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Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth 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 ...
. In the system of
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 those ...
, it has a value of 200. In general mathematics, uppercase Σ is used as an operator for summation. When used at the end of a letter-case word (one that does not use all caps), the final form (ς) is used. In ' (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the word-final sigma (ς) at the end. The
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
letter S derives from sigma while the Cyrillic letter Es derives from a lunate form of this letter.


History

The shape (Σς) and alphabetic position of sigma is derived from the Phoenician letter ( ''shin''). Sigma's original name may have been ''san'', but due to the complicated early history of the Greek epichoric alphabets, ''san'' came to be identified as a separate letter in the Greek alphabet, represented as Ϻ.
Herodotus Herodotus ( ; grc, , }; BC) was an ancient Greek historian and geographer from the Greek city of Halicarnassus, part of the Persian Empire (now Bodrum, Turkey) and a later citizen of Thurii in modern Calabria (Italy). He is known fo ...
reports that "san" was the name given by the
Dorians The Dorians (; el, Δωριεῖς, ''Dōrieîs'', singular , ''Dōrieús'') were one of the four major ethnic groups into which the Hellenes (or Greeks) of Classical Greece divided themselves (along with the Aeolians, Achaeans, and Ioni ...
to the same letter called "sigma" by the Ionians."the same letter, which the Dorians call "san", but the Ionians 'sigma'..." ranslated_from_Ancient_Greek:_"".html" ;"title="Ancient_Greek.html" ;"title="ranslated from Ancient Greek">ranslated from Ancient Greek: """>Ancient_Greek.html" ;"title="ranslated from Ancient Greek">ranslated from Ancient Greek: ""(
Herodotus Herodotus ( ; grc, , }; BC) was an ancient Greek historian and geographer from the Greek city of Halicarnassus, part of the Persian Empire (now Bodrum, Turkey) and a later citizen of Thurii in modern Calabria (Italy). He is known fo ...
1.139)
According to one hypothesis, the name "sigma" may continue that of Phoenician ''samekh'' (), the letter continued through Greek ''xi'', represented as Xi (letter), Ξ. Alternatively, the name may have been a Greek innovation that simply meant 'hissing', from the root of (, from Proto-Greek ''*sig-jō'' 'I hiss').


Lunate sigma

In handwritten Greek during the
Hellenistic period In Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in ...
(4th–3rd century BC), the epigraphic form of Σ was simplified into a C-like shape, which has also been found on coins from the 4th century BC onward. This became the universal standard form of sigma during late antiquity and the Middle Ages. Today, it is known as ''lunate sigma'' (uppercase Ϲ, lowercase ϲ), because of its crescent-like shape, and is still widely used in decorative typefaces in Greece, especially in religious and church contexts, as well as in some modern print editions of classical Greek texts. A dotted lunate sigma (''sigma periestigmenon'', Ͼ) was used by
Aristarchus of Samothrace Aristarchus of Samothrace ( grc-gre, Ἀρίσταρχος ὁ Σαμόθραξ ''Aristarchos o Samothrax''; c. 220 – c. 143 BC) was an ancient Greek grammarian, noted as the most influential of all scholars of Homeric poetry. He was the h ...
(220–143 BC) as an editorial sign indicating that the line marked as such is at an incorrect position. Similarly, a reversed sigma ('' antisigma'', Ͻ), may mark a line that is out of place. A dotted antisigma (''antisigma periestigmenon'', Ͽ) may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority. In Greek inscriptions from the late first century BC onwards, Ͻ was an abbreviation indicating that a man's father's name is the same as his own name, thus Dionysodoros son of Dionysodoros would be written Διονυσόδωρος Ͻ (''Dionysodoros Dionysodorou''). In
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
, the above variations of lunate sigma are encoded as ; , , and .


Derived alphabets

Sigma was adopted in the Old Italic alphabets beginning in the 8th century BC. At that time a simplified three-stroke version, omitting the lowermost stroke, was already found in Western Greek alphabets, and was incorporated into classical Etruscan and
Oscan Oscan is an extinct Indo-European language of southern Italy. The language is in the Osco-Umbrian or Sabellic branch of the Italic languages. Oscan is therefore a close relative of Umbrian. Oscan was spoken by a number of tribes, including t ...
, as well as in the earliest
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
epigraphy (early Latin '' S''), such as the
Duenos inscription The Duenos inscription is one of the earliest known Old Latin texts, variously dated from the 7th to the 5th century BC. It is inscribed on the sides of a ''kernos'', in this case a trio of small globular vases adjoined by three clay struts. It w ...
. The alternation between three and four (and occasionally more than four) strokes was also adopted into the early runic alphabet (early form of the ''s''-rune). Both the Anglo-Saxon runes and the Younger Futhark consistently use the simplified three-stroke version. The letter С of
Cyrillic script The Cyrillic script ( ), Slavonic script or the Slavic script, is a writing system used for various languages across Eurasia. It is the designated national script in various Slavic, Turkic, Mongolic, Uralic, Caucasian and Iranic-speaking c ...
originates in the lunate form of Sigma.


Uses


Language and linguistics

* In both Ancient and
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 ...
, the sigma represents the voiceless alveolar fricative . In Modern Greek, this sound is voiced to the voiced alveolar fricative when occurring before , , , or . * The uppercase form of sigma (Σ) was re-borrowed into the
Latin alphabet The Latin alphabet or Roman alphabet is the collection of letters originally used by the ancient Romans to write the Latin language. Largely unaltered with the exception of extensions (such as diacritics), it used to write English and the ...
—more precisely, the International African Alphabet—to serve as the uppercase of modern ''esh'' (lowercase: ʃ). * In
phonology Phonology is the branch of linguistics that studies how languages or dialects systematically organize their sounds or, for sign languages, their constituent parts of signs. The term can also refer specifically to the sound or sign system of a ...
, σ is used to represent syllables. * In
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Ling ...
, Σ represents the set of symbols that form an alphabet (see also
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
).


Science and mathematics


Mathematics

* In general mathematics, lowercase σ is commonly used to represent unknown angles, as well as serving as a shorthand for " countably", whereas Σ is regularly used as the operator for summation, e.g.: ** \sum_^5k= 0 + 1 + 2 + 3 + 4 + 5 = 15 *In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
, \Sigma^0_n is used to denote the set of formulae with bounded quantifiers beginning with existential quantifiers, alternating n-1 times between existential and universal quantifiers. This notation reflects an indirect analogy between the relationship of summation and products on one hand, and existential and universal quantifiers on the other. See the article on the
arithmetic hierarchy In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define the ...
. *In
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, σ represents the standard deviation of population or probability distribution (where mu or μ is used for the mean). *In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, σ-compact
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
is one that can be written as a countable union of compact subsets. *In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
and 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 ...
, there is a type of algebra of sets known as σ-algebra (aka σ-field). Sigma algebra also includes terms such as: **σ(''A''), denoting the generated sigma-algebra of a set ''A'' ** σ-finite measure (see measure theory) * In number theory, σ is included in various divisor functions, especially the sigma function or sum-of-divisors function. * In
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
, σ(''T'') denotes the spectrum of a linear map ''T''. * In complex analysis, σ is used in the Weierstrass sigma-function. *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 ...
and
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, Σ denotes the covariance matrix of a set of random variables, sometimes in the form \;, \!\!\!\Sigma to distinguish it from the summation operator. * Theoretical spectral analysis uses σ as standard deviation opposed to lowercase mu as the absolute mean value.


Biology, physiology, and medicine

* In
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary ...
, the sigma receptor (σ–receptors) is a type of cell surface receptor. * In
biochemistry Biochemistry or biological chemistry is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology and ...
, the σ factor (or specificity factor) is a protein found in RNA polymerase. * In bone physiology, the
bone remodeling period In bone physiology, the bone remodeling period describes the temporal duration (i.e. lifespan) of the basic multicellular unit (BMU) that is responsible for bone turnover. Historically, this was referred to as the sigma (σ) or sigma period, but t ...
—i.e., the life span of a basic multicellular unit—has historically been referred to as the sigma period * In early 20th-century
physiology Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemic ...
literature, σ had been used to represent milliseconds


Business, finance, and economics

* In finance, σ is the symbol used to represent volatility of stocks, usually measured by the standard deviation of logarithmic returns. *In accounting, Σ indicates the balance of invoice classes and the overall amount of debts and demands. * In macroeconomics, σ is used in equations to represent the elasticity of substitution between two inputs. * In the machine industry, Six Sigma (6σ) is a quality model based on the standard deviation.


Chemistry

* Sigma bonds (σ bonds) are the strongest type of
covalent A covalent bond is a chemical bond that involves the sharing of electrons to form electron pairs between atoms. These electron pairs are known as shared pairs or bonding pairs. The stable balance of attractive and repulsive forces between atom ...
chemical bond. * In
organic chemistry Organic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms.Clayden, J ...
, σ symbolizes the sigma constant of Hammett equation. * In
alchemy Alchemy (from Arabic: ''al-kīmiyā''; from Ancient Greek: χυμεία, ''khumeía'') is an ancient branch of natural philosophy, a philosophical and protoscientific tradition that was historically practiced in China, India, the Muslim wo ...
, Σ was sometimes used to represent
sugar Sugar is the generic name for sweet-tasting, soluble carbohydrates, many of which are used in food. Simple sugars, also called monosaccharides, include glucose, fructose, and galactose. Compound sugars, also called disaccharides or do ...
.


Engineering and computer science

* In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, Σ represents the set of symbols that form an alphabet (see also
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Ling ...
) * Relational algebra uses the values \sigma_( R ) and \sigma_( R ) to denote selections, which are a type of unary operation. * In machine learning, σ is used in the formula that derives the
Sigmoid function A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: :S(x) = \frac = \f ...
. * In
radar jamming Radar jamming and deception is a form of electronic countermeasures that intentionally sends out radio frequency signals to interfere with the operation of radar by saturating its receiver with noise or false information. Concepts that blanket ...
or electronic warfare, radar cross-sections (RCS) are commonly represented as σ when measuring the size of a target's image on radar. * In signal processing, σ denotes the damping ratio of a system parameter. *In theoretical computer science, Σ serves as the busy beaver function. *In civil engineering, σ refers to the normal stress applied on a material or structure.


Physics

* In
nuclear Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space *Nuclear ...
and particle physics, σ is used to denote cross sections in general (see also
RCS RCS may refer to: Organisations *Racing Club de Strasbourg Alsace * Radio Corporation of Singapore * Radcliffe Choral Society * Rawmarsh Community School *Red Crescent Society * Red Cross Society * Representation of Czechs and Slovaks, a football ...
), while Σ represents macroscopic cross sections /length * The symbol is to denote the Stefan–Boltzmann constant. * In relation to fundamental properties of material, σ is often used to signify electrical conductivity. * In electrostatics, σ represents surface charge density. * In continuum mechanics, σ is used to signify stress. * In condensed matter physics, Σ denotes
self-energy In quantum field theory, the energy that a particle has as a result of changes that it causes in its environment defines self-energy \Sigma, and represents the contribution to the particle's energy, or effective mass, due to interactions between ...
. * The symbol can be used to signify
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) t ...
(alternatively, '' γ'' or '' T'' are also used instead). * 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, ...
, σ is used to indicate Pauli matrices. * In
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, σ represents velocity dispersion. * In astronomy, the prefix Σ is used to designate double stars of the Catalogus Novus Stellarum Duplicium by Friedrich Georg Wilhelm von Struve. * In particle physics, Σ represents a class of baryons.


Organizations

* During the 1930s, an uppercase Σ was in use as the symbol of the Ação Integralista Brasileira, a fascist political party in
Brazil Brazil ( pt, Brasil; ), officially the Federative Republic of Brazil (Portuguese: ), is the largest country in both South America and Latin America. At and with over 217 million people, Brazil is the world's fifth-largest country by area ...
. *
Sigma Corporation is a Japanese company, manufacturing cameras, lenses, flashes and other photographic accessories. All Sigma products are produced in the company's own Aizu factory in Bandai, Fukushima, Japan. Although Sigma produces several camera models ...
uses the name of the letter but not the letter itself, but in many Internet forums, photographers refer to the company or its lenses using the letter. *
Sigma Aldrich Sigma-Aldrich (formally MilliporeSigma) is an American chemical, life science, and biotechnology company that is owned by the German chemical conglomerate Merck Group. Sigma-Aldrich was created in 1975 by the merger of Sigma Chemical Company a ...
incorporate both the name and the character in their logo.


Character encoding


Greek sigma

Unicode Code Charts
Greek and Coptic (Range: 0370-03FF)
/ref>


Coptic sima


Mathematical sigma

These characters are used only as mathematical symbols. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.


See also

* Antisigma * Greek letters used in mathematics, science, and engineering * Sampi *
Sho (letter) The letter (sometimes called sho or san) was a letter added to the Greek alphabet in order to write the Bactrian language.Everson, M. and Sims-Williams, N. (2002) �Proposal to add two Greek letters for Bactrian to the UCS��,ISO/IEC JTC1/SC2 ...
* − Template to produce the character * Stigma (letter) * Sibilant consonant * Summation (Σ) *Combining form "''sigm-''" (e.g. ''sigmodon'', ''sigmurethra'', etc.) *Derivative "'' sigmoid''" (e.g.
sigmoid sinus The sigmoid sinuses (sigma- or s-shaped hollow curve), also known as the , are venous sinuses within the skull that receive blood from posterior dural venous sinus veins. Structure The sigmoid sinus is a dural venous sinus situated within the ...
, sigmoid colon, sigmoidoscopy, etc.)


References


Notes


Citations

{{Reflist Greek letters Letters with final form