Phi (; uppercase Φ, lowercase φ or ϕ; grc, ϕεῖ ''pheî'' ;

Modern Greek Modern Greek (, , or , ''Kiní Neoellinikí Glóssa''), generally referred to by speakers simply as Greek (, ), refers collectively to the dialects of the Greek language spoken in the modern era, including the official standardized form of the ...

: ''fi'' ) is the 21st letter of the

Greek alphabet The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as we ...

. In

Archaic Archaic is a period of time preceding a designated classical period, or something from an older period of time that is also not found or used currently: *List of archaeological periods **Archaic Sumerian language, spoken between 31st - 26th cent ...

and

Classical Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic peri ...

(c. 9th century BC to 4th century BC), it represented an aspirated voiceless bilabial plosive (), which was the origin of its usual romanization as . During the later part of

Classical Antiquity Classical antiquity (also the classical era, classical period or classical age) is the period of cultural history between the 8th century BC and the 5th century AD centred on the Mediterranean Sea, comprising the interlocking civilizations of ...

, in

Koine Greek Koine Greek (; Koine el, ἡ κοινὴ διάλεκτος, hē koinè diálektos, the common dialect; ), also known as Hellenistic Greek, common Attic, the Alexandrian dialect, Biblical Greek or New Testament Greek, was the common supra-reg ...

(c. 4th century BC to 4th century AD), its pronunciation shifted to that of a

voiceless bilabial fricative The voiceless bilabial fricative is a type of consonantal sound, used in some spoken languages. The symbol in the International Phonetic Alphabet that represents this sound is . Features Features of the voiceless bilabial fricative: Occ ...

(), and by the

Byzantine Greek Medieval Greek (also known as Middle Greek, Byzantine Greek, or Romaic) is the stage of the Greek language between the end of classical antiquity in the 5th–6th centuries and the end of the Middle Ages, conventionally dated to the Ottoman co ...

period (c. 4th century AD to 15th century AD) it developed its modern pronunciation as a

voiceless labiodental fricative The voiceless labiodental fricative is a type of consonantal sound used in a number of spoken languages. The symbol in the International Phonetic Alphabet that represents this sound is . Some scholars also posit the voiceless labiodental approx ...

(). The

romanization Romanization or romanisation, in linguistics, is the conversion of text from a different writing system to the Roman (Latin) script, or a system for doing so. Methods of romanization include transliteration, for representing written text, and ...

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 before shifting to Classical Greek . In traditional

Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to tho ...

, phi has a value of 500 () or 500,000 (). The Cyrillic letter Ef (Ф, ф) descends from phi. As with other Greek letters, lowercase phi (encoded as the

Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...

character ) is used as a mathematical or scientific symbol. Some uses, such as the golden ratio, require the old-fashioned 'closed' glyph, which is separately encoded as the Unicode character .

Use as a symbol

The lowercase letter φ (or often its variant, ϕ) is often used to represent the following: *

Magnetic flux In physics, specifically electromagnetism, the magnetic flux through a surface is the surface integral of the normal component of the magnetic field B over that surface. It is usually denoted or . The SI unit of magnetic flux is the weber ( ...

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...

* The letter phi is commonly used in

to represent wave functions in

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

, such as in the

Schrödinger equation The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the ...

and

bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathema ...

\langle\phi, \psi\rangle

. * The golden ratio

\tfrac \approx

... in mathematics, art, and architecture. *

Euler's totient function In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...

''φ''(''n'') in number theory; also called ''Euler's phi function''. * The

cyclotomic polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primiti ...

functions Φ_''n''(''x'') of algebra. * The number of phases in a power system in

electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...

, for example 1ϕ for single phase, 3ϕ for three phase. * In algebra, group or

ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

s * In

probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, is the

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

of the standard normal distribution. * In

, is the characteristic function of a random variable ''X''. * An

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

, typically the second angle mentioned, after ''θ'' (

theta Theta (, ; uppercase: Θ or ; lowercase: θ or ; grc, ''thē̂ta'' ; Modern: ''thī́ta'' ) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth . In the system of Greek numerals, it has a value of 9. Gr ...

). It can denote: ** The argument of a

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

. ** 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

protein Proteins are large biomolecules and macromolecules that comprise one or more long chains of amino acid residues. Proteins perform a vast array of functions within organisms, including catalysing metabolic reactions, DNA replication, respo ...

s in a Ramachandran plot ** Internal or effective

angle of friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...

. * The

work function In solid-state physics, the work function (sometimes spelt workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" m ...

of a surface, in

solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...

. * A

shorthand Shorthand is an abbreviated symbolic writing method that increases speed and brevity of writing as compared to longhand, a more common method of writing a language. The process of writing in shorthand is called stenography, from the Greek ''ste ...

representation for an

aromatic In chemistry, aromaticity is a chemical property of cyclic ( ring-shaped), ''typically'' planar (flat) molecular structures with pi bonds in resonance (those containing delocalized electrons) that gives increased stability compared to satur ...

functional group In organic chemistry, a functional group is a substituent or moiety in a molecule that causes the molecule's characteristic chemical reactions. The same functional group will undergo the same or similar chemical reactions regardless of the rest ...

organic chemistry Organic chemistry is a subdiscipline within chemistry involving the scientific study of the struc