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Concept
Phi
Summary
Phi (faɪ; uppercase Φ, lowercase φ or φ; φεῖ pheî phéî̯; Modern Greek: φι fi fi) is the twenty-first letter of the Greek alphabet.
In Archaic and Classical Greek (c. 9th century BC to 4th century BC), it represented an aspirated voiceless bilabial plosive ([ph]), which was the origin of its usual romanization as . During the later part of Classical Antiquity, in Koine Greek (c. 4th century BC to 4th century AD), its pronunciation shifted to that of a voiceless bilabial fricative ([ɸ]), and by the Byzantine Greek period (c. 4th century AD to 15th century AD) it developed its modern pronunciation as a voiceless labiodental fricative ([f]).
The romanization of the Modern Greek phoneme is therefore usually .
It may be that phi originated as the letter qoppa (Ϙ, ϙ), and initially represented the sound /kwh/ before shifting to Classical Greek ph. In traditional Greek numerals, phi has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) descends from phi.
As with other Greek letters, lowercase phi (encoded as the Unicode character ) is used as a mathematical or scientific symbol. Some uses require the old-fashioned 'closed' glyph, which is separately encoded as the Unicode character .
The lowercase letter φ (or often its variant, φ) is often used to represent the following:
The letter phi is commonly used in physics to represent wave functions in quantum mechanics, such as in the Schrödinger equation and bra–ket notation: .
The golden ratio 1.618033988749894848204586834... in mathematics, art, and architecture.
Euler's totient function φ(n) in number theory; also called Euler's phi function.
The cyclotomic polynomial functions Φn(x) of algebra.
The number of electrical phases in a power system in electrical engineering, for example 1φ for single phase, 3φ for three phase.
In algebra, group or ring homomorphisms
In probability theory, is the probability density function of the standard normal distribution.
In probability theory, φX(t) = E[eitX] is the characteristic function of a random variable X.
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