Phi

From Textus Receptus

Phi (uppercase Φ, lowercase φ or math symbol ϕ), pronounced ˈfaɪ or sometimes /ˈfiː/ in English,^[1] 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 golden ratio <math>\tfrac{1 + \sqrt{5}}{2} \approx 1.61803\ldots</math> 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.
• Electric potential in physics.
• 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.

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.
• Philosophy.
• 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.^[2]

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

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

References