Epsilon

This article is about the Greek letter. For other uses, see Epsilon (disambiguation).

Greek alphabet
Αα Alpha Νν Nu Ββ Beta Ξξ Xi Γγ Gamma Οο Omicron Δδ Delta Ππ Pi Εε Epsilon Ρρ Rho Ζζ Zeta Σσς Sigma Ηη Eta Ττ Tau Θθ Theta Υυ Upsilon Ιι Iota Φφ Phi Κκ Kappa Χχ Chi Λλ Lambda Ψψ Psi Μμ Mu Ωω Omega
History
Archaic local variants Diacritics Ligatures Numerals ϛ (6) ϟ (90) ϡ (900)
Use in other languages
Bactrian Coptic Albanian
Related topics
Use as scientific symbols
Book Category Commons

Epsilon (uppercase Ε, lowercase ε or lunate ϵ; Greek: έψιλον) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel /e/. In the system of Greek numerals it also has the value five. It was derived from the Phoenician letter He . 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 εἶ (Ancient Greek: [êː]), but the name was changed to ἒ ψιλόν (e psilon "simple e") in the Middle Ages to distinguish the letter from the digraph αι, a former diphthong that had come to be pronounced the same as epsilon.

In essence, the uppercase form of epsilon looks identical to Latin E. 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 "3". The other, also known as lunate or uncial epsilon and inherited from earlier uncial writing, looks like a semicircle crossed by a horizontal bar. 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 Unicode, the character U+03F5 "Greek lunate epsilon symbol" (ϵ) is provided specifically for the lunate form. In TeX, \epsilon ($\epsilon \!$) denotes the lunate form, while \varepsilon ($\varepsilon \!$) 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 U+025B ("Latin small-letter open e", ɛ) and U+0190 ("Latin capital-letter open e", Ɛ) and is used as an IPA phonetic symbol. The lunate or uncial epsilon has also provided inspiration for the euro sign (€).

The lunate epsilon (ϵ) is not to be confused with the set membership symbol (∈); nor should the Latin uppercase epsilon (Ɛ) be confused with the Greek uppercase sigma (Σ). The symbol $\in$, first used in set theory and logic by Giuseppe Peano and now used in mathematics in general for set membership ("belongs to") did, however, evolve from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word "est". In addition, mathematicians often read the symbol $\in$ as "element of", as in "1 is an element of the natural numbers" for $1\in \mathbb {N}$, for example. As late as 1960, $\epsilon$ itself was used for set membership, while its negation "does not belong to" (now $\notin$) was denoted by $\epsilon '$ (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, $\backepsilon$, for the phrase "such that", although the abbreviation "s.t." is occasionally used in place of $\backepsilon$ in informal writing.

History

Origin

The letter Ε was taken over from the Phoenician letter He () 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.

Sound value

While the original pronunciation of the Phoenician letter He 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. Besides its classical Greek sound value, the short /e/ phoneme, it could initially also be used for other [e]-like sounds. For instance, in early Attic before c.500 B.C., it was used also both for the long, open /ɛː/, and for the long close /eː/. In the former role, it was later replaced in the classic Greek alphabet by Eta (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, 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/. The letter Beta, in turn, took the deviant shape .

In Sicyon, a variant glyph resembling an X () was used in the same function as Corinthian .

In Thespiai (Boeotia), a special letter form consisting of a vertical stem with a single rightward-pointing horizontal bar () was used for what was probably a raised variant of /e/ in pre-vocalic environments. This tack glyph was used elsewhere also as a form of "Heta", i.e. for the sound /h/.

Glyph variants

After the establishment of the canonical classic Greek alphabet, new glyph variants for Ε were introduced through handwriting. In the uncial script (used for literary papyrus manuscripts in late antiquity and then in early medieval vellum codices), the "lunate" shape () 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 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.

Uncial	Uncial variants	Cursive variants	Minuscule	Minuscule with ligatures

Uses

International Phonetic Alphabet

Despite its pronunciation as mid, in the International Phonetic Alphabet, the Latin epsilon /ɛ/ represents open-mid front unrounded vowel, as in the English word pet /pɛt/.

Symbol

The uppercase Epsilon is not commonly used outside of the Greek language because of its similarity to the Latin letter E however, it is commonly used in structural mechanics with Young's Modulus equations for calculating tensile, compressive and Areal strain.

The Greek lowercase epsilon ε, the lunate epsilon symbol ϵ, or the Latin lowercase epsilon ɛ (see above) is used in a variety of places:

• In engineering mechanics strain calculations ϵ=increase of length / original length. Usually this relates to extensometer testing of metallic materials.

• In mathematics (particularly calculus), an arbitrarily small positive quantity is commonly denoted ε; see (ε, δ)-definition of limit.
  • In reference to this, the late mathematician Paul Erdős also used the term "epsilons" to refer to children (Hoffman 1998, p. 4).
• In mathematics, Hilbert introduced epsilon terms $\epsilon x.\phi$ as an extension to first order logic; see epsilon calculus.
• In mathematics, it is used to represent the Levi-Civita symbol.
• In mathematics, it is used to represent dual numbers: a + bε, with ε² = 0 and ε ≠ 0.
• In mathematics, it is sometimes used to denote the Heaviside step function.
• In set theory, the epsilon numbers are ordinal numbers that satisfy the fixed point ε = ω^ε. The first epsilon number, ε₀, is the limit ordinal of the set {ω, ω^ω, ω^ωω, ...}.
• In computer science, it often represents the empty string, though different writers use a variety of other symbols for the empty string as well; usually the lower-case Greek letter lambda (λ).
• In computer science, the machine epsilon indicates the upper bound on the relative error due to rounding in floating point arithmetic.
• In physics, it indicates the permittivity of a medium; with the subscript 0 (ϵ₀) it is the permittivity of free space.
• In physics, it can also indicate the strain of a material (a ratio of extensions).
• In automata theory, it shows a transition that involves no shifting of an input symbol.
• In astronomy, it stands for the fifth-brightest star in a constellation (see Bayer designation).
• In astronomy, Epsilon is the name for Uranus' most distant and most visible ring.
• In planetary science, ε denotes the axial tilt.
• In chemistry, it represents the molar extinction coefficient of a chromophore.
• In economics, ε refers to elasticity.
• In statistics, it is used to refer to error terms.
• In statistics, it also can to refer to the degree of sphericity in repeated measures ANOVAs.
• In agronomy, it is used to represent the "photosynthetic efficiency" of a particular plant or crop.

Unicode

• Greek Epsilon

Character	Ε		ε		ϵ		϶
Unicode name	GREEK CAPITAL LETTER EPSILON		GREEK SMALL LETTER EPSILON		GREEK LUNATE EPSILON SYMBOL		GREEK REVERSED LUNATE EPSILON SYMBOL
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	917	U+0395	949	U+03B5	1013	U+03F5	1014	U+03F6
UTF-8	206 149	CE 95	206 181	CE B5	207 181	CF B5	207 182	CF B6
Numeric character reference	Ε	Ε	ε	ε	ϵ	ϵ	϶	϶
Named character reference	Ε		ε
DOS Greek	132	84	156	9C
DOS Greek-2	168	A8	222	DE
Windows 1253	197	C5	229	E5
TeX			\varepsilon		\epsilon

• Coptic Eie

Character	Ⲉ		ⲉ
Unicode name	COPTIC CAPITAL LETTER EIE		COPTIC SMALL LETTER EIE
Encodings	decimal	hex	decimal	hex
Unicode	11400	U+2C88	11401	U+2C89
UTF-8	226 178 136	E2 B2 88	226 178 137	E2 B2 89
Numeric character reference	Ⲉ	Ⲉ	ⲉ	ⲉ

• Latin Open E

Character	Ɛ		ɛ		ᶓ		ᵋ
Unicode name	LATIN CAPITAL LETTER OPEN E		LATIN SMALL LETTER OPEN E		LATIN SMALL LETTER OPEN E WITH RETROFLEX HOOK		MODIFIER LETTER SMALL OPEN E
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	400	U+0190	603	U+025B	7571	U+1D93	7499	U+1D4B
UTF-8	198 144	C6 90	201 155	C9 9B	225 182 147	E1 B6 93	225 181 139	E1 B5 8B
Numeric character reference	Ɛ	Ɛ	ɛ	ɛ	ᶓ	ᶓ	ᵋ	ᵋ

Character	ɜ		ɝ		ᶔ		ᶟ
Unicode name	LATIN SMALL LETTER REVERSED OPEN E		LATIN SMALL LETTER REVERSED OPEN E WITH HOOK		LATIN SMALL LETTER REVERSED OPEN E WITH RETROFLEX HOOK		MODIFIER LETTER SMALL REVERSED OPEN E
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	604	U+025C	605	U+025D	7572	U+1D94	7583	U+1D9F
UTF-8	201 156	C9 9C	201 157	C9 9D	225 182 148	E1 B6 94	225 182 159	E1 B6 9F
Numeric character reference	ɜ	ɜ	ɝ	ɝ	ᶔ	ᶔ	ᶟ	ᶟ

Character	ᴈ		ᵌ		ʚ		ɞ
Unicode name	LATIN SMALL LETTER TURNED OPEN E		MODIFIER LETTER SMALL TURNED OPEN E		LATIN SMALL LETTER CLOSED OPEN E		LATIN SMALL LETTER CLOSED REVERSED OPEN E
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	7432	U+1D08	7500	U+1D4C	666	U+029A	606	U+025E
UTF-8	225 180 136	E1 B4 88	225 181 140	E1 B5 8C	202 154	CA 9A	201 158	C9 9E
Numeric character reference	ᴈ	ᴈ	ᵌ	ᵌ	ʚ	ʚ	ɞ	ɞ

• Mathematical Epsilon

Character	𝚬		𝛆		𝛦		𝜀		𝜠		𝜺
Unicode name	MATHEMATICAL BOLD CAPITAL EPSILON		MATHEMATICAL BOLD SMALL EPSILON		MATHEMATICAL ITALIC CAPITAL EPSILON		MATHEMATICAL ITALIC SMALL EPSILON		MATHEMATICAL BOLD ITALIC CAPITAL EPSILON		MATHEMATICAL BOLD ITALIC SMALL EPSILON
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120492	U+1D6AC	120518	U+1D6C6	120550	U+1D6E6	120576	U+1D700	120608	U+1D720	120634	U+1D73A
UTF-8	240 157 154 172	F0 9D 9A AC	240 157 155 134	F0 9D 9B 86	240 157 155 166	F0 9D 9B A6	240 157 156 128	F0 9D 9C 80	240 157 156 160	F0 9D 9C A0	240 157 156 186	F0 9D 9C BA
UTF-16	55349 57004	D835 DEAC	55349 57030	D835 DEC6	55349 57062	D835 DEE6	55349 57088	D835 DF00	55349 57120	D835 DF20	55349 57146	D835 DF3A
Numeric character reference	𝚬	𝚬	𝛆	𝛆	𝛦	𝛦	𝜀	𝜀	𝜠	𝜠	𝜺	𝜺

Character	𝝚		𝝴		𝞔		𝞮
Unicode name	MATHEMATICAL SANS-SERIF BOLD CAPITAL EPSILON		MATHEMATICAL SANS-SERIF BOLD SMALL EPSILON		MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL EPSILON		MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL EPSILON
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	120666	U+1D75A	120692	U+1D774	120724	U+1D794	120750	U+1D7AE
UTF-8	240 157 157 154	F0 9D 9D 9A	240 157 157 180	F0 9D 9D B4	240 157 158 148	F0 9D 9E 94	240 157 158 174	F0 9D 9E AE
UTF-16	55349 57178	D835 DF5A	55349 57204	D835 DF74	55349 57236	D835 DF94	55349 57262	D835 DFAE
Numeric character reference	𝝚	𝝚	𝝴	𝝴	𝞔	𝞔	𝞮	𝞮

Character	∈		∊		∋		∉
Unicode name	ELEMENT OF		SMALL ELEMENT OF		CONTAINS AS MEMBER		NOT AN ELEMENT OF
Encodings	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	8712	U+2208	8714	U+220A	8715	U+220B	8713	U+2209
UTF-8	226 136 136	E2 88 88	226 136 138	E2 88 8A	226 136 139	E2 88 8B	226 136 137	E2 88 89
Numeric character reference	∈	∈	∊	∊	∋	∋	∉	∉
Named character reference	∈				∋		∉

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

• 

  Initial epsilon in Lectionary 226, folio 20 verso

• 

  folio 64 verso

• 

  folio 125 verso

Further reading

• Hoffman, Paul; The Man Who Loved Only Numbers. Hyperion, 1998. ISBN 0-7868-6362-5.