Phi

For the robot E-121 Phi, see Emerl.

Phi (uppercase Φ, lowercase φ or ϕ), pronounced [fi] in modern Greek and as [faɪ] in English, 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. In the system of Greek numerals it has a value of 500 (φʹ) or 500,000 (͵φ).

The lower-case letter $\phi$ (or often its variant, $\varphi$) is used as a symbol for:

• The golden ratio 1.618... in mathematics, art, and architecture.
• Euler's totient function φ(n) in number theory; also called Euler's phi function.
• The probability density function of the normal distribution in mathematics and statistics.
• 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
• Any function in mathematics.
• 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
• A sentence in first-order logic.

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

• 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.
• Strength (or resistance) reduction factor in structural engineering, used to account for statistical variabilities in materials and construction methods.
• The number of phases in a power system in electrical engineering, for example 1Φ for single phase, 3Φ for three phase.

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 phi phenomenon.

Computing

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

• lower case:
  • U+03C6 GREEK SMALL LETTER PHI (φ): letter phi, used in Greek texts.
  • U+03D5 GREEK PHI SYMBOL (ϕ): phi symbol, for mathematical and technical contexts.
  • U+0278 LATIN SMALL LETTER PHI (ɸ): Latin letter phi (IPA symbol).
• upper case:
  • U+03A6 GREEK CAPITAL LETTER PHI (Φ): Greek capital letter phi

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 ($\varphi$).