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]]

Mnemonics in trigonometry

2014-01-28T11:12:01Z

112.207.36.190: /* Mnemonic chart */

[[File:IELMehmetNadir.jpg|thumb|Mehmet Nadir Bey, with daughter Hadiye]]

'''Mehmet Nadir''' (1856–1927) was a [[Turkish people|Turkish]] mathematician and educationist

== Early life ==
He was born in [[Chios|Sakız]] island (modern Chios in [[Greece]]) then a part of the [[Ottoman Empire]], to a poor family. He was adopted by a sea captain (who’d be his father in law in the future). He studied in the military high school in [[Bursa]]. Then he came to [[İstanbul]] to study in military collage and then the naval academy. After completing his education with honours, he was appointed as the mathematics teacher of the naval academy and then [[Darüşşafaka High School]], one of the most prominent schools in İstanbul. In 1879, together with one of his friends he travelled to [[England]] for advanced studies. But the details of this period are not known. Probably, he also spent some time in [[Cyprus]].<ref name="İnönü">Erdal İnönü: ''Mehmet Nadir'', Tubitak, Ankara 1997, ISBN 975-403-074-X</ref>

== Educationist ==
After returning to İstanbul, he began teaching in private high schools and in 1884 he established a high school of his own. It was a junior high school ( ''Rüştiye''). But later he added senior high school section to his school (''İdadi'') . Although there were private junior high schools before 1884, he was the founder of the very first private senior high school in İstanbul.<ref name="İnönü"/> This school is now considered as the core of the modern [[Istanbul High School]] for Boys ({{lang-tr|İstanbul Erkek Lisesi}}). He also added a section for the girls and a branch school in [[Edirne]]. Another novelty he introduced was the first school newspaper in Istanbul.
<ref>[http://www.istanbullisesi.k12.tr/Okulumuz/Pages/Eski-Mudurlerimiz.aspx İstanbul Erkek Lisesi page {{tr icon}}]</ref>

== Politics and later life==
He secretly supported the [[Young Turks]] movement. But when the sultan [[Abdülhamit II]] arrested 350 Young Turk adherents on the charge of planning a coup in 1896, Mehmet Nadir was forced to resign. After working in a public school in İstanbul, in 1903 he was appointed to [[Aleppo]] (now in [[Syria]]) as the director of education. In 1908 Young Turks came to power and he was exiled to [[Tripoli]] (now in [[Libya]]) by the Young Turks who suspected of his betrayal back in 1896.<ref name="İnönü"/> After the [[Italo-Turkish War]] in which [[Italian army|Italians]] invaded Libya he returned to İstanbul. Although he was also sent to Edirne, he soon returned to İstanbul during the [[Balkan Wars]]. After he convinced Young Turks of his innocence, he was appointed as the mathematics professor in the newly established Girls' University in İstanbul. In 1919 soon after the [[World War I]] he began working in the [[Darülfünun]] (present day, [[İstanbul University]]) as a ''müderris'' (professor) of mathematics <ref>[http://www.istanbul.edu.tr/fen/mat/tarihce.php İstanbul University page]</ref> in the newly established branch of [[Number theory]]. He died on 27 December 1927.

== As a mathematician ==
Mehmet Nadir is considered as one of the first mathematicians of Turkey.<ref>[http://www.bilkent.edu.tr/~sertoz/turkler.htm The list in Bilkent University page]</ref> He actively participated in a group of mathematicians of a [[French language|French]] mathematics periodical named ''I’Intermediaire des Mathematiciens''. According to Professor [[Erdal İnönü]], total number of his messages to the periodical was 62. One example cited by Erdal İnönü is as follows<ref name="İnönü"/>
The whole number solution to the equation

:<math>x^2+y^2-z^2=u^5</math>

proposed by Mehmet Nadir is

:<math>x=b\cdot(a^2+b^2)\cdot (a^2-b^2)</math>
:<math>y=\frac{1}{2}\cdot((a^2-1)\cdot(a^2+b^2)^2-4\cdot b^2)</math>
:<math>z=\frac{1}{2}\cdot((a^2+1)\cdot(a^2+b^2)^2+4\cdot b^2)</math>
:<math>u=a^2+b^2</math>

where ''b'' is a pozitive integer and ''a'' is a positive odd integer.

In 1917 Mehmet Nadir published a book about the Number theory. In this book he proposed an alternative algorithm on divisibility. [[Felix Klein]], well known [[German people|German]] mathematics professor congratulated him for the algorithm.<ref name="İnönü"/>

== References ==
<references/>

{{Persondata 
| NAME = Nadir, Mehmet
| ALTERNATIVE NAMES =
| SHORT DESCRIPTION = Turkish mathematician
| DATE OF BIRTH = 1856
| PLACE OF BIRTH =
| DATE OF DEATH = 1927
| PLACE OF DEATH =
}}
{{DEFAULTSORT:Nadir, Mehmet}}
[[Category:1856 births]]
[[Category:1927 deaths]]
[[Category:Ottoman scientists]]
[[Category:Turkish mathematicians]]
[[Category:Year of birth unknown]]
[[Category:People from Chios]]

Square root of 2

2014-01-26T02:20:02Z

112.207.32.182:

{{Refimprove|date=March 2013}}
[[File:Gimbal 3 axes rotation.gif|thumb|Gimbal with 3 axes of rotation. A set of three gimbals mounted together to allow three degrees of freedom: roll, pitch and yaw. When two gimbals rotate around the same axis, the system loses one degree of freedom.]]
[[File:Gimbal lock still occurs with 4 axis.png|thumb|Adding a fourth rotational axis can solve the problem of gimbal lock, but it requires the outermost ring to be actively driven so that it stays 90 degrees out of alignment with the innermost axis (the flywheel shaft). Without active driving of the outermost ring, all four axes can become aligned in a plane as shown above, again leading to gimbal lock and inability to roll.]]

'''Gimbal lock''' is the loss of one [[degree of freedom (mechanics)|degree of freedom]] in a three-dimensional, three-[[gimbal]] mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into [[rotation]] in a degenerate two-dimensional space.

The word ''lock'' is misleading: no gimbal is restrained. All three gimbals can still rotate freely about their respective axes of suspension. Nevertheless, because of the parallel orientation of two of the gimbals axes there is no gimbal available to accommodate rotation along one axis.

==Gimbals==
{{Main|Gimbal}}

A gimbal is a ring that is so suspended as to rotate about an axis. Gimbals typically nest one within another to accommodate rotation about multiple axes.

They so appear in [[gyroscope]]s and in [[inertial measurement unit]]s as to allow the inner gimbal's orientation to remain fixed while the outer gimbal suspension assumes any orientation. In [[compasses]] and [[flywheel energy storage]] mechanisms they allow objects to remain upright. They are used to orient [[Rocket engine|thrusters]] on rockets.<ref>{{cite web|url=http://science.howstuffworks.com/gimbal.htm|title=What is a gimbal -- and what does it have to do with NASA?|author=Jonathan Strickland|year=2008}}</ref>

Some [[coordinate system]]s in mathematics behave as if real gimbals were used to measure the angles, notably [[Euler angles]].

For cases of three or fewer nested gimbals, gimbal lock inevitably occurs at some point in the system due to properties of [[covering space]]s (described below).

==Gimbal lock in mechanical engineering==
===Gimbal lock in two dimensions===
Gimbal lock can occur in gimbal systems with two degrees of freedom such as a [[theodolite]] with rotations about an [[azimuth]] and elevation in two dimensions. These systems can gimbal lock at [[zenith]] and [[nadir]], because at those points azimuth is not well-defined, and rotation in the azimuth direction does not change the direction the theodolite is pointing.

Consider tracking a helicopter flying towards the theodolite from the horizon. The theodolite is a telescope mounted on a tripod so that it can move in azimuth and elevation to track the helicopter. The helicopter flies towards the theodolite and is tracked by the telescope in elevation and azimuth. The helicopter flies immediately above the tripod (i.e. it is at zenith) when it changes direction and flies at 90 degrees to its previous course. The telescope cannot track this maneuver without a discontinuous jump in one or both of the gimbal orientations. There is no continuous motion that allows it to follow the target. It is in gimbal lock. So there is an infinity of directions around zenith that the telescope cannot continuously track all movements of a target.<ref>{{cite web|url= http://www.madsci.org/posts/archives/aug98/896993617.Eg.r.html |title= Re: What is meant by the term gimbal lock? |author= Adrian Popa |date=Thu Jun 4 10:47:03 1998}}</ref> Note that even if the helicopter does not pass through zenith, but only ''near'' zenith, so that gimbal lock does not occur, the system must still move exceptionally rapidly to track it, as it rapidly passes from one bearing to the other. The closer to zenith the nearest point is, the faster this must be done, and if it actually goes through zenith, the limit of these "increasingly rapid" movements becomes ''infinitely'' fast, namely discontinuous.

To recover from gimbal lock the user has to go around the zenith – explicitly: reduce the elevation, change the azimuth to match the azimuth of the target, then change the elevation to match the target.

Mathematically, this corresponds to the fact that [[spherical coordinates]] do not define a [[coordinate chart]] on the sphere at zenith and nadir. Alternatively, that the corresponding map ''T''2→''S''2 from the [[torus]] ''T''2 to the sphere ''S''2 (given by the point with given azimuth and elevation) is not a [[covering map]] at these points.

===Gimbal lock in three dimensions===
[[File:Gimbal lock airplane.gif|thumb|Gimbal locked airplane. When the pitch (green) and yaw (magenta) gimbals become aligned, changes to roll (blue) and yaw apply the same rotation to the airplane.]]
[[Image:no gimbal lock.png|thumb|Normal situation: the three gimbals are independent]]
[[Image:gimbal lock.png|thumb|Gimbal lock: two out of the three gimbals are in the same plane, one degree of freedom is lost]]

Consider a case of a level sensing platform on an aircraft flying due North with its three gimbal axes mutually perpendicular (i.e., [[Roll (flight)|roll]], [[Pitch (aviation)|pitch]] and [[Yaw angle|yaw]] angles each zero). If the aircraft pitches up 90 degrees, the aircraft and platform's Yaw axis gimbal becomes parallel to the Roll axis gimbal, and changes about yaw can no longer be compensated for.

===Solutions===
This problem may be overcome by use of a fourth gimbal, intelligently driven by a motor so as to maintain a large angle between roll and yaw gimbal axes. Another solution is to rotate one or more of the gimbals to an arbitrary position when gimbal lock is detected and thus reset the device.

Modern practice is to avoid the use of gimbals entirely. In the context of [[inertial navigation system]]s, that can be done by mounting the inertial sensors directly to the body of the vehicle (this is called a [[strapdown]] system)<ref>{{cite web|url=http://xenia.media.mit.edu/~verp/projects/smartpen/node8.html#SECTION00322000000000000000|title=Overview of Pen Design and Navigation Background|author=Chris Verplaetse|year=1995}}</ref> and integrating sensed rotation and acceleration digitally using [[quaternion]] methods to derive vehicle orientation and velocity. Another way to replace gimbals is to use fluid bearings or a flotation chamber.<ref>{{cite web|url=http://www.wipo.int/pctdb/en/wo.jsp?IA=US2005043537&DISPLAY=DESC|title=Articulated gas bearing support pads|author=Chappell, Charles, D.|year=2006}}</ref>

===Gimbal lock on Apollo 11===
A well-known gimbal lock incident happened in the [[Apollo 11]] Moon mission. On this spacecraft, a set of gimbals was used on an [[inertial measurement unit]] (IMU). The engineers were aware of the gimbal lock problem but had declined to use a fourth gimbal.<ref>{{cite web|url=http://www.hq.nasa.gov/alsj/e-1344.htm|title=Apollo Guidance and Navigation - Considerations of Apollo IMU Gimbal Lock - MIT Instrumentation Laboratory Document E-1344|author=David Hoag|year=1963}}</ref> Some of the reasoning behind this decision is apparent from the following quote:

{{Quote|"The advantages of the redundant gimbal seem to be outweighed by the equipment simplicity, size advantages, and corresponding implied reliability of the direct three degree of freedom unit."|David Hoag|''Apollo Lunar Surface Journal''}}

They preferred an alternate solution using an indicator that would be triggered when near to 85 degrees pitch.

{{Quote|"Near that point, in a closed stabilization loop, the torque motors could theoretically be commanded to flip the gimbal 180 degrees instantaneously. Instead, in the [[Apollo Lunar Module|LM]], the computer flashed a 'gimbal lock' warning at 70 degrees and froze the IMU at 85 degrees"|Paul Fjeld|''Apollo Lunar Surface Journal''}}

Rather than try to drive the gimbals faster than they could go, the system simply gave up and froze the platform. From this point, the spacecraft would have to be manually moved away from the gimbal lock position, and the platform would have to be manually realigned using the stars as a reference.<ref>{{cite web|url=http://www.hq.nasa.gov/alsj/gimbals.html|title=Gimbal Angles, Gimbal Lock, and a Fourth Gimbal for Christmas|author=Eric M. Jones and Paul Fjeld|year=2006}}</ref>

After the Lunar Module had landed, [[Michael Collins (astronaut)|Mike Collins]] aboard the Command Module joked "How about sending me a fourth gimbal for Christmas?"

===Robotics===
[[File:Automation of foundry with robot.jpg|thumb|right|Industrial robot operating in a foundry.]]
In robotics, gimbal lock is commonly referred to as "wrist flip", due to the use of a "triple-roll wrist" in [[robotic arm]]s, where three axes of the wrist, controlling yaw, pitch, and roll, all pass through a common point.

An example of a wrist flip, also called a wrist singularity, is when the path through which the robot is traveling causes the first and third axes of the robot's wrist to line up. The second wrist axis then attempts to spin 180° in zero time to maintain the orientation of the end effector. The result of a singularity can be quite dramatic and can have adverse effects on the robot arm, the end effector, and the process.

The importance of non-singularities in robotics has led the American National Standard for Industrial Robots and Robot Systems — Safety Requirements to define it as "a condition caused by the collinear alignment of two or more robot axes resulting in unpredictable robot motion and velocities".<ref>ANSI/RIA R15.06-1999</ref>

==Gimbal lock in applied mathematics==
{{No footnotes|section|date=July 2013}}
The problem of gimbal lock appears when one uses [[Euler angles]] in applied mathematics; developers of 3D [[computer program]]s, such as [[3D modeling]], [[inertial guidance system|embedded navigation systems]], and [[video game]]s must take care to avoid it.

In formal language, gimbal lock occurs because the map from Euler angles to rotations (topologically, from the 3-torus ''T''3 to the [[real projective space]] '''RP'''3) is not a [[covering map]] – it is not a [[local homeomorphism]] at every point, and thus at some points the [[Rank (differential topology)|rank]] (degrees of freedom) must drop below 3, at which point gimbal lock occurs. Euler angles provide a means for giving a numerical description of any [[rotation]] in three-dimensional space using three numbers, but not only is this description not unique, but there are some points where not every change in the target space (rotations) can be realized by a change in the source space (Euler angles). This is a topological constraint – there is no covering map from the 3-torus to the 3-dimensional real projective space; the only (non-trivial) covering map is from the 3-sphere, as in the use of [[quaternions]].

To make a comparison, all the [[translation (geometry)|translations]] can be described using three numbers <math>x</math>, <math>y</math>, and <math>z</math>, as the succession of three consecutive linear movements along three perpendicular axes <math>X</math>, <math>Y</math> and <math>Z</math> axes. That's the same for rotations, all the rotations can be described using three numbers <math>\alpha</math>, <math>\beta</math>, and <math>\gamma</math>, as the succession of three rotational movements around three axes that are perpendicular one to the next. This similarity between linear coordinates and angular coordinates makes Euler angles very [[intuition (knowledge)|intuitive]], but unfortunately they suffer from the gimbal lock problem.

===Loss of a degree of freedom with Euler angles===
A rotation in 3D space can be represented numerically with [[matrix (mathematics)|matrices]] in several ways. One of these representations is:
:<math>\begin{align}
R &= \begin{bmatrix}
\cos \alpha & -\sin \alpha & 0 \\
\sin \alpha & \cos \alpha & 0 \\
0 & 0 & 1 \end{bmatrix} \begin{bmatrix}
1 & 0 & 0 \\
0 & \cos \beta & -\sin \beta \\
0 & \sin \beta & \cos \beta \end{bmatrix} \begin{bmatrix}
\cos \gamma & -\sin \gamma & 0 \\
\sin \gamma & \cos \gamma & 0 \\
0 & 0 & 1 \end{bmatrix} \end{align}
</math>
with <math>\alpha</math> and <math>\gamma</math> constrained in the interval <math>[-\pi, \pi]</math>, and <math>\beta</math> constrained in the interval <math>[0, \pi]</math>.

Let's examine for example what happens when <math>\beta = 0</math>. Knowing that <math>\cos\, 0 = 1</math> and <math>\sin \,0 = 0</math>, the above expression becomes equal to:
:<math>\begin{align}
R &= \begin{bmatrix}
\cos \alpha & -\sin \alpha & 0 \\
\sin \alpha & \cos \alpha & 0 \\
0 & 0 & 1 \end{bmatrix} \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{bmatrix} \begin{bmatrix}
\cos \gamma & -\sin \gamma & 0 \\
\sin \gamma & \cos \gamma & 0 \\
0 & 0 & 1 \end{bmatrix} \end{align}
</math>

The second matrix is the [[identity matrix]] and has no effect on the product. Carrying out [[matrix multiplication]] of first and third matrices:
:<math>\begin{align}
R &= \begin{bmatrix}
\cos \alpha \cos \gamma -\sin \alpha \sin \gamma & -\cos \alpha \sin \gamma - \sin \alpha \cos \gamma & 0 \\
\sin \alpha \cos \gamma + \cos \alpha \sin \gamma & -\sin \alpha \sin \gamma + \cos \alpha \cos \gamma & 0 \\
0 & 0 & 1 \end{bmatrix} \end{align}
</math>

And finally using the [[trigonometry formulas#Angle_sum_and_difference_identities|trigonometry formulas]]:
:<math>\begin{align}
R &= \begin{bmatrix}
\cos ( \alpha + \gamma ) & -\sin (\alpha + \gamma) & 0 \\
\sin ( \alpha + \gamma ) & \cos (\alpha + \gamma) & 0 \\
0 & 0 & 1 \end{bmatrix} \end{align}
</math>

Changing the values of <math>\alpha</math> and <math>\gamma</math> in the above matrix has the same effects: the rotation angle <math>\alpha + \gamma</math> changes, but the rotation axis remains in the <math>Z</math> direction: the last column and the last row in the matrix won't change. Only one degree of freedom (corresponding to <math>\alpha + \gamma</math>) remains; one other (corresponding to <math>\alpha - \gamma</math>) has been lost (the third degree of freedom corresponding to the choice <math>\beta = 0</math>). The only solution for <math>\alpha</math> and <math>\gamma</math> to recover different roles is to change <math>\beta</math> to some value other than 0. A similar problem appears when <math>\beta = \pi</math>.

One can choose another convention for representing a rotation with a matrix using Euler angles than the '''Z-X-Z''' convention above, and also choose other variation intervals for the angles, but in the end there is always at least one value for which a degree of freedom is lost.

Note that the gimbal lock problem does not make Euler angles "invalid" (they always serve as a well-defined coordinate system), but it makes them unsuited for some practical applications.

===Alternate orientation representation===
The cause of gimbal lock is representing an orientation as 3 axial rotations with [[Euler angles]]. A potential solution therefore is to represent the orientation in some other way. This could be as a [[rotation matrix]], a [[rotation formalisms in three dimensions#Quaternions|quaternion]], or a similar orientation representation that treats the orientation as a value rather than 3 separate and related values. Given such a representation, the user stores the orientation as a value. To apply angular changes, the orientation is modified by a delta angle/axis rotation. The resulting orientation must be re-normalized to prevent floating-point error from successive transformations from accumulating. For matrices, re-normalizing the result requires converting the matrix into its [[orthonormal matrix#Nearest_orthogonal_matrix|nearest orthonormal representation]]. For quaternions, re-normalization requires [[unit quaternions|performing quaternion normalization]].

==See also==
* [[Flight dynamics]]
* [[Inertial navigation system]]
* [[Motion planning]]
* [[Quaternions and spatial rotation]]
* [[Charts on SO(3)]]
* [[Grid north]] (equivalent navigational problem on polar expeditions)

==References==
<references/>

==External links==
* Euler angles and gimbal lock (video) [http://guerrillacg.org/home/3d-rigging/the-rotation-problem Part 1], [http://guerrillacg.org/home/3d-rigging/euler-rotations-explained Part 2]
* [http://www.youtube.com/watch?v=rrUCBOlJdt4 Gimble Lock - Explained] at [[YouTube]]

{{DEFAULTSORT:Gimbal Lock}}
[[Category:Rotation in three dimensions]]
[[Category:Angle]]
[[Category:Gyroscopes]]
[[Category:Spaceflight concepts]]
[[Category:3D computer graphics]]

Area of a disk

2014-01-18T13:09:59Z

112.207.23.172:

In [[number theory]], the '''Elliott–Halberstam conjecture''' is a [[conjecture]] about the distribution of [[prime number]]s in [[arithmetic progression]]s. It has many applications in [[sieve theory]]. It is named for [[Peter D. T. A. Elliott]] and [[Heini Halberstam]].

To state the conjecture requires some notation. Let <math>\pi(x)</math> denote the number of primes less than or equal to ''x''. If ''q'' is a [[negative and positive numbers|positive]] [[integer]] and ''a'' is [[coprime]] to ''q'', we let <math>\pi(x;q,a)</math>, denote the number of primes less than or equal to ''x'' which are equal to ''a'' modulo ''q''. [[Dirichlet's theorem on primes in arithmetic progressions]] then tells us
that

:<math> \pi(x;q,a) \approx \frac{\pi(x)}{\varphi(q)}</math>

where ''a'' is coprime to ''q'' and <math>\varphi</math> is [[Euler's totient function]]. If we then define the error function

:<math> E(x;q) = \max_{(a,q) = 1} \left|\pi(x;q,a) - \frac{\pi(x)}{\varphi(q)}\right|</math>

where the max is taken over all ''a'' coprime to ''q'', then the Elliott–Halberstam conjecture is the assertion that
for every ''θ'' < 1 and ''A'' > 0 there exists a constant ''C'' > 0 such that

:<math> \sum_{1 \leq q \leq x^\theta} E(x;q) \leq \frac{C x}{\log^A x}</math>

for all ''x'' > 2.

This conjecture was proven for all ''θ'' < 1/2 by [[Enrico Bombieri]] and [[A. I. Vinogradov]] (the [[Bombieri–Vinogradov theorem]], sometimes known simply as "Bombieri's theorem"); this result is already quite useful, being an averaged form of the [[generalized Riemann hypothesis]]. It is known that the conjecture fails at the endpoint ''θ'' = 1.

The Elliott–Halberstam conjecture has several consequences. One striking one is the result announced by [[Dan Goldston]], [[János Pintz]], and [[Cem Yıldırım]],<ref>{{arxiv|math.NT/0508185}}; see also {{arxiv|math.NT/0505300}}, {{arxiv|math.NT/0506067}}.</ref> which shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 16. In November 2013, James Maynard showed that subject to the Elliott-Halberstam conjecture, one can show the existence of infinitely many pairs of consecutive primes that differ by at most 12.

==See also==
*[[Barban–Davenport–Halberstam theorem]]
*[[Barban–Montgomery theorem]]

==Notes==
{{Reflist}}

== References ==
*{{cite journal |first=E. |last=Bombieri |title=On the large sieve |journal=Mathematika |volume=12 |issue= |year=1965 |pages=201–225 |doi= }}
*{{cite journal |first=P. D. T. A. |last=Elliott |first2=H. |last2=Halberstam |title=A conjecture in prime number theory |journal=Symp. Math. |volume=4 |year=1968 |issue= |pages=59–72 }}
*{{cite journal |first=A. I. |last=Vinogradov |title=The density hypothesis for Dirichlet L-series |language=Russian |journal=Izv. Akad. Nauk SSSR Ser. Mat. |volume=29 |issue=4 |year=1965 |pages=903–934 |mr=197414 }}
*{{cite journal |first=K. |last=Soundararajan |authorlink=Kannan Soundararajan |title=Small gaps between prime numbers: The work of Goldston–Pintz–Yıldırım |journal=Bull. AMS |volume=44 |year=2007 |issue=1 |pages=1–18 |doi=10.1090/S0273-0979-06-01142-6 }}

{{DEFAULTSORT:Elliott-Halberstam conjecture}}
[[Category:Analytic number theory]]
[[Category:Conjectures about prime numbers]]

Template:Lyceum of the Philippines University

2014-01-17T03:11:22Z

112.207.72.157:

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Icosidodecahedron

2014-01-12T15:47:14Z

112.207.23.172: /* Cartesian coordinates */

{{About|theoretical attempts to define income|its definition in United States law|Income (United States legal definitions)}}
{{refimprove|date=October 2010}}
'''Income''' is the consumption and savings opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms.<ref name="Barr"/> However, for households and individuals, "income is the sum of all the wages, salaries, profits, interests payments, rents and other forms of earnings received... in a given period of time."<ref name="Case & Fair">Case, K. & Fair, R. (2007). ''Principles of Economics''. Upper Saddle River, NJ: Pearson Education. p. 54.</ref>

In the field of [[public economics]], the term may refer to the accumulation of both monetary and non-monetary consumption ability, with the former (monetary) being used as a proxy for total income.<ref name="Barr"/>
==Increase in income==
Income per capita has been increasing steadily in almost every country.<ref>http://www.gapminder.org/world/#$majorMode=chart$is;shi=t;ly=2003;lb=f;il=t;fs=11;al=30;stl=t;st=t;nsl=t;se=t$wst;tts=C$ts;sp=5.59290322580644;ti=2007$zpv;v=0$inc_x;mmid=XCOORDS;iid=0AkBd6lyS3EmpdHo5S0J6ekhVOF9QaVhod05QSGV4T3c;by=ind$inc_y;mmid=YCOORDS;iid=rdCufG2vozTpKw7TBGbyoWw;by=ind$inc_s;uniValue=8.21;iid=phAwcNAVuyj0XOoBL_n5tAQ;by=ind$inc_c;uniValue=255;gid=CATID0;by=grp$map_x;scale=log;dataMin=58;dataMax=108111$map_y;scale=lin;dataMin=26;dataMax=56$map_s;sma=49;smi=2.65$cd;bd=0$inds=</ref> Many factors contribute to people having a higher income such as [[Education]],<ref>http://www.gapminder.org/world/#$majorMode=chart$is;shi=t;ly=2003;lb=f;il=t;fs=11;al=30;stl=t;st=t;nsl=t;se=t$wst;tts=C$ts;sp=5.59290322580644;ti=2007$zpv;v=0$inc_x;mmid=XCOORDS;iid=0AkBd6lyS3EmpdHo5S0J6ekhVOF9QaVhod05QSGV4T3c;by=ind$inc_y;mmid=YCOORDS;iid=pyj6tScZqmEdrsBnj2ROXAg;by=ind$inc_s;uniValue=8.21;iid=phAwcNAVuyj0XOoBL_n5tAQ;by=ind$inc_c;uniValue=255;gid=CATID0;by=grp$map_x;scale=log;dataMin=58;dataMax=108111$map_y;scale=lin;dataMin=8.7;dataMax=100$map_s;sma=49;smi=2.65$cd;bd=0$inds=</ref> [[globalisation]] and favorable political circumstances such as [[Index of Economic Freedom|economic freedom]] and [[peace]]. Increase income also tends to lead to people choosing to work less [[working hours]].
[[Developed countries]] defined as countries with a "developed economy" have higher incomes as opposed to a [[developing countries]] tend to have lower incomes.

==Economic definitions==
In [[economics]], "[[factor income]]" is the return accruing for a person, or a nation, derived from the "factors of production": rental income, wages generated by labor, the interest created by capital, and profits from entrepreneurial ventures.<ref>{{cite web|title=factor income|url=http://www.businessdictionary.com/definition/factor-income.html|work=BusinessDictionary.com|publisher=WebFinance, Inc|accessdate=20 June 2012|author=Staff|year=2012}}</ref>

From labor services, as well as ownership of [[Land (economics)|land]] and [[Capital (economics)|capital]].{{citation needed|date=June 2012}}

In [[consumer theory]] 'income' is another name for the "budget constraint," an amount <math>Y</math> to be spent on different goods x and y in quantities <math>x</math> and <math>y</math> at prices <math>P_x</math> and <math>P_y</math>. The basic equation for this is
:<math>Y=P_x \cdot x + P_y \cdot y</math>
This equation implies two things. First buying one more unit of good x implies buying <math>\frac{P_x}{P_y}</math> less units of good y. So, <math>\frac{P_x}{P_y}</math> is the ''relative'' price of a unit of x as to the number of units given up in y. Second, if the price of x falls for a fixed <math>Y</math>, then its relative price falls. The usual hypothesis is that the quantity demanded of x would increase at the lower price, the [[law of demand]]. The generalization to more than two goods consists of modelling y as a [[composite good]].

The theoretical generalization to more than one period is a multi-period [[wealth (economics)|wealth]] and income constraint. For example the same person can gain more productive skills or acquire more productive income-earning assets to earn a higher income. In the multi-period case, something might also happen to the economy beyond the control of the individual to reduce (or increase) the flow of income. Changing measured income and its relation to consumption over time might be modeled accordingly, such as in the [[permanent income hypothesis]].

===Full and Haig-Simons income===
{{main|Haig-Simons income}}
Full income refers to the accumulation of both, monetary and non-monetary consumption ability of any given entity, such a person or household. According to the what economist [[Nicholas Barr]] describes as the "classical definition of income:" the 1938 Haig-Simons definition, "income may be defined as the... sum of (1) the market value of rights exercised in consumption and (2) the change in the value of the store of property rights..." Since the consumption potential of non-monetary goods, such as leisure, cannot be measured, monetary income may be thought of as a proxy for full income.<ref name="Barr"/> As such, however, it is criticized for being unreliable, ''i.e.'' failing to accurately reflect affluence and that is consumption opportunities of any given agent. It omits the utility a person may derive from non-monetary income and, on a macroeconomic level, fails to accurately chart social welfare. According to Barr, "in practice money income as a proportion of total income varies widely and unsystematically. Non-observability of full-income prevent a complete characterization of the individual opportunity set, forcing us to use the unreliable yardstick of money income." On the macro-economic level, national per-capita income, increases with the consumption of activities that produce harm and omits many variables of societal health.<ref name="Barr">Barr, N. (2004). Problems and definition of measurement. In ''Economics of the welfare state''. New York: Oxford University Press. pp. 121-124</ref>

== Income inequality ==
[[Income inequality]] refers to the extent to which income is distributed in an uneven manner. Within a society can be measured by various methods, including the [[Lorenz curve]] and the [[Gini coefficient]]. Economists generally agree that certain amounts of inequality are necessary and desirable but that excessive inequality leads to efficiency problems and social injustice.<ref name="Barr"/>

National income, measured by statistics such as the [[Net National Income]] (NNI), measures the total income of individuals, corporations, and government in the economy. For more information see [[measures of national income and output]].

== Income in philosophy and ethics ==
Throughout history, many have written about the impact of income on [[morality]] and [[society]]. [[Saint Paul]] wrote 'For the love of money is a root of all kinds of evil:' ([[wikisource:Bible (American Standard)/1 Timothy#6:10|1 Timothy 6:10]] ([[American Standard Version|ASV]])).

Some scholars have come to the conclusion that material progress and prosperity, as manifested in continuous income growth at both individual and national level, provide the indispensable foundation for sustaining any kind of morality. This argument was explicitly given by [[Adam Smith]] in his ''Theory of Moral Sentiments''{{Citation needed|date=April 2009}}, and has more recently been developed by Harvard economist [[Benjamin M. Friedman|Benjamin Friedman]] in his book ''The Moral Consequences of Economic Growth''.{{Citation needed|date=April 2009}}

==Accountancy==
The [[International Accounting Standards Board]] (IASB) uses the following definition: "Income is increases in economic benefits during the accounting period in the form of inflows or enhancements of assets or decreases of liabilities that result in increases in equity, other than those relating to contributions from equity participants." [F.70] (IFRS Framework)

== See also ==
{{Wiktionary|income}}
*[[Basic income]]
*[[Comprehensive income]]
*[[Income tax]]
*[[Unpaid workers]]

==References==
{{reflist}}
*D. Usher (1987). "real income," ''The [[New Palgrave: A Dictionary of Economics]]'', v. 4, pp. 104–05

[[Category:Generally Accepted Accounting Principles]]
[[Category:Income| ]]

Diameter

2013-12-30T14:40:16Z

112.207.127.74:

{{about||the notion of diameter in graph theory|Distance (graph theory)|the computer network protocol|Diameter (protocol)}}
{{distinguish|Demeter}}

[[File:Circle-withsegments.svg|thumb|200px|right|Circle illustration with [[circumference]] (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.]]
In [[geometry]], the '''diameter''' of a [[circle]] is any straight [[line segment]] that passes through the center of the circle and whose endpoints lie on the circle. It can be also defined as the longest [[chord (geometry)|chord]] of the circle. Both definitions are also valid for the diameter of a [[sphere]]. The word "diameter" is derived from [[Greek language|Greek]] ''διάμετρος'' (''diametros''), "diameter of a circle", from ''δια-'' (''dia-''), "across, through" + ''μέτρον'' (''metron''), "measure".<ref>[http://www.etymonline.com/index.php?term=diameter Online Etymology Dictionary]</ref> It is often abbreviated '''DIA''', '''dia''', '''d''', or '''[[#Diameter symbol|⌀]]'''.

In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of ''the'' diameter rather than ''a'' diameter (which refers to the line itself), because all diameters of a circle or sphere have the same length, this being twice the [[radius]] '''r'''.

: <math>d = 2r \quad \Rightarrow \quad r = \frac{d}{2}.</math>

For a [[convex set|convex shape]] in the plane, the diameter is defined to be the largest distance that can be formed between two opposite [[parallel lines]] tangent to its boundary, and the ''width'' is defined to be the smallest such distance. Both quantities can be calculated efficiently using [[rotating calipers]].<ref>{{cite journal
| author = Toussaint, Godfried T.
| title = Solving geometric problems with the rotating calipers
| publisher=Proc. MELECON '83, Athens
|url=http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.155.5671
|year = 1983
}}</ref> For a [[curve of constant width]] such as the [[Reuleaux triangle]], the width and diameter are the same because all such pairs of parallel tangent lines have the same distance. See also [[Tangent lines to circles]].

==Generalizations==
The definitions given above are only valid for circles, spheres and convex shapes. However, they are special cases of a more general definition that is valid for any kind of ''n''-dimensional convex or non-convex object, such as a [[hypercube]] or a set of scattered points. The '''diameter''' of a [[subset]] of a [[metric space]] is the [[supremum|least upper bound]] of the set of all distances between pairs of points in the subset. So, if ''A'' is the subset, the diameter is
:[[supremum|sup]] { d(''x'', ''y'') | ''x'', ''y'' ∈ ''A'' } .
If the [[distance function]] d is viewed here as having [[codomain]] '''R''' (the set of all [[real number]]s), this implies that the diameter of the [[empty set]] (the case {{nowrap|1=''A'' = ∅}}) equals −∞ ([[negative infinity]]). Some authors prefer to treat the empty set as a special case, assigning it a diameter equal to 0,<ref>[http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0860.0002 Re: diameter of an empty set]</ref> which corresponds to taking the codomain of d to be the set of nonnegative reals.

For any solid object or set of scattered points in n-dimensional [[Euclidean space]], the diameter of the object or set is the same as the diameter of its [[convex hull]].

In [[differential geometry]], the diameter is an important global [[Riemannian geometry|Riemannian]] [[invariant (mathematics)|invariant]].

In plane geometry, a diameter of a [[conic section]] is typically defined as any chord which passes through the conic's centre; such diameters are not necessarily of uniform length, except in the case of the circle, which has [[eccentricity (mathematics)|eccentricity]] ''e'' = 0.

In medical [[Idiom#Parlance|parlance]] the diameter of a [[lesion]] is the longest line segment whose endpoints are within the lesion.

==Diameter symbol==
[[Image:Technical Drawing Hole 01.png|thumb|122px|Sign {{Unicode|⌀}} in a technical drawing]]
[[Image:Sign diameter.png|thumb|150px|Sign {{Unicode|⌀}} from [[AutoCAD]] drawing]]
{{distinguish2|the Scandinavian letter "[[Ø]]" or the [[Empty set]] symbol {{italics correction|"}}''{{Unicode|∅}}''"}}

The [[symbol]] or [[variable (mathematics)|variable]] for diameter, {{Unicode|⌀}}, is [[Homoglyph|similar in size and design]] to [[ø]], the Latin small letter o with stroke. [[Unicode]] provides character number 8960 ([[hexadecimal]] 2300) for the symbol, which can be encoded in [[HTML]] webpages as <tt>&#8960;</tt> or <tt>&#x2300;</tt>. The character can be obtained in [[Microsoft Windows]] by holding the {{key press|[[Alt key|Alt]]}} key down while entering {{key press|8}}{{key press|9}}{{key press|6}}{{key press|0}} on the [[numeric keypad]]. On an [[Apple, Inc.|Apple]] [[Macintosh]], the diameter symbol can be entered via the character palette (this is opened by pressing {{key press|Opt}}{{key press|Cmd}}{{key press|T}} in most applications), where it can be found in the Technical Symbols category.

The character will sometimes not display correctly, however, since many [[Typeface|font]]s do not include it. In most situations the letter ø is acceptable, which is unicode 0248 (hexadecimal 00F8). It can be obtained in UNIX-like operating systems using a [[Compose key]] by pressing, in sequence, {{key press|[[Compose key|Compose]]}}{{key press|/}}{{key press|o}} and on a Macintosh by pressing {{key press|Opt}} {{key press|O}} (in both cases, that is the letter [[o]], not the number [[0 (number)|0]]).

In [[LaTeX]] the symbol is achieved with the command \diameter which is part of the wasysym package.

The diameter symbol {{Unicode|⌀}} is distinct from the [[empty set]] symbol {{Unicode|∅}}, from an ([[italic script|italic]]) uppercase [[Phi (letter)|phi]] ''Φ'', and from the Nordic vowel [[Ø]].<ref>{{citation|title=Unicode Explained|first=Jukka K.|last=Korpela|publisher=O'Reilly Media, Inc.|year=2006|isbn=978-0-596-10121-3|pages=23–24|url=http://books.google.com/books?id=lxndiWaFMvMC&pg=PA23}}.</ref>

==See also==
* [[Angular diameter]]
* [[Caliper]], [[micrometer]], tools for measuring diameters
* [[Eratosthenes]], who calculated the diameter of the [[Earth]] around 240 BC.
* [[Distance (graph theory)|Graph or network diameter]]
* [[Hydraulic diameter]]
* [[List_of_gear_nomenclature#Inside diameter|Inside diameter]]
* [[Jung's theorem]], an inequality relating the diameter to the radius of the [[circumradius|smallest enclosing ball]]
* [[Sauter mean diameter]]

==Notes==
{{reflist}}

[[Category:Elementary geometry]]
[[Category:Length]]
[[Category:Circles]]

Gradian

2013-12-27T13:58:55Z

112.207.127.74: /* Conversion of some common angles */

{{For||Trapezoid (disambiguation)}}
{{Infobox Polygon
| name = Trapezoid (AmE) Trapezium (BrE)
| image = Trapezoid.svg
| caption = Trapezoid
| type = [[quadrilateral]]
| edges = 4
| symmetry =
| area = <math>\tfrac{a + b}{2} h</math>
| properties = [[convex polygon|convex]]}}
In [[Euclidean geometry]], a [[convex set|convex]] [[quadrilateral]] with at least one pair of parallel sides is referred to as a '''trapezoid''' in [[American English|American]] and [[Canadian English]] but as a '''trapezium''' in English outside North America. The parallel sides are called the ''bases'' of the trapezoid and the other two sides are called the ''legs'' or the lateral sides (if they are not parallel; otherwise there are two pairs of bases). A ''scalene trapezoid'' is a trapezoid with no sides of equal measure, in contrast to the [[Trapezoid#Special cases|special cases]] below.

There is some disagreement whether [[parallelogram]]s, which have two pairs of parallel sides, should be counted as trapezoids. Some define a trapezoid as a quadrilateral having ''only'' one pair of parallel sides (the exclusive definition), thereby excluding parallelograms.<ref>{{cite web |url=http://www.math.com/school/glossary/defs/trapezoid.html |title=American School definition from "math.com" |accessdate=2008-04-14}}</ref> Others<ref name=Mathworld>{{MathWorld |title=Trapezoid |urlname=Trapezoid}}</ref> define a trapezoid as a quadrilateral with ''at least'' one pair of parallel sides (the inclusive definition<ref>Trapezoids, [http://www.math.washington.edu/~king/coursedir/m444a00/syl/class/trapezoids/Trapezoids.html], accessed 2012-02-24.</ref>), making the parallelogram a special type of trapezoid. The latter definition is consistent with its uses in higher mathematics such as calculus. The former definition would make such concepts as the [[trapezoidal approximation]] to a [[definite integral]] ill-defined. This article uses the inclusive definition and considers parallelograms as special cases of a trapezoid. This is also advocated in the [[Quadrilateral#Taxonomy|taxonomy of quadrilaterals]].

==Etymology==
The term ''trapezium'' has been in use in English since 1570, from Late Latin ''trapezium'', from Greek τραπέζιον (''trapézion''), literally "a little table", a diminutive of τράπεζα (''trápeza''), "a table", itself from τετράς (''tetrás''), "four" + πέζα (''péza''), "a foot, an edge". The first recorded use of the Greek word translated ''trapezoid'' (τραπέζοειδη, ''trapézoeide'', "table-like") was by Marinus [[Proclus]] (412 to 485 AD) in his Commentary on the first book of [[Euclid's Elements]].<ref>Oxford English Dictionary entry at ''trapezoid''.</ref>

This article uses the term ''trapezoid'' in the sense that is current in the United States and Canada. In all other languages using a word derived from the Greek for this figure, the form closest to ''trapezium'' (e.g. French ''trapèze'', Italian ''trapezio'', Spanish ''trapecio'', German ''Trapez'', Russian ''трапеция'') is used.{{citation needed|date=January 2013}}

==Special cases==
In an [[isosceles trapezoid]], the legs (''AD'' and ''BC'' in the figure above) have the same length, and the base angles have the same measure. In a '''right trapezoid''' (also called right-angled trapezoid), two adjacent angles are [[right angles]].<ref name=Mathworld/> A [[tangential trapezoid]] is a trapezoid that has an [[Tangential quadrilateral|incircle]].

A trapezoid with two pairs of parallel sides is a [[parallelogram]]. Under the inclusive definition, all parallelograms (including [[rhombus]]es, [[rectangle]]s and [[Square (geometry)|square]]s) are trapezoids.

==Characterizations==
Given a convex quadrilateral, the following properties are equivalent, and each implies that the quadrilateral is a trapezoid:
*It has two adjacent [[angle]]s that are [[supplementary angles|supplementary]], that is, they add up 180 [[degree (angle)|degree]]s.

*The angle between a side and a diagonal is equal to the angle between the opposite side and the same diagonal.

*The [[diagonal]]s cut each other in mutually the same [[ratio]] (this ratio is the same as that between the lengths of the parallel sides).

*The diagonals cut the quadrilateral into four triangles of which one opposite pair are [[similarity (geometry)|similar]].

*The diagonals cut the quadrilateral into four triangles of which one opposite pair have equal areas.<ref name=Josefsson>Martin Josefsson, [http://forumgeom.fau.edu/FG2013volume13/FG201305.pdf "Characterizations of trapezoids"], Forum Geometricorum, 13 (2013) 23-35.</ref>{{rp|Prop.5}}

*The product of the areas of the two triangles formed by one diagonal equals the product of the areas of the two triangles formed by the other diagonal.<ref name=Josefsson/>{{rp|Thm.6}}

*The areas ''S'' and ''T'' of some two opposite triangles of the four triangles formed by the diagonals verify the equation

::<math>\sqrt{K}=\sqrt{S}+\sqrt{T},</math>
:where ''K'' is the area of the quadrilateral.<ref name="Josefsson" />{{rp|Thm.8}}

*The midpoints of two opposite sides and the intersection of the diagonals are [[collinear]].<ref name="Josefsson" />{{rp|Thm.15}}

Additionally, the following properties are equivalent, and each implies that opposite sides ''a'' and ''b'' are parallel:
* The consecutive sides ''a'', ''c'', ''b'', ''d'' and the diagonals ''p'', ''q'' verify the equation <ref name="Josefsson" />{{rp|Cor.11}}
::<math>p^2+q^2=c^2+d^2+2ab.</math>

*The distance ''v'' between the midpoints of the diagonals verifies the equation<ref name="Josefsson" />{{rp|Thm.12}}
::<math>v=\frac{|a-b|}{2}.</math>

==Midsegment and height==
The ''midsegment'' (also called the median or midline) of a trapezoid is the segment that joins the [[midpoint]]s of the legs. It is parallel to the bases. Its length ''m'' is equal to the average of the lengths of the bases ''a'' and ''b'' of the trapezoid,<ref name=Mathworld/>
:<math>m = \frac{a + b}{2}.</math>

The midsegment of a trapezoid is one of the two [[Quadrilateral#Special line segments|bimedian]]s (the other bimedian divides the trapezoid into equal areas).

The ''height'' (or altitude) is the [[perpendicular]] distance between the bases. In the case that the two bases have different lengths (''a'' ≠ ''b''), the height of a trapezoid ''h'' can be determined by the length of its four sides using the formula<ref name=Mathworld/>
:<math>h= \frac{\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}}{2|b-a|}</math>

where ''c'' and ''d'' are the lengths of the legs. This formula also gives a way of determining when a trapezoid of consecutive sides ''a'', ''c'', ''b'', and ''d'' exists. There is such a trapezoid with bases ''a'' and ''b'' if and only if<ref>Quadrilateral Formulas, ''The Math Forum'', Drexel University, 2012, [http://mathforum.org/dr.math/faq/formulas/faq.quad.html].</ref>
:<math>\displaystyle h^2>0.</math>

==Area==
The area ''K'' of a trapezoid is given by<ref name=Mathworld/>
:<math>K = \frac{a + b}{2} \cdot h</math>

where ''a'' and ''b'' are the lengths of the parallel sides, and ''h'' is the height (the perpendicular distance between these sides.) In 499 AD [[Aryabhata]], a great [[mathematician]]-[[astronomer]] from the classical age of [[Indian mathematics]] and [[Indian astronomy]], used this method in the ''[[Aryabhatiya]]'' (section 2.8). This yields as a special case the well-known formula for the area of a [[triangle]], by considering a triangle as a degenerate trapezoid in which one of the parallel sides has shrunk to a point.

Therefore the area of a trapezoid is equal to the length of this midsegment multiplied by the height<ref name=Mathworld/>
:<math>K = mh\,</math>

From the formula for the height, it can be concluded that the area can be expressed in terms of the four sides as<ref name=Mathworld/>
:<math>K = \frac{a+b}{4|b-a|}\sqrt{(-a+b+c+d)(a-b+c+d)(a-b+c-d)(a-b-c+d)}.</math>

When one of the parallel sides has shrunk to a point (say ''a'' = 0), this formula reduces to [[Heron's formula]] for the area of a triangle.

Another equivalent formula for the area, which more closely resembles Heron's formula, is<ref name=Mathworld/>
:<math>K = \frac{a+b}{|b-a|}\sqrt{(s-b)(s-a)(s-b-c)(s-b-d)},</math>

where <math>s = \tfrac{1}{2}(a + b + c + d)</math> is the [[semiperimeter]] of the trapezoid. (This formula is similar to [[Brahmagupta's formula]], but it differs from it, in that a trapezoid might not be [[cyclic quadrilateral|cyclic]] (inscribed in a circle). The formula is also a special case of [[Bretschneider's formula]] for a general [[quadrilateral]]).

From Bretschneider's formula, it follows that
:<math>K= \sqrt{\frac{(ab^2-a^2 b-ad^2+bc^2)(ab^2-a^2 b-ac^2+bd^2)}{(2(b-a))^2} - \left(\frac{b^2+d^2-a^2-c^2}{4}\right)^2}.</math>

The line that joins the midpoints of the parallel sides, bisects the area.

==Diagonals==
[[File:Trapezium.svg|200px|right]]
The lengths of the diagonals are<ref name=Mathworld/>
:<math>p= \sqrt{\frac{ab^2-a^2b-ac^2+bd^2}{b-a}},</math>
:<math>q= \sqrt{\frac{ab^2-a^2b-ad^2+bc^2}{b-a}}</math>

where ''a'' and ''b'' are the bases, ''c'' and ''d'' are the other two sides, and ''a'' < ''b''.

If the trapezoid is divided into four triangles by its diagonals ''AC'' and ''BD'' (as shown on the right), intersecting at ''O'', then the area of {{trianglenotation|AOD}} is equal to that of {{trianglenotation|BOC}}, and the product of the areas of {{trianglenotation|AOD}} and {{trianglenotation|BOC}} is equal to that of {{trianglenotation|AOB}} and {{trianglenotation|COD}}. The ratio of the areas of each pair of adjacent triangles is the same as that between the lengths of the parallel sides.<ref name=Mathworld/>

Let the trapezoid have vertices ''A'', ''B'', ''C'', and ''D'' in sequence and have parallel sides ''AB'' and ''DC''. Let ''E'' be the intersection of the diagonals, and let ''F'' be on side ''DA'' and ''G'' be on side ''BC'' such that ''FEG'' is parallel to ''AB'' and ''CD''. Then ''FG'' is the [[harmonic mean]] of ''AB'' and ''DC'':<ref>''GoGeometry'', [http://www.gogeometry.com/problem/p747-trapezoid-diagonal-parallel-similarity-harmonic-mean-high-school-college.htm], Accessed 2012-07-08.</ref>
:<math>\frac{1}{FG}=\frac{1}{2} \left( \frac{1}{AB}+ \frac{1}{DC} \right).</math>

The line that goes through both the intersection point of the extended nonparallel sides and the intersection point of the diagonals, bisects each base.<ref name=Byer/>

==Other properties==
The center of area (center of mass for a uniform [[planar lamina|lamina]]) lies along the line joining the midpoints of the parallel sides, at a perpendicular distance ''x'' from the longer side ''b'' given by<ref>''efunda'', General Trapezoid, [http://www.efunda.com/math/areas/Trapezoid.cfm], Accessed 2012-07-09.</ref>
:<math>x = \frac{h}{3} \left( \frac{2a+b}{a+b}\right).</math>

If the angle bisectors to angles ''A'' and ''B'' intersect at ''P'', and the angle bisectors to angles ''C'' and ''D'' intersect at ''Q'', then<ref name=Byer>Owen Byer, Felix Lazebnik and Deirdre Smeltzer, ''Methods for Euclidean Geometry'', Mathematical Association of America, 2010, p. 55.</ref>
:<math>PQ=\frac{|AD+BC-AB-CD|}{2}.</math>

==More on terminology==
The term trapezium is sometimes defined in the USA as a [[quadrilateral]] with no parallel sides, though this shape is more usually called an irregular quadrilateral.<ref>[http://www.chambersharrap.co.uk/chambers/features/chref/chref.py/main?xref=21C44644&title=21st&query=trapezoid ''Chambers 21st Century Dictionary'' Trapezoid]</ref><ref>{{cite web |url=http://www.merriam-webster.com/dictionary/trapezium |title=1913 American definition of trapezium |work=Merriam-Webster Online Dictionary |accessdate=2007-12-10}}</ref> The term trapezoid was once defined as a quadrilateral without any parallel sides in Britain and elsewhere, but this does not reflect current usage. (The Oxford English Dictionary says "Often called by English writers in the 19th century".)<ref name="oed">''[[Oxford English Dictionary]]'' entries for trapezoid and trapezium.</ref>

According to the ''[[Oxford English Dictionary]]'', the sense of a figure with no sides parallel is the meaning for which [[Proclus]] introduced the term "trapezoid". This is retained in the French ''trapézoïde'', German ''Trapezoid'', and in other languages. A trapezium in Proclus' sense is a quadrilateral having one pair of its opposite sides parallel. This was the specific sense in England in 17th and 18th centuries, and again the prevalent one in recent use. A trapezium as any quadrilateral more general than a [[parallelogram]] is the sense of the term in [[Euclid]]. The sense of a trapezium as an irregular quadrilateral having no sides parallel was sometimes used in England from c. 1800 to c. 1875, but is now obsolete. This sense is the one that is sometimes quoted in the US, but in practice quadrilateral is used rather than trapezium.<ref name="oed"/>

==Architecture==
[[Image:Temple of Dendur- night.jpg|right|250px|thumb|The [[Temple of Dendur]] in the Metropolitan Museum of Art, New York]]
In architecture the word is used to refer to symmetrical doors, windows, and buildings built wider at the base, tapering towards the top, in Egyptian style. If these have straight sides and sharp angular corners, their shapes are usually [[isosceles trapezoid]]s. This was the standard style for the doors and windows of the [[Inca Empire|Inca]]<nowiki/>s.<ref>http://gogeometry.com/MachuPicchu.htm</ref>

==See also==
*[[Polite number]], also known as a trapezoidal number
*[[Trapezoidal rule]]

==References==
{{reflist}}

==External links==
*[http://www.mathopenref.com/trapezoid.html Trapezoid definition]   [http://www.mathopenref.com/trapezoidarea.html Area of a trapezoid]   [http://www.mathopenref.com/trapezoidmedian.html Median of a trapezoid] With interactive animations
*[http://www.elsy.at/kurse/index.php?kurs=Trapezoid+%28North+America%29&status=public Trapezoid (North America)] at elsy.at: Animated course (construction, circumference, area)
*[http://numericalmethods.eng.usf.edu/topics/trapezoidal_rule.html] on ''Numerical Methods for Stem Undergraduate''
*Autar Kaw and E. Eric Kalu, ''Numerical Methods with Applications'', (2008) [http://www.autarkaw.com/books/numericalmethods/index.html]

[[Category:Quadrilaterals]]
[[Category:Elementary shapes]]