Phi

This article is about the letter of the Greek alphabet. For other uses, see Phi (disambiguation).

Greek alphabet
Αα	Alpha	Νν	Nu
Ββ	Beta	Ξξ	Xi
Γγ	Gamma	Οο	Omicron
Δδ	Delta	Ππ	Pi
Εε	Epsilon	Ρρ	Rho
Ζζ	Zeta	Σσς	Sigma
Ηη	Eta	Ττ	Tau
Θθ	Theta	Υυ	Upsilon
Ιι	Iota	Φφ	Phi
Κκ	Kappa	Χχ	Chi
Λλ	Lambda	Ψψ	Psi
Μμ	Mu	Ωω	Omega
History
Archaic local variants
Ligatures (ϛ, ȣ, ϗ) Diacritics
Numerals (6) (90) (900)
In other languages
Bactrian Coptic Albanian
Scientific symbols
Book Category Commons

Phi (uppercase Φ, lowercase φ, or math symbol ϕ), pronounced /faɪ/ FY or sometimes /fiː/ FEE 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 φ (or often its variant, ϕ) is often used to represent the following:

• The golden ratio $\tfrac{1 + \sqrt{5}}{2} \approx 1.618033988749894848204586834\ldots$ in mathematics, art, and architecture.
• Euler's totient function φ(n) in number theory; also called Euler's phi function.
• In algebra, group or ring homomorphisms
• In probability theory, ϕ(x) = (2π)^−½e^−x2/2 is the probability density function of the normal distribution.
• In probability theory, φ_X(t) = E[e^itX] is the characteristic function of a random variable X.
• An angle, typically the second angle mentioned, after θ (theta). Especially:
  • The argument of a complex number.
  • The phase of a wave in signal processing.
  • In spherical coordinates, mathematicians usually refer to phi as the polar angle (from the z-axis). The convention in physics is to use phi as the azimuthal angle (from the x-axis).
  • One of the dihedral angles in the backbones of proteins.
  • Internal or effective angle of friction.
• The work function in electronics.
• A shorthand representation for an aromatic functional group in organic chemistry.
• The fugacity coefficient in thermodynamics.
• The ratio of free energy destabilizations of protein mutants in phi value analysis.
• In cartography and navigation, latitude.
• In combustion engineering, equivalence ratio. The ratio between the actual fuel air ratio to the stoichiometric fuel air ratio.
• A sentence in first-order logic.
• Porosity in geology and hydrology.
• Strength (or resistance) reduction factor in structural engineering, used to account for statistical variabilities in materials and construction methods.
• The symbol for a voiceless bilabial fricative in the International Phonetic Alphabet (using the straight line variant character)
• In economics, this is usually an additive term.
• In flight dynamics, the roll angle.
• In philosophy, φ is often used as shorthand for a generic act. (Also in upper-case.)

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

• The golden ratio conjugate −0.618... in mathematics.
• The magnetic flux and electric flux in physics, with subscripts distinguishing the two.
• The cumulative distribution function of the normal distribution in mathematics and statistics.
• In philosophy, Φ is often used as shorthand for a generic act. (Also in lower-case.)
• The number of phases in a power system in electrical engineering, for example 1Φ for single phase, 3Φ for three phase.
• A common symbol for the parametrization of a surface in vector calculus.
• In Lacanian algebra, Φ stands for the imaginary phallus and also represents phallic signification. -Φ stands in for castration.

The diameter symbol in engineering, ⌀, is often incorrectly^{[citation needed]} 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.

Computing

In Unicode, there are multiple forms of the phi letter:

Character	Name	Correct appearance	Your browser	Usage
U+03A6	GREEK CAPITAL LETTER PHI	$\Phi\,\!$	Φ	used in Greek texts
U+03C6	GREEK SMALL LETTER PHI	$\varphi\,\!$ or $\phi\,\!$	φ	used in Greek texts (contrary to what is shown here, it is normally not italicized)
U+03D5	GREEK PHI SYMBOL	$\phi\,\!$	ϕ	used in mathematical and technical contexts
U+0278	LATIN SMALL LETTER PHI		ɸ	used in IPA to symbolise a voiceless bilabial fricative

In some older fonts that are not yet compatible with Unicode 3.0 from 1998, the U+03D5 GREEK PHI SYMBOL might be represented by the "loopy" $\varphi$ symbol instead. This is no longer a correct representation. The U+03C6 GREEK SMALL LETTER PHI may be presented as either the "stroked" $\phi\,\!$ glyph, but preferably as the "loopy" $\varphi$ glyph.

In HTML/XHTML, the upper and lower case phi character entity references are Φ (Φ) and φ (φ) respectively.

In LaTeX, the math symbols are \Phi ($\Phi\,\!$), \phi ($\phi\,\!$), and \varphi ($\varphi\,\!$).

See also

• F, f - Ef (Latin)
• Ф, ф - Ef (Cyrillic)
• Psi and phi type figurine
• Փ (Armenian letter)
• Deposition (geology)