View source for Delta (letter)

{{Short description|Fourth letter in the Greek alphabet}}
{{Redirect|Δ|other uses|∆ (disambiguation)}}{{Distinguish|text=ẟ, the [[Latin delta]]}}
{{Refimprove|date=February 2015}}
{{Greek Alphabet|letter=delta}}'''Delta''' ({{IPAc-en|'|d|ɛ|l|t|ə|audio=LL-Q1860 (eng)-Flame, not lame-Delta.wav}} {{respell|DEL|tə}};<ref>{{OED|delta}}</ref> uppercase '''Δ''', lowercase '''δ'''; {{langx|el|δέλτα}}, ''délta'', {{IPA|el|ˈðelta|}})<ref>{{cite web|url=http://www.greek-language.gr/greekLang/modern_greek/tools/lexica/triantafyllides/search.html?lq=%CE%B4%CE%AD%CE%BB%CF%84%CE%B1&dq=|title=Dictionary of Standard Modern greek|publisher=Centre for the Greek Language}}</ref> is the fourth letter of the [[Greek alphabet]]. In the system of [[Greek numerals]], it has a value of four. It was derived from the [[Phoenician alphabet|Phoenician letter]] [[Dalet (letter)|dalet]] 𐤃.<ref>{{cite web|url=http://www.merriam-webster.com/dictionary/delta|title=Definition of DELTA|website=www.merriam-webster.com|access-date=26 October 2017}}</ref> Letters that come from delta include the [[Latin alphabet|Latin]] [[D]] and the [[Cyrillic script|Cyrillic]] [[De (Cyrillic)|Д]].

A [[river delta]] (originally, the [[Nile Delta|delta]] of the [[Nile River]]) is named so because its shape approximates the triangular uppercase letter delta. Contrary to a popular legend,{{Vague|reason=Which legend?|date=January 2025}} this use of the word ''delta'' was not coined by [[Herodotus]].<ref>{{cite journal|last=Celoria|first=Francis|title=Delta as a geographical concept in Greek literature|journal=Isis|year=1966|volume=57|issue=3|pages=385–388|doi=10.1086/350146|jstor=228368|s2cid=143811840}}</ref>

==Pronunciation==
In [[Ancient Greek]], delta represented a [[voiced dental plosive]] {{IPA|el|d|IPA}}. In [[Modern Greek]], it represents a [[voiced dental fricative]] {{IPA|el|ð|IPA}}, like the "''th''" in "''that''" or "''this''" (while {{IPA|el|d|IPA}} in foreign words is instead commonly transcribed as ντ). Delta is [[romanization of Greek|romanized]] as ''d'' or ''dh''.

==Uppercase==
The uppercase letter Δ is used to denote:
* Change of any changeable quantity, in mathematics and the sciences (in particular, the [[difference operator]]<ref name="Richardson1954">{{cite book
|author=Clarence H. Richardson
|title=''An Introduction to the Calculus of Finite Differences''
|url=https://archive.org/details/introductiontoca00rich
|url-access=registration
|year=1954
|publisher=Van Nostrand
|at=Chapter 1, pp. 1—3}}[http://babel.hathitrust.org/cgi/pt?id=mdp.39015000982945;view=1up;seq=5 online copy]</ref><ref name="Comenetz2002">{{cite book
|author=Michael Comenetz
|title=Calculus: The Elements
|year=2002
|publisher=World Scientific
|isbn=978-981-02-4904-5
|pages=73–74}}</ref>); for example, in <math>
\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta y}{\Delta x}
</math>, the average change of ''y'' per unit ''x'' (i.e. the change of ''y'' over the change of ''x''). Delta is the initial letter of the Greek word {{lang|el|διαφορά}}, ''diaphorá'', "difference". (The small [[Latin]] letter '''d''' is used in much the same way for the notation of [[derivative]]s and [[differential (mathematics)|differentials]], which also describe change by [[infinitesimal]] amounts.)
* The [[Laplace operator]]:
*: <math>\Delta f=\sum_{i=1}^n{\frac{\partial^2f}{\partial x_i^2}}</math>.
* The [[discriminant]] of a [[polynomial]] equation, especially the [[quadratic equation]]:<ref>{{cite book
|title=Solving polynomial equations: foundations, algorithms, and applications
|first1=Alicia
|last1=Dickenstein
|author1-link=Alicia Dickenstein
|first2=Ioannis Z.
|last2=Emiris
|publisher=Springer
|year=2005
|isbn=978-3-540-24326-7
|url=https://books.google.com/books?id=rSs-pQNrO_YC&pg=PA26
|at=Example 2.5.6, p. 120}}
</ref><ref>{{cite book
|title=Integers, polynomials, and rings
|first1=Ronald S.
|last1=Irving
|publisher=Springer-Verlag New York, Inc.
|year=2004
|isbn=978-0-387-40397-7
|url=https://books.google.com/books?id=B4k6ltaxm5YC&pg=PA154
|at=Ch. 10.1, pp. 145}}</ref>
*: <math>\Delta=b^2-4ac</math>.
* The [[area of a triangle]]:
*: <math>\Delta=\tfrac{1}{2}ab\sin{C}</math>.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Triangle Area |url=https://mathworld.wolfram.com/TriangleArea.html |access-date=2025-01-23 |website=mathworld.wolfram.com |language=en}}</ref>
* The [[symmetric difference]] of two sets.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Symmetric Difference |url=https://mathworld.wolfram.com/SymmetricDifference.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en |quote=The symmetric difference of sets A and B is variously written as A ⊖ B, A∇ B, A+B (Borowski and Borwein 1991) or AΔB (Harris and Stocker 1998, p. 3). All but the first notation should probably be deprecated since each of the other symbols has a common meaning in other areas of mathematics.}}</ref>
* A [[derivative|macroscopic change]] in the value of a variable in mathematics or science.
* [[Uncertainty]] in a physical variable as seen in the [[uncertainty principle]].
* An interval of possible values for a given quantity.
* Any of the [[Delta baryon|delta particles]] in particle physics.
* The determinant of the matrix of coefficients of a set of linear equations (see [[Cramer's rule]]).
* That an associated [[locant]] number represents the location of a [[covalent bond]] in an [[organic compound]], the position of which is variant between [[isomer]]ic forms.
* A [[simplex]], [[simplicial complex]], or [[convex hull]].
* In chemistry, the addition of heat in a reaction.
* In [[legal shorthand]], it represents a [[defendant]].<ref>{{cite book |last1=Tepper |first1=Pamela |url=https://books.google.com/books?id=z9wbCgAAQBAJ |title=The Law of Contracts and the Uniform Commercial Code |publisher=Cengage Learning |year=2014 |page=32 |isbn=978-1285448947 |access-date=2018-04-30 }}</ref>
* In the financial markets, one of the [[Greeks (finance)|Greeks]], describes the rate of change of an option price for a given change in the underlying benchmark.<ref>{{Cite web |title=Black-Scholes Formulas (d1, d2, Call Price, Put Price, Greeks) - Macroption |url=https://www.macroption.com/black-scholes-formula/ |access-date=2025-01-23 |website=www.macroption.com |quote=Delta is the first derivative of option price with respect to underlying price S.}}</ref>
* A [[major seventh chord]] in jazz music notation.<ref>{{Cite web |title=Every chord symbol found on lead sheets |url=https://jazz-library.com/articles/chord-symbols/ |access-date=2025-01-23 |website=Jazz-Library |language=en-US |quote=Chord with a triangle Major 7th “C△”}}</ref>
* In [[genetics]], it can stand for a [[Deletion (genetics)|gene deletion]] (e.g. the [[CCR5-Δ32]], a 32 nucleotide/bp deletion within CCR5).
* The [[American Dental Association]] cites it (together with [[omicron]] for "odont") as the symbol of dentistry.<ref>{{cite web |url=http://www.ada.org/2517.aspx?currentTab=2 |title=Caduceus, the emblem of dentistry |website=American Dental Association |access-date=26 October 2017 |archive-url=https://web.archive.org/web/20121112073815/http://www.ada.org/2517.aspx?currentTab=2 |archive-date=12 November 2012 |url-status=dead}}</ref>
* The anonymous signature of [[James David Forbes]].<ref>[[Proceedings of the Royal Society]], vol. XIX, p. ii.</ref>
* Determinacy (having a definite truth-value) in [[philosophical logic]].<ref>{{Cite journal |last1=Shramko |first1=Yaroslav |last2=Wansing |first2=Heinrich |date=2010-03-30 |title=Truth Values |url=https://plato.stanford.edu/archIves/fall2015/entries/truth-values/}}</ref>
* In mathematics, the symbol ≜ (delta over equals) is occasionally used to define a new variable or function.<ref>{{cite web |url=https://hsm.stackexchange.com/questions/5166/who-first-defined-the-equal-delta-or-delta-over-equal-triangleq-symbol |title=Who first defined the "equal-delta" or "delta over equal" symbol? |access-date=2 October 2022 |archive-url=https://web.archive.org/web/20220306070901/https://hsm.stackexchange.com/questions/5166/who-first-defined-the-equal-delta-or-delta-over-equal-triangleq-symbol |archive-date=6 March 2022 |url-status=dead}}</ref>

==Lowercase==
[[File:NAMA Alphabet grec.jpg|thumb|right|The [[alphabet]] on a [[black figure]] vessel, with a D-shaped delta.]]

The lowercase letter δ (or 𝛿) can be used to denote:
* A change in the value of a variable in [[calculus]].
* A [[functional derivative]] in [[functional calculus]].
* The [[(ε, δ)-definition of limit]]s, in mathematics and more specifically in [[calculus]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Epsilon-Delta Proof |url=https://mathworld.wolfram.com/Epsilon-DeltaProof.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en}}</ref>
* The [[Kronecker delta]] in mathematics.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Kronecker Delta |url=https://mathworld.wolfram.com/KroneckerDelta.html |access-date=2025-01-23 |website=mathworld.wolfram.com |language=en}}</ref>
* The [[Finite difference|central difference]] for a function.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Central Difference |url=https://mathworld.wolfram.com/CentralDifference.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en}}</ref>
* The degree of a [[vertex (graph theory)|vertex]] in [[graph theory]].
* The [[Dirac delta function]] in mathematics.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Delta Function |url=https://mathworld.wolfram.com/DeltaFunction.html |access-date=2025-01-23 |website=mathworld.wolfram.com |language=en}}</ref>
* The transition function in [[Finite-state machine|automata]].
* [[Deflection (engineering)|Deflection]] in engineering mechanics.
* The [[Compound interest|force of interest]] in actuarial science.
* The [[chemical shift]] of [[nuclear magnetic resonance]] in chemistry.
* The relative [[electronegativity]] of different atoms in a molecule, δ<sup>−</sup> being more electronegative than δ<sup>+</sup>.
* Text requiring deletion in [[proofreading]]; the usage is said to date back to classical times.
* In some of the manuscripts written by Dr. [[John Dee (mathematician)|John Dee]], the character of delta is used to represent Dee.<ref>{{Cite web |title=Greek Alphabet |url=https://www.ancient-symbols.com/greek-alphabet |access-date=2025-01-31 |website=Ancient Symbols |language=en-us |quote=Dr. John Dee, a mathematician, used the lowercase Delta symbol to represent himself in manuscripts.}}</ref>
* A subunit of the F1 sector of the [[F-ATPase]].
* The [[declination]] of an object in the [[equatorial coordinate system]] of [[astronomy]].<ref>{{Cite web |title=Greek Alphabet |url=https://www.ancient-symbols.com/greek-alphabet |access-date=2025-01-31 |website=Ancient Symbols |language=en-us |quote=In astronomy, the symbol is used to represent the declination of an object.}}</ref>
* The dividend yield in the [[Black–Scholes]] option pricing formula.
* Ratios of [[environmental isotopes]], such as [[Oxygen-18|<sup>18</sup>O]]/[[Oxygen-16|<sup>16</sup>O]] and [[Deuterium|D]]/[[Hydrogen-1|<sup>1</sup>H]] from water are displayed using [[Isotope geochemistry|delta notation]]&nbsp;– [[δ18O|δ<sup>18</sup>O]] and [[δD]], respectively.
* The rate of depreciation of the aggregate capital stock of an economy in an [[exogenous growth model]] in [[macroeconomics]].<ref>{{cite web|url=http://econ.duke.edu/people?subpage=unit&Gurl=/aas/Economics&cname=Masters+Students|title=Faculty - Economics Department|website=econ.duke.edu|access-date=26 October 2017}}</ref>
* In a system that exhibits [[electrical reactance]], the angle between [[voltage]] and [[Electric current|current]].
* [[Partial charge]] in chemistry.<ref>{{Citation |title=Declination |encyclopedia=The Encyclopedia of Astronomy and Astrophysics |date=2001 |url=https://doi.org/10.1888/0333750888/4541 |access-date=2025-01-31 |publisher=IOP Publishing Ltd |doi=10.1888/0333750888/4541 |isbn=0-333-75088-8|url-access=subscription }}</ref>
* The maximum [[birefringence]] of a crystal in [[optical mineralogy]].<ref>{{Cite book|title=Mineralogia Óptica|last=MACHADO, Fábio Braz|first=NARDY, Antônio José Ranalli|publisher=Oficina de Textos|year=2018|isbn=9788579752452|location=São Paulo|pages=85}}</ref>
*  An [[Old Irish]] voiced dental or alveolar [[fricative]] of uncertain articulation, the ancestor of the sound represented by [[irish_language|Modern Irish]] ''dh''.
*  [[Silver ratio]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Silver Ratio |url=https://mathworld.wolfram.com/SilverRatio.html |access-date=2025-01-31 |website=mathworld.wolfram.com |language=en}}</ref>

==Unicode==

* {{unichar|018D|html=}}
* {{unichar|0394|html=}} ({{tt|\Delta}} in [[TeX]])
* {{unichar|03B4|html=}} ({{tt|\delta}} in TeX)
* {{unichar|1D5F|html=}}
* {{unichar|1E9F|html=}}
* {{unichar|2207|html=}}
* {{unichar|225C|html=}}
* {{unichar|234B|html=}}
* {{unichar|234D|html=}}
* {{unichar|2359|html=}}
* {{unichar|2C86|html=}}
* {{unichar|2C87|html=}}
* {{unichar|1D6AB|html=}}{{efn|The {{sc|mathematical}} codes should only be used in math.  Stylized Greek text should be encoded using the normal Greek letters, with markup and formatting to indicate text style.}}
* {{unichar|1D6C5|html=}}
* {{unichar|1D6E5|html=}}
* {{unichar|1D6FF|html=}}
* {{unichar|1D71F|html=}}
* {{unichar|1D739|html=}}
* {{unichar|1D759|html=}}
* {{unichar|1D773|html=}}
* {{unichar|1D793|html=}}
* {{unichar|1D7AD|html=}}
{{notelist}}

==See also==
{{Wiktionary|Δ|δ}}
* [[Arrow (symbol)]]
* [[Chevron (insignia)]]
* [[∆ (disambiguation)]]
* [[D|D, d]]
* [[De (Cyrillic)|Д, д]]
* [[ẟ]]&nbsp;– Latin delta
* [[∂]]&nbsp;– the [[partial derivative]] symbol, a curved ''d'', sometimes mistaken for a lowercase Greek letter Delta.
* [[ð]]&nbsp;– the small [[eth]] appears similar to a small delta and also represents a ''d'' sound in some contexts
* [[Th (digraph)]]
* [[Thorn (letter)]]
* [[Greek letters used in mathematics, science, and engineering]]
* ∇&nbsp;– [[Nabla symbol]]
* [[Delta Air Lines]]
* [[SARS-CoV-2 Delta variant]]

==References==
<references />

[[Category:Greek letters]]