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Pronunciation

In Ancient Greek, delta represented a voiced dental plosive IPA: [d]. In Modern Greek, it represents a voiced dental fricative IPA: [ð], like the "th" in "that" or "this" (while IPA: [d] in foreign words is instead commonly transcribed as ντ). Delta is 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 operator⁵⁶); for example, in y 2 − y 1 x 2 − x 1 = Δ y Δ x {\displaystyle {\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\frac {\Delta y}{\Delta x}}} , 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 διαφορά, diaphorá, "difference". (The small Latin letter d is used in much the same way for the notation of derivatives and differentials, which also describe change by infinitesimal amounts.)
• The Laplace operator: Δ f = ∑ i = 1 n ∂ 2 f ∂ x i 2 {\displaystyle \Delta f=\sum _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}} .
• The discriminant of a polynomial equation, especially the quadratic equation:⁷⁸ Δ = b 2 − 4 a c {\displaystyle \Delta =b^{2}-4ac} .
• The area of a triangle: Δ = 1 2 a b sin ⁡ C {\displaystyle \Delta ={\tfrac {1}{2}}ab\sin {C}} .⁹
• The symmetric difference of two sets.¹⁰
• A 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 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 isomeric forms.
• A simplex, simplicial complex, or convex hull.
• In chemistry, the addition of heat in a reaction.
• In legal shorthand, it represents a defendant.¹¹
• In the financial markets, one of the Greeks, describes the rate of change of an option price for a given change in the underlying benchmark.¹²
• A major seventh chord in jazz music notation.¹³
• In genetics, it can stand for a 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.¹⁴
• The anonymous signature of James David Forbes.¹⁵
• Determinacy (having a definite truth-value) in philosophical logic.¹⁶
• In mathematics, the symbol ≜ (delta over equals) is occasionally used to define a new variable or function.¹⁷

Lowercase

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 limits, in mathematics and more specifically in calculus.¹⁸
• The Kronecker delta in mathematics.¹⁹
• The central difference for a function.²⁰
• The degree of a vertex in graph theory.
• The Dirac delta function in mathematics.²¹
• The transition function in automata.
• Deflection in engineering mechanics.
• The 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− being more electronegative than δ+.
• 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, the character of delta is used to represent Dee.²²
• A subunit of the F1 sector of the F-ATPase.
• The declination of an object in the equatorial coordinate system of astronomy.²³
• The dividend yield in the Black–Scholes option pricing formula.
• Ratios of environmental isotopes, such as 18O/16O and D/1H from water are displayed using delta notation – δ18O and δD, respectively.
• The rate of depreciation of the aggregate capital stock of an economy in an exogenous growth model in macroeconomics.²⁴
• In a system that exhibits electrical reactance, the angle between voltage and current.
• Partial charge in chemistry.²⁵
• The maximum birefringence of a crystal in optical mineralogy.²⁶
• An Old Irish voiced dental or alveolar fricative of uncertain articulation, the ancestor of the sound represented by Modern Irish dh.
• Silver ratio²⁷

Unicode

• U+018D ƍ LATIN SMALL LETTER TURNED DELTA
• U+0394 Δ GREEK CAPITAL LETTER DELTA (Δ) (\Delta in TeX)
• U+03B4 δ GREEK SMALL LETTER DELTA (δ) (\delta in TeX)
• U+1D5F ᵟ MODIFIER LETTER SMALL DELTA
• U+1E9F ẟ LATIN SMALL LETTER DELTA
• U+2207 ∇ NABLA (∇, ∇)
• U+225C ≜ DELTA EQUAL TO (≜, ≜)
• U+234B ⍋ APL FUNCTIONAL SYMBOL DELTA STILE
• U+234D ⍍ APL FUNCTIONAL SYMBOL QUAD DELTA
• U+2359 ⍙ APL FUNCTIONAL SYMBOL DELTA UNDERBAR
• U+2C86 Ⲇ COPTIC CAPITAL LETTER DALDA
• U+2C87 ⲇ COPTIC SMALL LETTER DALDA
• U+1D6AB 𝚫 MATHEMATICAL BOLD CAPITAL DELTA²⁸
• U+1D6C5 𝛅 MATHEMATICAL BOLD SMALL DELTA
• U+1D6E5 𝛥 MATHEMATICAL ITALIC CAPITAL DELTA
• U+1D6FF 𝛿 MATHEMATICAL ITALIC SMALL DELTA
• U+1D71F 𝜟 MATHEMATICAL BOLD ITALIC CAPITAL DELTA
• U+1D739 𝜹 MATHEMATICAL BOLD ITALIC SMALL DELTA
• U+1D759 𝝙 MATHEMATICAL SANS-SERIF BOLD CAPITAL DELTA
• U+1D773 𝝳 MATHEMATICAL SANS-SERIF BOLD SMALL DELTA
• U+1D793 𝞓 MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL DELTA
• U+1D7AD 𝞭 MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL DELTA

See also

• Arrow (symbol)
• Chevron (insignia)
• ∆ (disambiguation)
• D, d
• Д, д
• ẟ – Latin delta
• ∂ – the partial derivative symbol, a curved d, sometimes mistaken for a lowercase Greek letter Delta.
• ð – 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
• ∇ – Nabla symbol
• Delta Air Lines
• SARS-CoV-2 Delta variant

References

1. "delta". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.) https://www.oed.com/search/dictionary/?q=delta ↩

2. "Dictionary of Standard Modern greek". Centre for the Greek Language. 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= ↩

3. "Definition of DELTA". www.merriam-webster.com. Retrieved 26 October 2017. http://www.merriam-webster.com/dictionary/delta ↩

4. Celoria, Francis (1966). "Delta as a geographical concept in Greek literature". Isis. 57 (3): 385–388. doi:10.1086/350146. JSTOR 228368. S2CID 143811840. /wiki/Doi_(identifier) ↩

5. Clarence H. Richardson (1954). An Introduction to the Calculus of Finite Differences. Van Nostrand. Chapter 1, pp. 1—3.online copy https://archive.org/details/introductiontoca00rich ↩

6. Michael Comenetz (2002). Calculus: The Elements. World Scientific. pp. 73–74. ISBN 978-981-02-4904-5. 978-981-02-4904-5 ↩

7. Dickenstein, Alicia; Emiris, Ioannis Z. (2005). Solving polynomial equations: foundations, algorithms, and applications. Springer. Example 2.5.6, p. 120. ISBN 978-3-540-24326-7. 978-3-540-24326-7 ↩

8. Irving, Ronald S. (2004). Integers, polynomials, and rings. Springer-Verlag New York, Inc. Ch. 10.1, pp. 145. ISBN 978-0-387-40397-7. 978-0-387-40397-7 ↩

9. Weisstein, Eric W. "Triangle Area". mathworld.wolfram.com. Retrieved 2025-01-23. https://mathworld.wolfram.com/TriangleArea.html ↩

10. Weisstein, Eric W. "Symmetric Difference". mathworld.wolfram.com. Retrieved 2025-01-31. 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. https://mathworld.wolfram.com/SymmetricDifference.html ↩

11. Tepper, Pamela (2014). The Law of Contracts and the Uniform Commercial Code. Cengage Learning. p. 32. ISBN 978-1285448947. Retrieved 2018-04-30. 978-1285448947 ↩

12. "Black-Scholes Formulas (d1, d2, Call Price, Put Price, Greeks) - Macroption". www.macroption.com. Retrieved 2025-01-23. Delta is the first derivative of option price with respect to underlying price S. https://www.macroption.com/black-scholes-formula/ ↩

13. "Every chord symbol found on lead sheets". Jazz-Library. Retrieved 2025-01-23. Chord with a triangle Major 7th "C△" https://jazz-library.com/articles/chord-symbols/ ↩

14. "Caduceus, the emblem of dentistry". American Dental Association. Archived from the original on 12 November 2012. Retrieved 26 October 2017. https://web.archive.org/web/20121112073815/http://www.ada.org/2517.aspx?currentTab=2 ↩

15. Proceedings of the Royal Society, vol. XIX, p. ii. /wiki/Proceedings_of_the_Royal_Society ↩

16. Shramko, Yaroslav; Wansing, Heinrich (2010-03-30). "Truth Values". {{cite journal}}: Cite journal requires |journal= (help) https://plato.stanford.edu/archIves/fall2015/entries/truth-values/ ↩

17. "Who first defined the "equal-delta" or "delta over equal" symbol?". Archived from the original on 6 March 2022. Retrieved 2 October 2022. https://web.archive.org/web/20220306070901/https://hsm.stackexchange.com/questions/5166/who-first-defined-the-equal-delta-or-delta-over-equal-triangleq-symbol ↩

18. Weisstein, Eric W. "Epsilon-Delta Proof". mathworld.wolfram.com. Retrieved 2025-01-31. https://mathworld.wolfram.com/Epsilon-DeltaProof.html ↩

19. Weisstein, Eric W. "Kronecker Delta". mathworld.wolfram.com. Retrieved 2025-01-23. https://mathworld.wolfram.com/KroneckerDelta.html ↩

20. Weisstein, Eric W. "Central Difference". mathworld.wolfram.com. Retrieved 2025-01-31. https://mathworld.wolfram.com/CentralDifference.html ↩

21. Weisstein, Eric W. "Delta Function". mathworld.wolfram.com. Retrieved 2025-01-23. https://mathworld.wolfram.com/DeltaFunction.html ↩

22. "Greek Alphabet". Ancient Symbols. Retrieved 2025-01-31. Dr. John Dee, a mathematician, used the lowercase Delta symbol to represent himself in manuscripts. https://www.ancient-symbols.com/greek-alphabet ↩

23. "Greek Alphabet". Ancient Symbols. Retrieved 2025-01-31. In astronomy, the symbol is used to represent the declination of an object. https://www.ancient-symbols.com/greek-alphabet ↩

24. "Faculty - Economics Department". econ.duke.edu. Retrieved 26 October 2017. http://econ.duke.edu/people?subpage=unit&Gurl=/aas/Economics&cname=Masters+Students ↩

25. "Declination", The Encyclopedia of Astronomy and Astrophysics, IOP Publishing Ltd, 2001, doi:10.1888/0333750888/4541, ISBN 0-333-75088-8, retrieved 2025-01-31 0-333-75088-8 ↩

26. MACHADO, Fábio Braz, NARDY, Antônio José Ranalli (2018). Mineralogia Óptica. São Paulo: Oficina de Textos. p. 85. ISBN 9788579752452.{{cite book}}: CS1 maint: multiple names: authors list (link) 9788579752452 ↩

27. Weisstein, Eric W. "Silver Ratio". mathworld.wolfram.com. Retrieved 2025-01-31. https://mathworld.wolfram.com/SilverRatio.html ↩

28. The 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. ↩