Epsilon (, ;
uppercase
Letter case is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation of certain languages. The writing ...
,
lowercase
Letter case is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation of certain languages. The writing ...
or
lunate
Lunate is a crescent or moon-shaped microlith. In the specialized terminology of lithic reduction, a lunate flake is a small, crescent-shaped flake removed from a stone tool during the process of pressure flaking.
In the Natufian period, a lun ...
; el, έψιλον) is the fifth 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 w ...
, corresponding phonetically to a
mid front unrounded vowel
The mid front unrounded vowel is a type of vowel sound that is used in some spoken languages. There is no dedicated symbol in the International Phonetic Alphabet that represents the exact mid front unrounded vowel between close-mid and open-mid ...
or . 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 tho ...
it also has the value five. It was derived from the
Phoenician letter He ![Phoenician he](https://upload.wikimedia.org/wikipedia/commons/8/8a/Phoenician_he.svg)
. Letters that arose from epsilon include the Roman
E,
Ë and
Ɛ, and Cyrillic
Е,
È,
Ё,
Є and
Э.
The name of the letter was originally (), but it was later changed to ( 'simple e') in the Middle Ages to distinguish the letter from the
digraph , a former
diphthong that had come to be pronounced the same as epsilon.
The uppercase form of epsilon is identical to Latin E but has its own
code point
In character encoding terminology, a code point, codepoint or code position is a numerical value that maps to a specific character. Code points usually represent a single grapheme—usually a letter, digit, punctuation mark, or whitespace—but ...
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, wh ...
: . The lowercase version has two typographical variants, both inherited from
medieval Greek handwriting. One, the most common in modern typography and inherited from medieval minuscule, looks like a reversed number "3" and is encoded . The other, also known as
lunate
Lunate is a crescent or moon-shaped microlith. In the specialized terminology of lithic reduction, a lunate flake is a small, crescent-shaped flake removed from a stone tool during the process of pressure flaking.
In the Natufian period, a lun ...
or
uncial
Uncial is a majuscule Glaister, Geoffrey Ashall. (1996) ''Encyclopedia of the Book''. 2nd edn. New Castle, DE, and London: Oak Knoll Press & The British Library, p. 494. script (written entirely in capital letters) commonly used from the 4th to ...
epsilon and inherited from earlier uncial writing,
[Nick Nicholas]
Letters
, 2003–2008. (''Greek Unicode Issues'') looks like a semicircle crossed by a horizontal bar: it is encoded . While in normal typography these are just alternative font variants, they may have different meanings as mathematical symbols: computer systems therefore offer distinct encodings for them.
In
TeX
Tex may refer to:
People and fictional characters
* Tex (nickname), a list of people and fictional characters with the nickname
* Joe Tex (1933–1982), stage name of American soul singer Joseph Arrington Jr.
Entertainment
* ''Tex'', the Italian ...
,
\epsilon
(
) denotes the lunate form, while
\varepsilon
(
) denotes the reversed-3 form.
There is also a '
Latin epsilon', or "open e", which looks similar to the Greek lowercase epsilon. It is encoded in Unicode as and and is used as an
IPA phonetic symbol. The lunate or uncial epsilon provided inspiration for the
euro sign, .
The lunate epsilon, , is not to be confused with the
set membership symbol ; nor should the Latin uppercase epsilon, , be confused with the Greek uppercase (
sigma). The symbol
, first used in set theory and logic by
Giuseppe Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
and now used in mathematics in general for set membership ("belongs to") evolved from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word . In addition, mathematicians often read the symbol as "element of", as in "1 is an element of the natural numbers" for
, for example. As late as 1960, ε itself was used for set membership, while its negation "does not belong to" (now ) was denoted by (epsilon prime). Only gradually did a fully separate, stylized symbol take the place of epsilon in this role. In a related context, Peano also introduced the use of a backwards epsilon, , for the phrase "such that", although the abbreviation ''s.t.'' is occasionally used in place of ϶ in informal cardinals.
History
Origin
The letter Ε was adopted from the
Phoenician letter
He (
![Phoenician he](https://upload.wikimedia.org/wikipedia/commons/8/8a/Phoenician_he.svg)
) when Greeks first adopted alphabetic writing. In archaic Greek writing, its shape is often still identical to that of the Phoenician letter. Like other Greek letters, it could face either leftward or rightward (
![Greek Epsilon left](https://upload.wikimedia.org/wikipedia/commons/c/ca/Greek_Epsilon_left.svg)
![Greek Epsilon archaic](https://upload.wikimedia.org/wikipedia/commons/8/8e/Greek_Epsilon_archaic.svg)
), depending on the current writing direction, but, just as in Phoenician, the horizontal bars always faced in the direction of writing. Archaic writing often preserves the Phoenician form with a vertical stem extending slightly below the lowest horizontal bar. In the classical era, through the influence of more cursive writing styles, the shape was simplified to the current E glyph.
Sound value
While the original pronunciation of the Phoenician letter ''He'' was , the earliest Greek sound value of Ε was determined by the vowel occurring in the Phoenician letter name, which made it a natural choice for being reinterpreted from a consonant symbol to a vowel symbol denoting an sound.
[Jeffery, ''Local scripts'', p. 24.] Besides its classical Greek sound value, the short phoneme, it could initially also be used for other -like sounds. For instance, in early
Attic before c. 500 BC, it was used also both for the long,
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' ( ...
, and for the long
close
Close may refer to:
Music
* ''Close'' (Kim Wilde album), 1988
* ''Close'' (Marvin Sapp album), 2017
* ''Close'' (Sean Bonniwell album), 1969
* "Close" (Sub Focus song), 2014
* "Close" (Nick Jonas song), 2016
* "Close" (Rae Sremmurd song), 201 ...
. In the former role, it was later replaced in the classic Greek alphabet by
Eta
Eta (uppercase , lowercase ; grc, ἦτα ''ē̂ta'' or ell, ήτα ''ita'' ) is the seventh letter of the Greek alphabet, representing the close front unrounded vowel . Originally denoting the voiceless glottal fricative in most dialects, ...
(Η), which was taken over from eastern
Ionic alphabets, while in the latter role it was replaced by the
digraph spelling ΕΙ.
Epichoric alphabets
Some dialects used yet other ways of distinguishing between various e-like sounds.
In
Corinth
Corinth ( ; el, Κόρινθος, Kórinthos, ) is the successor to an ancient city, and is a former municipality in Corinthia, Peloponnese (region), Peloponnese, which is located in south-central Greece. Since the 2011 local government refor ...
, the normal function of Ε to denote and was taken by a glyph resembling a pointed B (
![Greek Beta archaic](https://upload.wikimedia.org/wikipedia/commons/0/0c/Greek_Beta_archaic.svg)
), while Ε was used only for long close .
[Jeffery, ''Local scripts'', p. 114.] The letter
Beta, in turn, took the deviant shape
![Greek Beta Corinth 1](https://upload.wikimedia.org/wikipedia/commons/7/79/Greek_Beta_Corinth_1.svg)
.
In
Sicyon, a variant glyph resembling an X (
![Greek Epsilon X-shaped](https://upload.wikimedia.org/wikipedia/commons/8/83/Greek_Epsilon_X-shaped.svg)
) was used in the same function as Corinthian
![Greek Beta archaic](https://upload.wikimedia.org/wikipedia/commons/0/0c/Greek_Beta_archaic.svg)
.
[Jeffery, ''Local scripts'', p. 138.]
In
Thespiai (
Boeotia
Boeotia ( ), sometimes Latinized as Boiotia or Beotia ( el, Βοιωτία; modern: ; ancient: ), formerly known as Cadmeis, is one of the regional units of Greece. It is part of the region of Central Greece. Its capital is Livadeia, and its ...
), a special letter form consisting of a vertical stem with a single rightward-pointing horizontal bar (
![Greek Eta tack](https://upload.wikimedia.org/wikipedia/commons/3/3e/Greek_Eta_tack.svg)
) was used for what was probably a
raised variant of in pre-vocalic environments.
[Jeffery, ''Local scripts'', p. 89.] This tack glyph was used elsewhere also as a form of "
Heta", i.e. for the sound .
Glyph variants
After the establishment of the canonical classical Ionian (Euclidean)
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 w ...
, new glyph variants for Ε were introduced through handwriting. In the
uncial script
Uncial is a majuscule Glaister, Geoffrey Ashall. (1996) ''Encyclopedia of the Book''. 2nd edn. New Castle, DE, and London: Oak Knoll Press & The British Library, p. 494. script (written entirely in capital letters) commonly used from the 4th ...
(used for literary
papyrus
Papyrus ( ) is a material similar to thick paper that was used in ancient times as a writing surface. It was made from the pith of the papyrus plant, '' Cyperus papyrus'', a wetland sedge. ''Papyrus'' (plural: ''papyri'') can also refer to a ...
manuscripts in late antiquity and then in early medieval
vellum
Vellum is prepared animal skin or membrane, typically used as writing material. Parchment is another term for this material, from which vellum is sometimes distinguished, when it is made from calfskin, as opposed to that made from other anima ...
codices), the "
lunate
Lunate is a crescent or moon-shaped microlith. In the specialized terminology of lithic reduction, a lunate flake is a small, crescent-shaped flake removed from a stone tool during the process of pressure flaking.
In the Natufian period, a lun ...
" shape (
![Greek uncial Epsilon](https://upload.wikimedia.org/wikipedia/commons/1/12/Greek_uncial_Epsilon.svg)
) became predominant. In
cursive handwriting, a large number of shorthand glyphs came to be used, where the cross-bar and the curved stroke were linked in various ways.
Some of them resembled a modern lowercase Latin "e", some a "6" with a connecting stroke to the next letter starting from the middle, and some a combination of two small "c"-like curves. Several of these shapes were later taken over into
minuscule
Letter case is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation of certain languages. The writing ...
book hand. Of the various minuscule letter shapes, the inverted-3 form became the basis for lower-case Epsilon in Greek typography during the modern era.
Uses
International Phonetic Alphabet
Despite its pronunciation as
front unrounded vowel">mid, in the
International Phonetic Alphabet
The International Phonetic Alphabet (IPA) is an alphabetic system of phonetic notation based primarily on the Latin script. It was devised by the International Phonetic Association in the late 19th century as a standardized representation ...
, the Latin epsilon represents
open-mid front unrounded vowel
The open-mid front unrounded vowel, or low-mid front unrounded vowel, is a type of vowel sound used in some spoken languages. The symbol in the International Phonetic Alphabet that represents this sound is a Latinised variant of the Greek lower ...
, as in the English word ''pet'' .
Symbol
The uppercase Epsilon is not commonly used outside of the Greek language because of its similarity to the
Latin
Latin (, or , ) is a classical language belonging to the 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 the power of the ...
letter
E. However, it is commonly used in
structural mechanics with
Young's Modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied le ...
equations for calculating tensile, compressive and areal
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
.
The Greek lowercase epsilon , the lunate epsilon symbol , and the
Latin lowercase epsilon (see above) are used in a variety of places:
* In
engineering mechanics
Applied mechanics is the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments. In short, when mechanics concepts surpass being theoretical and are applied and e ...
, strain calculations ϵ = increase of length / original length. Usually this relates to extensometer testing of metallic materials.
* In
mathematics
** (particularly
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
), an infinitesimally small positive quantity is commonly denoted ε; see
(ε, δ)-definition of limit
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
.
**
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
introduced epsilon terms
as an extension to
first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
; see
epsilon calculus Hilbert's epsilon calculus is an extension of a formal language by the epsilon operator, where the epsilon operator substitutes for quantifiers in that language as a method leading to a proof of consistency for the extended formal language. The ' ...
.
** it is used to represent the
Levi-Civita symbol
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the n ...
.
** it is used to represent
dual number
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
s:
, with
and
.
** it is sometimes used to denote the
Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argum ...
.
** in
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, the
epsilon numbers are
ordinal numbers that satisfy the
fixed point ε = ω
ε. The first epsilon number, ε
0, is the
limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...
of the set .
** in numerical analysis and statistics it is used as the
error term In mathematics and statistics, an error term is an additive type of error. Common examples include:
* errors and residuals in statistics, e.g. in linear regression
* the error term in numerical integration
In analysis, numerical integration ...
** in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
it is used as the
idempotent
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
group when e is in use as a variable name
* 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 practical disciplines (includi ...
** it often represents the
empty string
In formal language theory, the empty string, or empty word, is the unique string of length zero.
Formal theory
Formally, a string is a finite, ordered sequence of characters such as letters, digits or spaces. The empty string is the special cas ...
, though different writers use a variety of other symbols for the empty string as well; usually the lower-case Greek letter
lambda
Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave ri ...
(λ).
** the
machine epsilon indicates the upper bound on the relative error due to rounding in floating point arithmetic.
* In
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 ...
,
** it indicates the
permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
of a medium; with the subscript 0 (ε
0) it is the
permittivity of free space.
** it can also indicate the
strain
Strain may refer to:
Science and technology
* Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes
* Strain (chemistry), a chemical stress of a molecule
* Strain (injury), an injury to a mu ...
of a material (a ratio of extensions).
* In
automata theory
Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word ''automata'' comes from the Greek word αὐτόματο ...
, it shows a transition that involves no shifting of an input symbol.
* 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 ...
,
** it stands for the fifth-brightest star in a
constellation (see
Bayer designation).
** Epsilon is the name for the most distant and most visible ring of
Uranus
Uranus is the seventh planet from the Sun. Its name is a reference to the Greek god of the sky, Uranus ( Caelus), who, according to Greek mythology, was the great-grandfather of Ares (Mars), grandfather of Zeus (Jupiter) and father of ...
.
** In
planetary science, ε denotes the
axial tilt
In astronomy, axial tilt, also known as obliquity, is the angle between an object's rotational axis and its orbital axis, which is the line perpendicular to its orbital plane; equivalently, it is the angle between its equatorial plane and orb ...
.
* In
chemistry, it represents the
molar extinction coefficient of a
chromophore.
* In
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
, ε refers to
elasticity.
* In
statistics,
** it is used to refer to
error terms.
** it also can to refer to the degree of
sphericity
Sphericity is a measure of how closely the shape of an object resembles that of a perfect sphere. For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed a ...
in
repeated measures ANOVAs.
* In
agronomy, it is used to represent the "
photosynthetic efficiency" of a particular plant or crop.
Unicode
* Greek Epsilon
* Coptic Eie
*
Latin Open E
* Mathematical Epsilon
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.
Initial
File:Lectionary 226 GA 0020b Epsilon.JPG, Initial
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...
epsilon in Lectionary 226, folio 20 verso
' is the "right" or "front" side and ''verso'' is the "left" or "back" side when text is written or printed on a leaf of paper () in a bound item such as a codex, book, broadsheet, or pamphlet.
Etymology
The terms are shortened from Latin ...
File:Lectionary 226 (GA) Epsilon 0064b.JPG, folio 64 verso
File:Lectionary 226 GA 0125b Epsilon.JPG, folio 125 verso
References
Further reading
*
Hoffman, Paul; ''
The Man Who Loved Only Numbers
''The Man Who Loved Only Numbers'' is a biography of the famous mathematician Paul Erdős written by Paul Hoffman. The book was first published on July 15, 1998, by Hyperion Books as a hardcover edition. A paperback edition appeared in 1999. ...
''. Hyperion, 1998. {{ISBN, 0-7868-6362-5.
Greek letters
Vowel letters