Theuppercase form of epsilon is identical to Latin⟨E⟩ but has its owncode point inUnicode:U+0395ΕGREEK CAPITAL LETTER EPSILON. Thelowercase version has two typographical variants, both inherited frommedieval Greek handwriting. One, the most common in modern typography and inherited from medievalminuscule, looks like a reversed number "3" and is encodedU+03B5εGREEK SMALL LETTER EPSILON. The other, also known aslunate oruncial epsilon and inherited from earlier uncial writing,[3][4] looks like a semicircle crossed by a horizontal bar: it is encodedU+03F5ϵGREEK LUNATE EPSILON SYMBOL. 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.[3] InTeX,\epsilon ( ) denotes the lunate form, while\varepsilon ( ) denotes theepsilon number. Unicode versions 2.0.0 and onwards useɛ as the lowercase Greek epsilon letter,[5] but in version 1.0.0,ϵ was used.[6] The lunate or uncial epsilon provided inspiration for theeuro sign,€.[7]
There is also a 'Latin epsilon',⟨ɛ⟩ or "open e", which looks similar to the Greek lowercase epsilon. It is encoded in Unicode asU+025BɛLATIN SMALL LETTER OPEN E andU+0190ƐLATIN CAPITAL LETTER OPEN E and is used as anIPA phonetic symbol. This Latin uppercase epsilon,Ɛ, is not to be confused with the Greek uppercaseΣ (sigma)
The lunate epsilon,⟨ϵ⟩, is not to be confused with theset membership symbol∈. The symbol, first used in set theory and logic byGiuseppe Peano 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 wordest. 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,ɛ itself was used for set membership, while its negation "does not belong to" (now∉) was denoted byε' (epsilon prime).[8] 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 abbreviations.t. is occasionally used in place of϶ in informal cardinals.
The letter⟨Ε⟩ was adopted from thePhoenician letterHe () 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 (), 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.[9]
While the original pronunciation of the Phoenician letterHe was[h], 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[e] sound.[10] Besides its classical Greek sound value, the short/e/ phoneme, it could initially also be used for other[e]-like sounds. For instance, in earlyAttic beforec. 500 BC, it was used also both for the long,open/ɛː/, and for the longclose/eː/. In the former role, it was later replaced in the classic Greek alphabet byEta (⟨Η⟩), which was taken over from easternIonic alphabets, while in the latter role it was replaced by thedigraph ⟨ΕΙ⟩.
Some dialects used yet other ways of distinguishing between various e-like sounds.
InCorinth, the normal function of⟨Ε⟩ to denote/e/ and/ɛː/ was taken by a glyph resembling a pointed B (), while⟨Ε⟩ was used only for long close/eː/.[11] The letterBeta, in turn, took the deviant shape.
InSicyon, a variant glyph resembling an⟨X⟩ () was used in the same function as Corinthian.[12]
InThespiai (Boeotia), a special letter form consisting of a vertical stem with a single rightward-pointing horizontal bar () was used for what was probably araised variant of/e/ in pre-vocalic environments.[13][14] This tack glyph was used elsewhere also as a form of "Heta", i.e. for the sound/h/.
After the establishment of the canonical Ionian (Euclidean)Greek alphabet, new glyph variants for Ε were introduced through handwriting. In theuncial script (used for literarypapyrus manuscripts in late antiquity and then in early medievalvellum codices), the "lunate" shape () became predominant. Incursive handwriting, a large number of shorthand glyphs came to be used, where the cross-bar and the curved stroke were linked in various ways.[15] 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 intominuscule 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.
The uppercase Epsilon is not commonly used outside of the Greek language because of its similarity to theLatin letterE. However, it is commonly used instructural mechanics withYoung's Modulus equations for calculating tensile, compressive and arealstrain.
The Greek lowercase epsilonε, the lunate epsilon symbolϵ, and theLatin lowercase epsilonɛ (see above) are used in a variety of places:
Inengineering mechanics, strain calculations ϵ = increase of length / original length. Usually this relates to extensometer testing of metallic materials.
(Inanalysis) By extension, a quantity thought of as "small," "negligible," or, especially, "arbitrarily small," is often denoted ε. For instance, quantities subject to alimit which takes them towards zero are often denoted ε; see(ε, δ)-definition of limit.[17]
it often represents theempty string, though different writers use a variety of other symbols for the empty string as well; usually the lower-case Greek letterlambda (λ).
themachine epsilon indicates the upper bound on the relative error due to rounding in floating point arithmetic.[20]
U+1D75A𝝚MATHEMATICAL SANS-SERIF BOLD CAPITAL EPSILON
U+1D774𝝴MATHEMATICAL SANS-SERIF BOLD SMALL EPSILON
U+1D78A𝞊MATHEMATICAL SANS-SERIF BOLD EPSILON SYMBOL
U+1D794𝞔MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL EPSILON
U+1D7AE𝞮MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL EPSILON
U+1D7C4𝟄MATHEMATICAL SANS-SERIF BOLD ITALIC EPSILON SYMBOL
^TheMATHEMATICAL symbols are used only in math. Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.
^Colwell, Ernest C. (1969). "A chronology for the letters Ε, Η, Λ, Π in the Byzantine minuscule book hand".Studies in methodology in textual criticism of the New Testament. Leiden: Brill. p. 127.
^Thompson, Edward M. (1911).An Introduction to Greek and Latin palaeography. Oxford: Clarendon. pp. 191–194.
^Weisstein, Eric W."Epsilon".mathworld.wolfram.com. Retrieved30 January 2025.In mathematics, a small positive infinitesimal quantity, usually denotedε orϵ, whose limit is usually taken asϵ->0.
^Weisstein, Eric W."Limit".mathworld.wolfram.com. Retrieved30 January 2025.
^Weisstein, Eric W."Dual Number".mathworld.wolfram.com. Retrieved30 January 2025.
^Weisstein, Eric W."Delta Function".mathworld.wolfram.com. Retrieved19 February 2019.
^Überhuber, Christoph W. (1997).Numerical Computation 1: Methods, Software, and Analysis. SpringerLink Bücher. Berlin, Heidelberg: Springer. p. 140.ISBN978-3-540-62058-7.eps frequently denotes his upper bound on the relative rounding error and is referred to as themachine epsilon.
^Montenari, Michael, ed. (2018).Cyclostratigraphy and astrochronology. Stratigraphy and Timescales (1st ed.). London San Diego, Calif. Cambridge, Mass. Oxford: Academic Press, an imprint of Elsevier. p. 84.ISBN978-0-12-815098-6.The Earth's orbital obliquity or axial tilt (ε) is the angle between the Earth's equatorial plane and its orbital plane,
^Free, Rhona C. (2010).21st century economics: a reference handbook. Thousand Oaks (Calif.): Sage. pp. 93–94.ISBN978-1-4129-6142-4.
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