_2},\!\,</math>

where the force on this mass is given by the force between the two bodies.

===Properties===

The reduced mass is always less than or equal to the mass of each body:

:<math>m_\text{red} \leq m_1, \quad m_\text{red} \leq m_2 \!\,</math>

and has the reciprocal additive property:

:<math>\frac{1}{m_\text{red}} = \frac{1}{m_1} + \frac{1}{m_2} \,\!</math>

which by re-arrangement is equivalent to half of the [[harmonic mean]].

==Derivation==

The equation can be derived as follows.

===Newtonian mechanics===

{{main|Newtonian mechanics}}

Using [[Newton's second law]], the force exerted by body 2 on body 1 is
:<math>\bold{F}_{12} = m_1 \bold{a}_1. \!\,</math>

The force exerted by body 1 on body 2 is
:<math>\bold{F}_{21} = m_2 \bold{a}_2. \!\,</math>

According to [[Newton's third law]], the force that body 2 exerts on body 1 is equal and opposite to the force that body 1 exerts on body 2:
:<math>\bold{F}_{12} = - \bold{F}_{21}.\!\,</math>

Therefore,
:<math>m_1 \bold{a}_1 = - m_2 \bold{a}_2. \!\,</math>

and

:<math>\bold{a}_2=-{m_1 \over m_2} \bold{a}_1. \!\,</math>

The relative acceleration '''a'''rel between the two bodies is given by

:<math>\bold{a}_{\rm rel}= \bold{a}_1-\bold{a}_2 = \left(1+\frac{m_1}{m_2}\right) \bold{a}_1 = \frac{m_2+m_1}{m_1 m_2} m_1 \bold{a}_1 = \frac{\bold{F}_{12}}{m_{\rm red}}. </math>

So we conclude that body 1 moves with respect to the position of body 2 as a body of mass equal to the reduced mass.

===Lagrangian mechanics===

{{main|Lagrangian mechanics}}

Alternatively, a Lagrangian description of the two-body problem gives a [[Lagrangian]] of

:<math>L = {1 \over 2} m_1 \mathbf{\dot{r}}_1^2 + {1 \over 2} m_2 \mathbf{\dot{r}}_2^2 - V(| \mathbf{r}_1 - \mathbf{r}_2 | ) \!\,</math>

where '''r''' is the position vector of mass ''mi'' (of particle ''<math>i</math>''). The potential energy ''V'' is a function as it is only dependent on the absolute distance between the particles. If we define
:<math>\mathbf{r} = \mathbf{r}_1 - \mathbf{r}_2 </math>
and let the centre of mass coincide with our origin in this reference frame, i.e.
:<math> m_1 \mathbf{r}_1 + m_2 \mathbf{r}_2 = 0 </math>,
then
:<math> \mathbf{r}_1 = \frac{m_2 \mathbf{r}}{m_1 + m_2} , \mathbf{r}_2 = \frac{-m_1 \mathbf{r}}{m_1 + m_2}.</math>

Then substituting above gives a new Lagrangian

:<math> L = {1 \over 2}m_\text{red} \mathbf{\dot{r}}^2 - V(r), </math>

where

:<math>m_\text{red} = \frac{m_1 m_2}{m_1 + m_2} </math>

is the reduced mass. Thus we have reduced the two-body problem to that of one body.

==Applications==

Reduced mass occurs in a multitude of two-body problems, where classical mechanics is applicable.

===Collisions of particles===

In a collision with a [[coefficient of restitution]] ''e'', the change in kinetic energy can be written as
:<math>\Delta K = \frac{1}{2}\mu v^2_{\rm rel}(e^2-1)</math>,
where vrel is the relative velocity of the bodies before [[collision]].

For typical applications in nuclear physics, where one particle's mass is much larger than the other the reduced mass can be approximated as the smaller mass of the system. The limit of the reduced mass formula as one mass goes to infinity is the smaller mass, thus this approximation is used to ease calculations, especially when the larger particles exact mass is not known.

===Motions of masses in gravitational fields===

In the case of the gravitational potential energy
:<math>V(| \mathbf{r}_1 - \mathbf{r}_2 | ) = - \frac{G m_1 m_2}{| \mathbf{r}_1 - \mathbf{r}_2 |} \, ,</math>
we find that the position of the first body with respect to the second is governed by the same differential equation as the position of a body with the reduced mass orbiting a body with a mass equal to the sum of the two masses, because

:<math>m_1 m_2 = (m_1+m_2) m_\text{red}\!\,</math>

===Non-relativistic quantum mechanics===

Consider the [[electron]] (mass ''me'') and [[proton]] (mass ''mp'') in the [[hydrogen atom]].<ref>Molecular Quantum Mechanics Parts I and II: An Introduction to QUANTUM CHEMISRTY (Volume 1), P.W. Atkins, Oxford University Press, 1977, ISBN 0-19-855129-0</ref> They orbit each other about a common centre of mass, a two body problem. To analyze the motion of the electron, a one-body problem, the reduced mass replaces the electron mass

:<math>m_e \rightarrow \frac{m_em_p}{m_e+m_p} </math>

and the proton mass becomes the sum of the two masses

:<math>m_p \rightarrow m_e + m_p </math>

This idea is used to set up the [[Schrödinger equation]] for the hydrogen atom.

===Other uses===

"Reduced mass" may also refer more generally to an [[algebra]]ic term of the form {{Citation needed|date=December 2011}}

:<math>x_\text{red} = {1 \over {1 \over x_1} + {1 \over x_2}} = {x_1 x_2 \over x_1 + x_2}\!\,</math>

that simplifies an equation of the form

:<math>\ {1\over x_\text{eq}} = \sum_{i=1}^n {1\over x_i} = {1\over x_1} + {1\over x_2} + \cdots+ {1\over x_n}.\!\,</math>

The reduced mass is typically used as a relationship between two system elements in parallel, such as [[resistors]]; whether these be in the electrical, thermal, hydraulic, or mechanical domains. This relationship is determined by the physical properties of the elements as well as the [[continuity equation]] linking them.

==See also==

*[[Center-of-momentum frame]]
*[[Momentum conservation]]
*[[Defining equation (physics)]]
*[[Harmonic oscillator]]

==References==

{{reflist}}

==External links==
*[http://hyperphysics.phy-astr.gsu.edu/hbase/orbv.html#rm ''Reduced Mass'' on HyperPhysics]

[[Category:Mechanics]]
[[Category:Mass]]

Equilateral triangle

2014-02-02T09:35:58Z

112.207.1.119: /* Using trigonometry */

{{About|the number||9 (disambiguation)}}
{{Infobox number
| number = 9
| numeral = [[nonary]]
| divisors = 1, 3, 9
| unicode = Ⅸ, ⅸ
| greek prefix = [[Wiktionary:ennea-|ennea-]]
| latin prefix = [[Wiktionary:nona-|nona-]]

| lang1 = [[Amharic language|Amharic]]
| lang1 symbol = ፱
| lang2 = [[Arabic language|Arabicl]]
| lang2 symbol = ٩
| lang3 = [[Urdu]]
| lang3 symbol = {{Urdu numeral|9|20}}
| lang4 = [[Armenian numerals|Armenian numeral]]
| lang4 symbol = Թ
| lang5 = [[Bengali language|Bengali]]
| lang5 symbol = ৯
| lang6 = [[Chinese numerals|Chinese/Japanese /Korean numeral]]
| lang6 symbol = 九 (jiu) 玖 (formal writing)
| lang7 = [[Devanāgarī]]
| lang7 symbol = ९ (nau)
| lang8 = [[Greek numerals|Greek numeral]]
| lang8 symbol = θ´
| lang9 = [[Hebrew numerals|Hebrew numeral]]
| lang9 symbol = ט (Tet)
| lang10 = [[Tamil numerals]]
| lang10 symbol = ௯
| lang11 = [[Khmer numerals|Khmer]]
| lang11 symbol = ៩
| lang12 = [[Telugu language|Telugu numeral]]
| lang12 symbol = ౯
| lang13 = [[Thai numerals|Thai numeral]]
| lang13 symbol = ๙
}}
{{Wiktionary|nine}}

'''9''' ('''nine''' {{IPAc-en|'|n|ai|n}}) is the [[natural number]] following [[8 (number)|8]] and preceding [[10 (number)|10]].

==Alphabets and codes==
*In the [[NATO phonetic alphabet]], the digit 9 is called "Niner".
*Five-digit [[produce]] [[Price Look-Up code|PLU codes]] that begin with 9 are [[Organic food|organic]].

==Commerce==
*Common terminal digit in [[psychological pricing]]
===Companies===
*[[9Lives|Nine Lives]] cat food; its name is derived from the legend that a cat has nine lives
*[[Nine Network]] a.k.a. Channel 9, an Australian free-to-air television station
*[[Nine West]], a clothing brand <ref>''Nine West'' http://www.ninewest.com/</ref>

==Culture and mythology==
===Chinese culture===
*Nine (九 [[pinyin]] jiǔ) is considered a good [[Numbers in Chinese culture|number in Chinese culture]] because it sounds the same as the word "longlasting" ([[久]] [[pinyin]] jiǔ){{Citation needed|date=April 2008}}.

*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 117 scales - 81 [[yin and yang|yang]] (masculine, heavenly) and 36 [[yin and yang|yin]] (feminine, earthly). All three numbers are multiples of 9 (9×13=117, 9×9=81, 9×4=36)<ref>{{cite book|title=Myths of China And Japan|author=Donald Alexander Mackenzie|url=http://books.google.com/books?id=vzbeLy4TBa4C&pg=PA46&dq=chinese+dragon+scales+yin+36|publisher=Kessinger|year=2005|isbn=1-4179-6429-4}}</ref> as well as having the same [[digital root]] of 9.

*The dragon often symbolizes the [[Emperor of China|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.

*The name of the area called ''[[Kowloon]]'' in [[Hong Kong]] literally means: ''nine [[Chinese dragon|dragons]]''.

*The [[nine-rank system]] was a civil service nomination system used during certain Chinese dynasties.

===Ancient Egypt===
*The [[nine bows]] is a term used in Ancient Egypt to represent the traditional enemies of Egypt

===European culture===
*The [[Nine Worthies]] are nine historical, or semi-legendary figures who, in the Middle Ages, were believed to personify the ideals of chivalry

===Greek Mythology===
* The nine [[muses]] in Greek mythology are [[Calliope]] (epic poetry), [[Clio]] (history), [[Erato]] (erotic poetry), [[Euterpe]] (lyric poetry), [[Melpomene]] (tragedy), [[Polyhymnia]] (song), [[Terpsichore]] (dance), [[Thalia (muse)|Thalia]] (comedy), and [[Urania]] (astronomy).
* It takes nine days (for an anvil) to fall from heaven to earth, and nine more to fall from earth to [[Tartarus]]--a place of torment in the [[Greek_underworld|underworld]].

===Japanese culture===
*The Japanese consider nine to be unlucky because in Japanese the word for nine sounds similar to the word for "pain" or "distress" ({{linktext|苦}}, ''kyū''){{Citation needed|date=April 2008}}.

== Evolution of the glyph ==
{{See also|Hindu-Arabic numeral system}}
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 (number)|0]].<ref>{{cite book|title=From One to Zero: A Universal History of Numbers|year=1985|author=Georges Ifrah|publisher=Viking|isbn=0-670-37395-8}}</ref>

[[File:Evo9glyph.svg|x50px|right]]
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 (number)|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 (typography)|ascender]] in most modern [[typeface]]s, in typefaces with [[text figures]] the character usually has a [[descender]], as, for example, in [[File:TextFigs196.png]].

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 often handwritten with a straight stem.

==Idioms and popular phrases==
*"A [[cat-o'-nine-tails]] suggests perfect punishment and atonement." --[[Robert Ripley]].
"A stitch in time saves Nine"
*The word "K-9" pronounces the same as ''canine'' and is used in many [[United States|U.S.]] police departments to denote the [[police dog]] unit. Despite not sounding like the translation of the word ''canine'' in other languages, many police and military units around the world use the same designation.
*Someone dressed "to the nines" is dressed up as much as they can be.
*In [[urban culture]], "nine" is a [[slang]] word for a [[9mm]] [[pistol]] or [[homicide]], the latter from the Illinois Criminal Code for homicide.

==Internet==
*''[[The 9 on Yahoo!]]'', hosted by [[Maria Sansone]], was a daily video compilation show, or vlog, on Yahoo! featuring the nine top "web finds" of the day.
*[[Cirno]] from Touhou Project is represented by ⑨

==Literature==
*There are [[Divine Comedy#The Circles of Hell|nine circles of Hell]] in Dante's ''[[Divine Comedy]]''.
*The [[Nine Bright Shiners]], characters in Garth Nix's [[Old Kingdom trilogy]]. ''The Nine Bright Shiners'' was a 1930s book of poems by Anne Ridler<ref>{{cite book|title=Women's Poetry of the 1930s: A Critical Anthology|author=Jane Dowson|year=1996|publisher=Routledge|isbn=0-415-13095-6|url=http://books.google.com/books?id=fVTQPI3ZIHcC&pg=RA1-PA103&dq=nine-bright-shiners+ridler&ie=ISO-8859-1#PRA1-PA103,M1}}</ref> and a 1988 fiction book by Anthea Fraser;<ref>{{cite book|title=The Nine Bright Shiners|author=Anthea Fraser|publisher=Doubleday|year=1988|isbn=0-385-24323-5|url=}}</ref> the name derives from "a very curious old semi-pagan, semi-Christian" song.<ref>{{cite book|title=Recollections of an Eton Colleger, 1898-1902|author=Charles Herbert Malden|publisher=Spottiswoode|year=1905|url=http://books.google.com/books?id=EKB9T4pPcfkC&pg=PA182&dq=nine-bright-shiners#PPA180,M1}}</ref>
*''[[The Nine Tailors]]'' is a 1934 [[mystery novel]] by [[United Kingdom|British]] writer [[Dorothy L. Sayers]], her ninth featuring sleuth [[Lord Peter Wimsey]]
*[[Nine Unknown Men]] are, in occult legend, the custodians of the sciences of the world since ancient times
*In [[J.R.R. Tolkien's]] [[Middle-earth]], there are nine rings of power given to men, and consequently, nine [[Nazgul|ringwraiths]]
**Additionally, ''[[The Fellowship of the Ring]]'' consists of nine companions, representing the free races and also as a positive mirror of the nine ringwraiths

==Mathematics==
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 (p2) 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 = 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.<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

!40

!50
!60
!70
!80
!90
!100
|-
!110
!120
!130
!140
!150

!200
!250
!500
!1000
!10000
!100000
!1000000
|
|
|-
|rowspan="2"|<math>x_{9} \ </math>
|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===
{| 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 −log10 0.01 = 2 nines of probability.
[[0 (number)|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, [[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%.

==Organizations==
* Divine Nine—The [[National Pan-Hellenic Council]] (NPHC) is a collaborative organization of nine historically African American, international Greek lettered fraternities and sororities.

==Places and thoroughfares==
*[[List of highways numbered 9]]
*[[Ninth Avenue (Manhattan)|Ninth Avenue]] is a major avenue in Manhattan.
* Outside the Norwegian coast, east of the Barents sea [http://www.imr.no/galleri/v/mareano/terrengmodeller_bhavet/ number 9], in a 260 meters depth.

==Religion and philosophy==
[[Image:Bahai star.svg|150px|right|A nine-pointed star]]
*Nine, as the highest single-digit number (in [[decimal|base ten]]), symbolizes completeness in the [[Bahá'í Faith]]. In addition, the word Bahá' in the [[Abjad numerals|Abjad notation]] has a value of 9, and a 9-pointed star is used to [[Bahá'í symbols|symbolize]] the religion.
*The number 9 is revered in Hinduism and considered a complete, perfected and divine number because it represents the end of a cycle in the [[decimal]] system, which originated from the Indian subcontinent as early as [[30th century BC|3000]] BC.
*Important Buddhist rituals usually involve nine monks.
*The first nine days of the [[Hebrew calendar|Hebrew month]] of [[Av (month)|Av]] are collectively known as "The Nine Days" (''Tisha HaYamim''), and are a period of semi-mourning leading up to [[Tisha B'Av]], the ninth day of Av on which both [[Temple in Jerusalem|Temples in Jerusalem]] were destroyed.
*Nine is a significant number in [[Norse Mythology]]. [[Odin]] hung himself on an ash tree for nine days to learn the runes.
*The [[Fourth Way Enneagram]] is one system of knowledge which shows the correspondence between the 9 integers and the circle.
*In the [[Christian angelic hierarchy]] there are 9 choirs of angels.
*[[Ramadan (calendar month)|Ramadan]], the month of fasting and prayer, is the ninth month of the [[Islamic calendar]].

==Science==
===Astronomy===
*Before 2006 (when Pluto was [[Pluto#2006: IAU classification|officially designated as a non-planet]]), there were nine [[planet]]s in the [[solar system]].
*[[Messier object]] [[Messier 9|M9]] is a magnitude 9.0 [[globular cluster]] in the constellation [[Ophiuchus]].
* The [[New General Catalogue]] [http://www.ngcic.org/ object] [[NGC 9]], a [[spiral galaxy]] in the [[constellation]] [[Pegasus (constellation)|Pegasus]]
*The Saros [http://sunearth.gsfc.nasa.gov/eclipse/LEsaros/LEsaros1-175.html number] of the [[lunar eclipse]] series which began on [[-2501]] [[June 26]] and ended on [[-1149]] [[September 16]]. The duration of Saros series 9 was 1352.2 years, and it contained 76 lunar eclipses.
*The [[Saros cycle|Saros]] [http://sunearth.gsfc.nasa.gov/eclipse/SEsaros/SEsaros1-175.html number] of the [[solar eclipse]] series which began on [[-2568]] [[February 6]] and ended on [[-1252]] [[April 4]]. The duration of Saros series 9 was 1316.2 years, and it contained 74 solar eclipses.

===Chemistry===
*The purity of chemicals (see [[Nine (purity)]])
*Nine is the [[atomic number]] of [[fluorine]].

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

==Sports==
[[File:9ball rack 2.jpg|thumb|right|200px|A [[Nine-ball]] [[Rack (billiards)|rack]] with the 9 ball at the center]]

===Auto racing===
* A car in the [[Sprint Cup Series]] currently owned by [[Richard Petty Motorsports]]. The number was most notably borne by the car that [[Bill Elliott]] drove to the Cup Series title in [[1988 NASCAR Winston Cup Series|1988]] with [[Melling Racing]]. Evernham Motorsports, the predecessor team to Richard Petty Motorsports, acquired the number in [[2001 NASCAR Winston Cup Series|2001]] when Elliott joined that team after a brief stint [[Bill Elliott Racing|as a driver-owner]]. Elliott used this number again through the [[2003 NASCAR Winston Cup Series|2003 season]]. [[Kasey Kahne]] has driven the 9 car since [[2004 NASCAR Nextel Cup Series|2004]]. Currently, the 9 car is driven by [[Marcos Ambrose]]

===Baseball===
* In [[baseball]], nine represents the [[right fielder]]'s position.
*The number of [[Innings#Baseball|innings]] in a regulation, non-tied game of baseball.
*The number of players on the field including the pitcher.
*The number worn by [[Roy Hobbs]] in the movie ''[[The Natural]]''.
*''NINE: A Journal of Baseball History and Culture'' published by the [[University of Nebraska Press]]<ref>{{cite web|url=http://nine.iweb.bsu.edu/|title=Web site for NINE: A Journal of Baseball History & Culture|accessdate=20 February 2013}}</ref>

===Billiards===
*[[Nine-ball]] is the standard professional pocket [[billiards]] variant played in the United States.

===Rugby===
*In [[rugby league]], the jersey number assigned to the [[Hooker (rugby league)|hooker]].
*In [[rugby union]], the number worn by the starting [[Scrum-half (rugby union)|scrum-half]].

===Soccer===
* In association [[soccer|football]] (soccer) the centre-forward/striker traditionally (since at least the fifties) wears the number 9 shirt.

===All sports===
The jersey number 9 has been retired by several [[Major professional sports leagues in the United States and Canada|North American sports]] teams in honor of past playing greats (or in one case, an owner):
* In [[Major League Baseball]]:
** The [[Boston Red Sox]], for [[National Baseball Hall of Fame and Museum|Hall of Famer]] [[Ted Williams]].
** The [[Chicago White Sox]], for [[Minnie Miñoso]].
** The [[New York Yankees]], for [[Roger Maris]].
** The [[Oakland Athletics]], for Hall of Famer [[Reggie Jackson]].
** The [[Pittsburgh Pirates]], for Hall of Famer [[Bill Mazeroski]].
** The [[St. Louis Cardinals]], for Hall of Famer [[Enos Slaughter]].
* In the [[National Basketball Association|NBA]]:
** The [[Atlanta Hawks]], for [[Naismith Memorial Basketball Hall of Fame|Hall of Famer]] [[Bob Pettit]].
** The [[Phoenix Suns]], for [[Dan Majerle]].
** The [[Utah Jazz]], for owner [[Larry H. Miller|Larry Miller]].
* In the [[National Hockey League|NHL]]:
** The [[Boston Bruins]], for [[Hockey Hall of Fame|Hall of Famer]] [[Johnny Bucyk]].
** The [[Calgary Flames]], for Hall of Famer [[Lanny McDonald]].
** The [[Chicago Blackhawks]], for Hall of Famer [[Bobby Hull]].
** The [[Dallas Stars]], for [[Mike Modano]].
** The [[Detroit Red Wings]], for Hall of Famer [[Gordie Howe]].
** The [[Edmonton Oilers]], for Hall of Famer [[Glenn Anderson]].
** The [[Montreal Canadiens]], for Hall of Famer [[Maurice Richard]].
** The [[New York Islanders]], for Hall of Famer [[Clark Gillies]].
** The [[New York Rangers]], for Hall of Famer [[Andy Bathgate]] and [[Adam Graves]].
** The [[Toronto Maple Leafs]] have a policy of not retiring numbers unless the player honoured either died or suffered a career-ending incident while a member of the team. Other players whose numbers would otherwise be retired instead have their numbers enshrined by the team as "Honoured Numbers", which remain in circulation for future players. The number 9 is currently honoured for Hall of Famers [[Ted Kennedy (ice hockey)|Ted Kennedy]] and [[Charlie Conacher]].
** The first NHL incarnation of the [[Winnipeg Jets (1972–96)|Winnipeg Jets]], also for Hull. Although the Jets moved from Winnipeg to become the [[Phoenix Coyotes]], the Coyotes continue to honor all numbers retired by the Jets. The Coyotes briefly took the number out of retirement for Hull's son [[Brett Hull]], also a Hall of Famer, in 2005–06 until the younger Hull retired five games into that season. The current [[Winnipeg Jets]] have yet to officially retire any numbers, but [[Evander Kane]] received Bobby Hull's blessing to wear the number.
* No [[National Football League|NFL]] team has yet retired #9.

==Technology==
[[Image:Seven-segment 9.svg|25px|right]]
[[Image:Seven-segment 9 alt.svg|25px|right]]
* [[ISO 9]] is the [[International Organization for Standardization|ISO]]'s standard for the transliteration of [[Cyrillic]] characters into [[Latin]] characters
* In the [[Rich Text Format]] specification, 9 is the language code for the [[English language]]. All codes for regional variants of English are congruent to 9 mod 256.
* The [[seven-segment display]] allows the number 9 to be constructed two ways, either with a hook at the end of its stem or without one. Most [[liquid crystal display|LCD]] calculators use the former, but some [[vacuum fluorescent display|VFD]] models use the latter.
* [[The9]] Limited (owner of [http://the9.com the9.com]) is a company in the video-game game industry, including former ties to the extremely popular [[MMORPG]] [[World of Warcraft]]

==Other fields==
[[Image:ICS Niner.svg|right|thumb|100px|[[International maritime signal flag]] for 9]]
[[Image:9 playing cards.jpg|thumb|250px|[[Playing card]]s showing the 9 of all four suits]]
*Nine justices sit on the [[United States Supreme Court]].
*[[Stanine]]s, a method of scaling test scores, range from 1 to 9.

==See also==
{{Portal|Mathematics}}
*[[9 (disambiguation)]]
*[[0.999...]]
*[[wikt:cloud nine|Cloud Nine]]

==References==
{{Reflist|35em}}

==Further reading==
*Cecil Balmond, "Number 9, the search for the sigma code" 1998, Prestel 2008, ISBN 3-7913-1933-7, ISBN 978-3-7913-1933-9

{{Integers|zero}}

{{DEFAULTSORT:9 (Number)}}
[[Category:Integers|09]]

Isosceles triangle

2014-02-02T09:34:56Z

112.207.1.119: /* See also */

{{ for|A kind of [[skipper (butterfly)]]|Caprona ransonnetti}}

[[File:Golden Angle.svg|right|thumb|The golden angle is the angle subtended by the smaller (red) arc when two arcs that make up a circle are in the [[golden ratio]]]]

In [[geometry]], the '''golden angle''' is the smaller of the two [[angle]]s created by sectioning the circumference of a circle according to the [[golden section]]; that is, into two [[Arc (geometry)|arc]]s such that the ratio of the length of the larger arc to the length of the smaller arc is the same as the ratio of the full circumference to the length of the larger arc.

Algebraically, let ''a+b'' be the circumference of a [[circle]], divided into a longer arc of length ''a'' and a smaller arc of length ''b'' such that

:<math> \frac{a + b}{a} = \frac{a}{b}. </math>

The golden angle is then the angle [[subtend]]ed by the smaller arc of length ''b''. It measures approximately 137.508°, or about 2.39996 [[radian]]s.

The name comes from the golden angle's connection to the [[golden ratio]] ''φ''; the exact value of the golden angle is

: <math>360\left(1 - \frac{1}{\varphi}\right) = 360(2 - \varphi) = \frac{360}{\varphi^2} = 180(3 - \sqrt{5})\text{ degrees}</math>

or

: <math> 2\pi \left( 1 - \frac{1}{\varphi}\right) = 2\pi(2 - \varphi) = \frac{2\pi}{\varphi^2} = \pi(3 - \sqrt{5})\text{ radians},</math>

where the equivalences follow from well-known algebraic properties of the golden ratio.

== Derivation ==
The golden ratio is equal to ''φ'' = ''a''/''b'' given the conditions above.

Let ''&fnof;'' be the fraction of the circumference subtended by the golden angle, or equivalently, the golden angle divided by the angular measurement of the circle.

:<math> f = \frac{b}{a+b} = \frac{1}{1+\varphi}.</math>

But since

: <math>{1+\varphi} = \varphi^2,</math>

it follows that

:<math> f = \frac{1}{\varphi^2} </math>

This is equivalent to saying that ''φ'' 2 golden angles can fit in a circle.

The fraction of a circle occupied by the golden angle is therefore

:<math>f \approx 0.381966. \,</math>

The golden angle ''g'' can therefore be numerically approximated in [[Degree (angle)|degrees]] as:

:<math>g \approx 360 \times 0.381966 \approx 137.508^\circ,\,</math>

or in radians as :

:<math> g \approx 2\pi \times 0.381966 \approx 2.39996. \,</math>

== Golden angle in nature ==
[[File:Goldener Schnitt Blattstand.png|thumb|right|300px|The angle between successive florets in some flowers is the golden angle.]]

The golden angle plays a significant role in the theory of [[phyllotaxis]]. Perhaps most notably, the golden angle is the angle separating the [[floret]]s on a [[sunflower]].

== References ==
{{refbegin}}
*{{Cite journal
| last =Vogel
| first =H
| title =A better way to construct the sunflower head
| journal =Mathematical Biosciences
| issue =44
| pages =179–189
| year =1979
| doi =10.1016/0025-5564(79)90080-4
| volume =44
}}
*{{cite book
| last =Prusinkiewicz
| first =Przemysław
| authorlink =Przemysław Prusinkiewicz
| coauthors =[[Aristid Lindenmayer|Lindenmayer, Aristid]]
| title =The Algorithmic Beauty of Plants
| publisher =Springer-Verlag
| date =1990
| location =
| pages =101–107
| url =http://algorithmicbotany.org/papers/#webdocs
| doi =
| isbn = 978-0-387-97297-8 }}
{{refend}}

== External links ==
* [http://mathworld.wolfram.com/GoldenAngle.html Golden Angle] at [[MathWorld]]

[[Category:Elementary geometry]]
[[Category:Golden ratio]]
[[Category:Angle]]

Rectangle

2014-02-02T08:37:27Z

112.207.1.119: /* Formulae */

'''Satisficing''' is a [[decision-making]] strategy or cognitive [[heuristic]] that entails searching through the available alternatives until an acceptability threshold is met.<ref>{{cite book |first=Andrew |last=Colman |year=2006 |title=A Dictionary of Psychology |location=New York |publisher=Oxford University Press |page=670 |isbn=019861035}}</ref> This is contrasted with [[optimal decision]] making, an approach that specifically attempts to find the best alternative available. The term ''satisficing'', a [[portmanteau]] of ''satisfy'' and ''suffice'',<ref>{{cite book |first=Ken |last=Manktelow |year=2000 |title=Reasoning and Thinking |location=Hove |publisher=Psychology Press |page=221 |isbn=0863777082 }}</ref> was introduced by [[Herbert A. Simon]] in 1956,<ref>{{cite journal |title=Rational Choice and the Structure of the Environment |last=Simon |first=H. A. |journal=[[Psychological Review]] |volume=63 |issue=2 |year=1956 |pages=129–138 |doi=10.1037/h0042769 }} (page 129: "Evidently, organisms adapt well enough to ‘satisfice’; they do not, in general, ‘optimize’."; page 136: "A ‘satisficing’ path, a path that will permit satisfaction at some specified level of all its needs.")</ref> although the concept "was first posited in ''[[Administrative Behavior]]'', published in 1947."<ref name=Brown2004>{{cite journal|last1=Brown|first1=Reva|title=Consideration of the Origin of Herbert Simon's Theory of 'Satisficing' (1933-1947)|journal=Management Decision|volume=42|issue=10|year=2004|pages=1240–1256|doi=10.1108/00251740410568944}}</ref><ref name=AB1947>{{cite book | last = Simon | first = Herbert A. | authorlink = Herbert A. Simon | title = [[Administrative Behavior]]: a Study of Decision-Making Processes in Administrative Organization | publisher = Macmillan | location = New York | year = 1947 | edition=1st | oclc = 356505}}</ref> Simon used satisficing to explain the behavior of decision makers under circumstances in which an optimal solution cannot be determined. He pointed out that human beings lack the cognitive resources to [[optimization|optimize]]: We can rarely evaluate all outcomes with sufficient precision, usually do not know the relevant probabilities of outcomes, and possess only limited memory. Simon formulated the concept within a novel approach to rationality, which takes into account these limitations. He referred to this approach as [[bounded rationality]]. Notice furthermore that some [[consequentialism|consequentialist]] theories in [[moral philosophy]] use the concept of satisficing in the same sense, though most call for optimization instead.

== In decision-making ==
In decision making, satisficing explains the tendency to select the first option that meets a given need or select the option that seems to address most needs rather than the “optimal” solution.

:Example: A task is to sew a patch onto a pair of jeans. The best needle to do the threading is a 4 inch long needle with a 3 millimeter eye. This needle is hidden in a haystack along with 1000 other needles varying in size from 1 inch to 6 inches. Satisficing claims that the first needle that can sew on the patch is the one that should be used. Spending time searching for that one specific needle in the haystack is a waste of energy and resources.

Satisficing also occurs in consensus building when the group looks towards a solution everyone can agree on even if it may not be the best.

:Example: A group spends hours projecting the next fiscal year's budget. After hours of debating they eventually reach a consensus, only to have one person speak up and ask if the projections are correct. When the group becomes upset at the question, it is not because this person is wrong to ask, but rather because the group has already come up with a solution that works. The projection may not be what will actually come, but the majority agrees on one number and thus the projection is good enough to close the book on the budget.

In many circumstances, the individual may be uncertain about what constitutes a satisfactory outcome.

:Example: An individual who only seeks a satisfactory retirement income may not know what level of wealth is required—given uncertainty about future prices—to ensure a satisfactory income. In this case, the individual can only evaluate outcomes on the basis of their probability of being satisfactory. If the individual chooses that outcome which has the maximum chance of being satisfactory, then this individual's behavior is theoretically indistinguishable from that of an optimizing individual under certain conditions.<ref>{{cite journal |last=Castagnoli |first=E. |first2=M. |last2=LiCalzi |year=1996 |title=Expected Utility without Utility |journal=Theory and Decision |volume=41 |issue=3 |pages=281–301 |doi=10.1007/BF00136129 }}</ref><ref>{{cite journal |last=Bordley |first=R. |first2=M. |last2=LiCalzi |year=2000 |title=Decision Analysis Using Targets Instead of Utility Functions |journal=Decisions in Economics & Finance |volume=23 |issue=1 |pages=53–74 |doi=10.1007/s102030050005 }}</ref><ref>{{cite journal |last=Bordley |first=R. |first2=C. |last2=Kirkwood |year=2004 |title=Preference Analysis with Multiattribute Performance Targets |journal=Operations Research |volume=52 |issue=6 |pages=823–835 |doi=10.1287/opre.1030.0093 }}</ref>

Satisficing is often a good option when making a decision, but it can also be detrimental if used the wrong way.

:Example: When considering a medical issue such as a diagnosis, satisficing is not the best decision making strategy to use. On the other hand, when choosing an outfit or an option from a menu, it can be helpful. When there is an unlimited amount of information available and it is necessary to eliminate options, satisficing is beneficial because it helps the person making the decision effectively and efficiently reach a conclusion.<ref>{{cite book |last=Sternberg |first=R. J. |year=2009 |title=Cognitive Psychology |edition=5th |location=Belmont, CA |publisher=Wadsworth |isbn=049550629X }}</ref>

=== Satisficing and optimization ===
One definition of satisficing is that it is [[Optimization (mathematics)|optimization]] where ''all'' costs, including the cost of the optimization calculations themselves and the cost of getting information for use in those calculations, are considered. As a result, the eventual choice is usually sub-optimal in regard to the main goal of the optimization, i.e., different from the optimum in the case that the costs of choosing are not taken into account.

==== Satisficing as a form of optimization ====
Alternatively, satisficing can be considered to be just [[constraint satisfaction]], the process of finding a solution satisfying a set of constraints, without concern for finding an optimum. Any such satisficing problem can be formulated as an (equivalent) optimization problem using the [[Indicator function]] of the satisficing requirements as an [[objective function]]. More formally, if {{math|<var>X</var>}} denotes the set of all options and {{math| <var>S</var> &sube; <var>X</var>}} denotes the set of "satisficing" options, then selecting a satisficing solution (an element of {{math|<var>S</var>}}) is equivalent to the following optimization problem

: <math>\max_{s\in X} I_{S}(s)</math>

where {{math|Is}} denotes the [[Indicator function]] of {{math|<var>S</var>}}, that is

:<math>I_{S}(s):=\begin{cases} \begin{array}{ccc} 1 &,& s\in S\\
0 &,& s\notin S
\end{array}
\end{cases} \ , \ s\in X</math>

A solution {{math|<var>s</var> ∈ <var>X</var>}} to this optimization problem is optimal if, and only if, it is a satisficing option (an element of {{math|<var>S</var>}}). Thus, from a decision theory point of view, the distinction between "optimizing" and "satisficing" is essentially a stylistic issue (that can nevertheless be very important in certain applications) rather than a substantive issue. What is important to determine is '''what''' should be optimized and '''what''' should be satisficed. The following quote from Jan Odhnoff's 1965 paper is appropriate:<ref>{{cite journal |last=Odhnoff |first=Jan |year=1965 |title=On the Techniques of Optimizing and Satisficing |journal=The Swedish Journal of Economics |volume=67 |issue=1 |pages=24–39 |jstor=3439096 }}</ref>

{{quote|In my opinion there is room for both 'optimizing' and 'satisficing' models in business economics. Unfortunately, the difference between 'optimizing' and 'satisficing' is often referred to as a difference in the quality of a certain choice. It is a triviality that an optimal result in an optimization can be an unsatisfactory result in a satisficing model. The best things would therefore be to avoid a general use of these two words.}}

==== Satisficing applied to the utility framework ====
In [[economics]], '''satisficing''' is a [[behavior]] which attempts to achieve at least some [[minimum]] level of a particular [[Variable (mathematics)|variable]], but which does not necessarily maximize its value.<ref>{{cite book |chapterurl=http://huwdixon.org/SurfingEconomics/chapter7.pdf |chapter=Artificial Intelligence and Economic Theory |title=Surfing Economics: Essays for the Inquiring Economist |authorlink=Huw Dixon |first=Huw |last=Dixon |year=2001 |location=New York |publisher=Palgrave |isbn=0333760611 }}</ref> The most common application of the concept in economics is in the behavioral [[theory of the firm]], which, unlike traditional accounts, postulates that producers treat [[Profit (economics)|profit]] not as a goal to be maximized, but as a constraint. Under these theories, a critical level of profit must be achieved by firms; thereafter, priority is attached to the attainment of other goals.

More formally, as before if {{math|<var>X</var>}} denotes the set of all options {{math|<var>s</var>}}, and we have the payoff function '''{{math|<var>U</var>(<var>s</var>)}}''' which gives the payoff enjoyed by the agent for each option. Suppose we define the optimum payoff {{math|<var>U</var>*}} the solution to

<math>\max_{s\in X} U(s)</math>

with the optimum actions being the set '''{{math|<var>O</var>}}''' of options such that {{math|<var>U</var>(<var>s*</var>) {{=}} ''U''*}} (i.e. it is the set of all options that yield the maximum payoff). Assume that the set '''{{math|<var>O</var>}}''' has at least one element.

We now introduce the idea of the '''Aspiration level''' as introduced by [[Herbert Simon]] and developed in economics by Richard Cyert and James march in their 1963 book "A [[Behavioral theory of the firm]]".<ref>{{cite book |last=Cyert |first=Richard |last2=March |first2=James G. |year=1992 |title=A Behavioral Theory of the Firm |edition=2nd |publisher=Wiley-Blackwell |isbn=0-631-17451-6 }}</ref> The aspiration level is the payoff that the agent aspires to: if the agent achieves at least this level it is satisfied, and if it does not achieve it, the agent is not satisfied. Let us define the aspiration level '''{{math|''A''}}''' and assume that {{math|''A'' ≤ ''U''*}}. Clearly, whilst it is possible that someone can aspire to something that is better than the optimum, it is in a sense irrational to do so. So, we require the aspiration level to be at or below the optimum payoff.

We can then define the set of satisficing options '''{{math|<var>S</var>}}''' as all those options that yield at least '''{{math|<var>A</var>}}''': {{math|<var>s</var> ∈ <var>S</var>}} '''if and only if''' {{math|<var>A</var> ≤ <var>U</var>(<var>s</var>)}}. Clearly since {{math|<var>A</var> ≤ <var>U</var>*}}, it follows that {{math|<var>O</var> &sube; S}}. That is, the set of optimum actions is a subset of the set of satisficing options. So, when an agent satisfices, then she will choose from a larger set of actions than the agent who optimizes. One way of looking at this is that the satisficing agent is not putting in the effort to get to the precise optimum or is unable to exclude actions that are below the optimum but still above aspiration.

An equivalent way of looking at satisficing is '''epsilon-optimization''' (that means you choose your actions so that the payoff is within epsilon of the optimum). If we define the "gap" between the optimum and the aspiration as '''{{math|ε}}''' where {{math|ε {{=}} ''U''* − ''A''}}. Then the set of satisficing options '''{{math|<var>S</var>(ε)}}''' can be defined as all those options '''{{math|<var>s</var>}}''' such that {{math|<var>U</var>(<var>s</var>) ≥ <var>U</var>* − ε}}.

==== Other applications in economics ====
Apart from the behavioral theory of the firm, Applications of the idea of satisficing behavior in economics include the Akerlof and Yellen model of [[Menu costs]] popular in [[New Keynesian macroeconomics]].<ref>{{cite journal |last=Akerlof |first=George A. |last2=Yellen |first2=Janet L. |year=1985 |title=Can Small Deviations from Rationality Make Significant Differences to Economic Equilibria? |journal=[[American Economic Review]] |volume=75 |issue=4 |pages=708–720 |jstor=1821349 }}</ref><ref>{{cite journal |last=Akerlof |first=George A. |last2=Yellen |first2=Janet L. |year=1985 |title=A Near-rational Model of the Business Cycle, with Wage and Price Intertia |journal=[[The Quarterly Journal of Economics]] |volume=100 |issue=5 |pages=823–838 |doi=10.1093/qje/100.Supplement.823 }}</ref> Also, in economics and [[Game theory]] there is the notion of an [[Epsilon equilibrium]], which is a generalization of the standard [[Nash equilibrium]] in which each player is within '''{{math|ε}}''' of his or her optimal payoff (the standard Nash-equilibrium being the special case where '''{{math|ε {{=}} 0}}''').

=== Endogenous aspiration levels ===
What determines the aspiration level? This can come from past experience (some function of an agent's or firm's previous payoffs), or some organizational or market institutions. For example, if we think of managerial firms, the managers will be expected to earn [[normal profits]] by their shareholders. Other institutions may have specific targets imposed externally (for example state-funded universities in the UK have targets for student recruitment).

An economic example is the [[Huw Dixon|Dixon]] model of an economy consisting of many firms operating in different industries, where each industry is a [[oligopoly|duopoly]].<ref>{{cite journal |last=Dixon |first=H. |year=2000 |title=Keeping Up with the Joneses: Competition and the Evolution of Collusion |journal=Journal of Economic Behaviour and Organization |volume=43 |issue=2 |pages=223–238 |doi=10.1016/S0167-2681(00)00117-7 }}</ref> The endogenous aspiration level is the average profit in the economy. This represents the power of the financial markets: in the long-run firms need to earn normal profits or they die (as [[Armen Alchian]] once said “This is the criterion by which the economic system selects survivors: those who realize positive proﬁts are the survivors; those who suffer losses disappear”<ref>{{cite journal |last=Alchian |first=A. |year=1950 |title=Uncertainty, Evolution and Economic Theory |journal=[[Journal of Political Economy]] |volume=58 |issue=3 |pages=211–222 |jstor=1827159 }}</ref>). We can then think what happens over time. If firms are earning profits at or above their aspiration level, then they just stay doing what they are doing (unlike the optimizing firm which would always strive to earn the highest profits possible). However, if the firms are earning below aspiration, then they try something else, until they get into a situation where they attain their aspiration level. it can be shown that in this economy, '''satisficing''' leads to [[collusion]] amongst firms: competition between firms leads to lower profits for one or both of the firms in a duopoly. This means that competition is unstable: one or both of the firms will fail to achieve their aspirations and hence try something else. The only situation which is stable is one where all firms achieve their aspirations, which can only happen when all firms earn average profits. In general, this will only happen if all firms earn the joint-profit maximizing or collusive profit.<ref>Dixon (2000), Theorem 1 page 228. for a non-technical explanation see [http://www.huwdixon.org/SurfingEconomics/chapter8.pdf Chapter 8],[http://www.huwdixon.org/SurfingEconomics/index.html Surfing Economics] by Dixon H</ref>

== In psychology ==

=== As a personality trait ===
Some research has suggested that satisficing/maximizing and other decision-making strategies, like [[personality]] traits, have a strong genetic component and endure over time. This genetic influence on decision-making behaviors has been found through classical [[twin studies]], in which decision-making tendencies are self-reported by pairs of twins and then compared between monozygotic and dizygotic twins.<ref>{{cite journal |last=Simonson |first=I. |last2=Sela |first2=A. |year=2011 |title=On the heritability of consumer decision making: An exploratory approach for studying genetic effects on judgment and choice |journal=Journal of Consumer Research |volume=37 |issue=6 |pages=951–966 |doi=10.1086/657022 }}</ref> This implies that people can be categorized into "maximizers" and "satisficers", with some people landing in between.

=== Relationship with happiness ===
The distinction between satisficing and maximizing not only differs in the decision-making process, but also in the post-decision evaluation. Maximizers tend to use a more exhaustive approach to their decision-making process: they seek and evaluate more options than satisficers do to achieve greater satisfaction. However, whereas satisficers tend to be relatively pleased with their decisions, maximizers tend to be less happy with their decision outcomes. This is thought to be due to limited cognitive resources people have when their options are vast, forcing maximizers to not make an optimal choice. Because maximization is unrealistic and usually impossible in everyday life, maximizers often feel regretful in their post-choice evaluation.<ref>{{cite journal |last=Schwartz |first=B. |last2=Ward |first2=A. |last3=Monterosso |first3=J. |last4=Lyubomirsky |first4=S. |last5=White |first5=K. |last6=Lehman |first6=D. R. |year=2002 |title=Maximizing versus satisficing: Happiness is a matter of choice |journal=Journal of Personality and Social Psychology |volume=83 |issue=5 |pages=1178–1197 |doi=10.1037/0022-3514.83.5.1178 }}</ref>

== In survey methodology ==
As an example of satisficing, in the field of [[social cognition]], [[Jon Krosnick]] proposed a theory of [[statistical survey]] '''satisficing''' which says that optimal question answering by a survey respondent involves a great deal of [[cognitive]] work and that some people would use satisficing to reduce that burden. Some people may shortcut their cognitive processes in two ways:
* Weak satisficing: Respondent executes all cognitive steps involved in optimizing, but less completely and with [[bias]].
* Strong satisficing: Respondent offers responses that will seem reasonable to the interviewer without any memory search or information integration.

Likelihood to satisfice is linked to respondent ability, respondent [[motivation]] and task difficulty

Regarding survey answers, satisficing manifests in:
* choosing explicitly offered no-opinion response option
* choosing socially desirable responses
* non-differentiation when a battery of questions asks for ratings of multiple objects on the same response scale
* acquiescence response bias, which is the tendency to agree with any assertion, regardless of its content

== See also ==
{{div col|4}}
* [[Alpha-beta pruning]]
* [[Decision theory]]
* [[Flipism]]
* [[Frame problem]]
* [[Homo economicus]]
* [[Optimism bias]]
* [[Portmanteau]]
* [[Principle of good enough]]
* [[Rationality]]
* [[Rational ignorance]]
* [[Satisfiability]]
* [[Utility maximization problem]]
{{div col end}}

== References ==
{{Reflist|30em}}

== Further reading ==
*{{cite journal |first=Michael |last=Byron |title=Satisficing and Optimality |journal=[[Ethics (journal)|Ethics]] |volume=109 |issue=1 |year=1998 |pages=67–93 |doi=10.1086/233874 }} A paper on satisficing considered from a [[philosophy|philosophical]] viewpoint.
*{{cite book |last=Byron |first=M. |year=2004 |title=Satisficing and Maximizing: Moral Theorists on Practical Reason |location=New York |publisher=Cambridge University Press |isbn=052181149X }}
*{{cite book |first=J. N. |last=Bearden |first2=T. |last2=Connolly |chapter=On Optimal Satisficing: How simple policies can achieve excellent results |editor1-first=T. |editor1-last=Kugler |editor2-first=J. C. |editor2-last=Smith |editor3-first=T. |editor3-last=Connolly |editor4-first=Y. J. |editor4-last=Son |title=Decision Modeling in Uncertain and Complex Environments |publisher=Springer |location=New York |year=2008 |isbn=9780387771311 }}
*{{cite book |first=Huw |last=Dixon |chapterurl=http://huwdixon.org/SurfingEconomics/chapter8.pdf |chapter=Donut world and the duopoly archipelago |title=Surfing Economics: Essays for the Inquiring Economist |publisher=Palgrave |location=New York |year=2001 |isbn=0333760611 }}
*{{cite journal |last=Holbrook |first=A. |last2=Green |first2=M. |last3=Krosnick |first3=J. |year=2003 |title=Telephone versus Face-to-Face Interviewing of National Probability Samples with Long Questionnaires: Comparisons of Respondent Satisficing and Social Desirability Response Bias |journal=[[Public Opinion Quarterly]] |volume=67 |issue=1 |pages=79–125 |doi=10.1086/346010 }}
*{{cite journal |last=Krosnick |first=J. |year=1991 |title=Response Strategies for coping with the cognitive demands of attitude measures in surveys |journal=Applied Cognitive Psychology |volume=5 |issue=3 |pages=213–236 |doi=10.1002/acp.2350050305 }}
*{{cite book |last=Simon |first=H. A. |year=1957 |title=Models of Man: Social and Rational |location=New York |publisher=Wiley }}
*{{cite journal |last=Simon |first=H. A. |year=1978 |title=Rationality as a Process and Product of Thought |journal=[[American Economic Review]] |volume=68 |issue=1 |pages=1–16 |jstor=1816653 }}
*{{cite book |last=Simon |first=H. A. |year=1983 |title=Reason in Human Affairs |location=Stanford |publisher=Stanford University Press |isbn=0804711798 }}

== External links ==
*[http://pespmc1.vub.ac.be/ASC/SATISFICING.html ''Web Dictionary of Cybernetics and Systems'' definition of "satisficing"]
*[http://www.moshe-online.com/satisficing/ A web page dedicated to a discussion on the "satisficing" vs "optimizing" debate.]
*[http://video.google.com/videoplay?docid=6127548813950043200 Schwartz's Tech Talk ("The Paradox of Choice - Why More Is Less") given at Google on April 27, 2006]

[[Category:Underlying principles of microeconomic behavior]]
[[Category:Heuristics]]
[[Category:Organizational behavior]]