Phi

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[[Image:Phi.png|thumb|150px|Phi]]
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'''Phi''' (uppercase Φ, lowercase φ or math symbol ϕ), pronounced ˈfaɪ or sometimes /ˈfiː/ in English,<sup>[]</sup> and ˈfi in [[Modern Greek language|modern Greek]], is the 21st letter of the [[Greek alphabet]]. In modern Greek, it represents [f], a [[voiceless labiodental fricative]]. In Ancient Greek it represented [pʰ], an [[Aspiration (phonetics)|aspirated]] [[voiceless bilabial plosive]] (from which English ultimately inherits the spelling "[[ph (digraph)|ph]]" in words derived from Greek).  In the system of [[Greek numerals]] it has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter [[Ef (Cyrillic)|Ef]] (Ф, ф) arose from Φ.
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== Use as a symbol ==
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The lower-case letter <math>\varphi \,</math> (or often its variant, <math>\phi \,</math>) is often used to represent the following:
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*The [[golden ratio]] <math>\tfrac{1 + \sqrt{5}}{2} \approx 1.61803\ldots</math> in mathematics, art, and architecture.
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*[[Euler's totient function]] ''φ''(''n'') in number theory; also called ''Euler's phi function''.
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*In algebra, [[Group (mathematics)|group]] or [[Ring (mathematics)|ring]] [[homomorphism]]s
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* In [[probability theory]], {{nowrap|1=''ϕ''(''x'') = (2π)<sup>−½</sup>''e''<sup>−''x''<sup>2</sup>/2</sup>}} is the [[probability density function]] of the [[normal distribution]].
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* In [[probability theory]], {{nowrap|1=''φ''<sub>''X''</sub>(''t'') = E[&thinsp;''e''<sup>''itX''</sup>&thinsp;]}} is the [[characteristic function (probability theory)|characteristic function]] of a random variable ''X''.
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*An [[angle]], typically the second angle mentioned, after ''θ'' ([[theta]]).  Especially:
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**The argument of a [[complex number]].
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**The [[phase (waves)|phase of a wave]] in signal processing.
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**In [[spherical coordinates]], mathematicians usually refer to phi as the polar angle (from the ''z''-[[Coordinate axis|axis]]).  The convention in physics is to use phi as the azimuthal angle (from the ''x''-axis).
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**One of the [[dihedral angle]]s in the backbones of [[protein]]s.
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**Internal or effective [[angle of friction]].
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*[[Electric potential]] in physics.
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*The [[work function]] in electronics.
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*A [[shorthand]] representation for an [[aromatic]] [[functional group]] in [[organic chemistry]].
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*The [[fugacity]] coefficient in [[thermodynamics]].
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*The ratio of [[Thermodynamic free energy|free energy]] destabilizations of protein [[mutant]]s in [[phi value analysis]].
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*In [[cartography]] and [[navigation]], [[latitude]].
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*In combustion engineering, equivalence ratio. The ratio between the actual fuel air ratio to the stoichiometric fuel air ratio.
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*A sentence in [[first-order logic]].
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*[[Porosity]] in geology and [[hydrology]].
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*Strength (or resistance) reduction factor in [[structural engineering]], used to account for statistical variabilities in materials and construction methods.
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*The symbol for a [[voiceless bilabial fricative]] in the [[International Phonetic Alphabet]] (using the straight line variant character)
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*In economics, this is usually an additive term.{{Clarify|date=October 2009}}
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*In [[flight dynamics]], the roll angle.
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The upper-case letter Φ is used as a symbol for:
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*The [[Golden_ratio#Golden_ratio_conjugate|golden ratio conjugate]] −0.618... in mathematics.
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*The [[magnetic flux]] and [[electric flux]] in physics, with subscripts distinguishing the two.
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*The [[cumulative distribution function]] of the [[normal distribution]] in [[mathematics]] and [[statistics]].
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*[[Philosophy]].
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*The number of phases in a power system in [[electrical engineering]], for example 1Φ for [[single phase]], 3Φ for [[three phase]].
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*A common symbol for the parametrization of a surface in vector [[calculus]].
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*In [[Lacanian algebra]], Φ stands for the imaginary phallus and also represents phallic signification. -Φ stands in for castration.<sup>[]</sup>
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The [[diameter]] symbol in [[engineering]], [[⌀]], is often incorrectly referred to as "phi". This symbol is used to indicate the diameter of a circular section, for example "⌀14", means the diameter of the [[circle]] is 14 [[Units of measurement|units]].
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== See also ==
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*[[F|F, f - Ef (Latin)]]
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*[[Ef (Cyrillic)|Ф, ф - Ef (Cyrillic)]]
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*[[Psi and phi type figurine]]
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* [[Փ]] ([[Armenian alphabet|Armenian letter]])
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==References==
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{{DEFAULTSORT:Phi (Letter)}}
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[[Category:Greek letters]]

Revision as of 03:55, 2 April 2011

Phi
Phi

Phi (uppercase Φ, lowercase φ or math symbol ϕ), pronounced ˈfaɪ or sometimes /ˈfiː/ in English,[] and ˈfi in modern Greek, is the 21st letter of the Greek alphabet. In modern Greek, it represents [f], a voiceless labiodental fricative. In Ancient Greek it represented [pʰ], an aspirated voiceless bilabial plosive (from which English ultimately inherits the spelling "ph" in words derived from Greek). In the system of Greek numerals it has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) arose from Φ.

Use as a symbol

The lower-case letter <math>\varphi \,</math> (or often its variant, <math>\phi \,</math>) is often used to represent the following:

The upper-case letter Φ is used as a symbol for:

The diameter symbol in engineering, ⌀, is often incorrectly referred to as "phi". This symbol is used to indicate the diameter of a circular section, for example "⌀14", means the diameter of the circle is 14 units.

See also

References

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