Delta (letter) explained

Delta (; uppercase Δ, lowercase δ; el|δέλτα, délta, pronounced as /el/)^[1] 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 letter dalet .^[2] Letters that come from delta include the Latin D and the Cyrillic Д.

A river delta (originally, the delta of the Nile River) is named so because its shape approximates the triangular uppercase letter delta. Contrary to a popular legend, this use of the word delta was not coined by Herodotus.^[3]

Pronunciation

In Ancient Greek, delta represented a voiced dental plosive pronounced as /el/. In Modern Greek, it represents a voiced dental fricative pronounced as /el/, like the "th" in "that", "this", or "though" (while pronounced as /el/ in foreign words is instead commonly transcribed as ντ, nt). Delta is romanized as d or (in Modern Greek) 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^{[4] [5]}); for example, in

, 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 Greek, Modern (1453-);: 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:

}.

• The discriminant of a polynomial equation, especially the quadratic equation:^{[6] [7]}

.

• The area of a triangle:

.^[8]

• The symmetric difference of two sets.^[9]
• 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.^[10]
• 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.^[11]
• A major seventh chord in jazz music notation.^[12]
• 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.^[13]
• The anonymous signature of James David Forbes.^[14]
• Determinacy (having a definite truth-value) in philosophical logic.^[15]
• In mathematics, the symbol ≜ (delta over equals) is occasionally used to define a new variable or function.^[16]

Lowercase

The lowercase letter δ (or) 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.^[17]
• The Kronecker delta in mathematics.^[18]
• The central difference for a function.^[19]
• The degree of a vertex in graph theory.
• The Dirac delta function in mathematics.^[20]
• 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.^[21]
• A subunit of the F1 sector of the F-ATPase.
• The declination of an object in the equatorial coordinate system of astronomy.^[22]
• 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.^[23]
• In a system that exhibits electrical reactance, the angle between voltage and current.
• Partial charge in chemistry.
• The maximum birefringence of a crystal in optical mineralogy.^[24]
• An Old Irish voiced dental or alveolar fricative of uncertain articulation, the ancestor of the sound represented by Modern Irish dh.
• Silver ratio^[25]

Unicode

• • (in TeX)
• (in TeX)

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. Web site: Dictionary of Standard Modern greek. Centre for the Greek Language.
2. Web site: Definition of DELTA. www.merriam-webster.com. 26 October 2017.
3. Celoria. Francis. Delta as a geographical concept in Greek literature. Isis. 1966. 57. 3. 385–388. 10.1086/350146. 228368. 143811840.
4. Book: Clarence H. Richardson. An Introduction to the Calculus of Finite Differences. registration. 1954. Van Nostrand. Chapter 1, pp. 1—3. online copy
5. Book: Michael Comenetz. Calculus: The Elements. 2002. World Scientific. 978-981-02-4904-5. 73–74.
6. Book: Solving polynomial equations: foundations, algorithms, and applications. Alicia. Dickenstein. Alicia Dickenstein. Ioannis Z.. Emiris. Springer. 2005. 978-3-540-24326-7. Example 2.5.6, p. 120.
7. Book: Integers, polynomials, and rings. Ronald S.. Irving. Springer-Verlag New York, Inc.. 2004. 978-0-387-40397-7. Ch. 10.1, pp. 145.
8. Web site: Weisstein . Eric W. . Triangle Area . 2025-01-23 . mathworld.wolfram.com . en.
9. Web site: Weisstein . Eric W. . Symmetric Difference . 2025-01-31 . mathworld.wolfram.com . en . 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..
10. Book: Tepper . Pamela . The Law of Contracts and the Uniform Commercial Code . Cengage Learning . 2014 . 32 . 978-1285448947 . 2018-04-30 .
11. Web site: Black-Scholes Formulas (d1, d2, Call Price, Put Price, Greeks) - Macroption . 2025-01-23 . www.macroption.com . Delta is the first derivative of option price with respect to underlying price S..
12. Web site: Every chord symbol found on lead sheets . 2025-01-23 . Jazz-Library . en-US . Chord with a triangle Major 7th “C△”.
13. Web site: Caduceus, the emblem of dentistry . American Dental Association . 26 October 2017 . https://web.archive.org/web/20121112073815/http://www.ada.org/2517.aspx?currentTab=2 . 12 November 2012 . dead.
14. [Proceedings of the Royal Society]
15. Shramko . Yaroslav . Wansing . Heinrich . 2010-03-30 . Truth Values .
16. Web site: Who first defined the "equal-delta" or "delta over equal" symbol? . 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 . 6 March 2022 . dead.
17. Web site: Weisstein . Eric W. . Epsilon-Delta Proof . 2025-01-31 . mathworld.wolfram.com . en.
18. Web site: Weisstein . Eric W. . Kronecker Delta . 2025-01-23 . mathworld.wolfram.com . en.
19. Web site: Weisstein . Eric W. . Central Difference . 2025-01-31 . mathworld.wolfram.com . en.
20. Web site: Weisstein . Eric W. . Delta Function . 2025-01-23 . mathworld.wolfram.com . en.
21. Web site: Greek Alphabet . 2025-01-31 . Ancient Symbols . en-us . Dr. John Dee, a mathematician, used the lowercase Delta symbol to represent himself in manuscripts..
22. Web site: Greek Alphabet . 2025-01-31 . Ancient Symbols . en-us . In astronomy, the symbol is used to represent the declination of an object..
23. Web site: Faculty - Economics Department. econ.duke.edu. 26 October 2017.
24. Book: MACHADO, Fábio Braz, NARDY, Antônio José Ranalli. Mineralogia Óptica. Oficina de Textos. 2018. 9788579752452. São Paulo. 85.
25. Web site: Weisstein . Eric W. . Silver Ratio . 2025-01-31 . mathworld.wolfram.com . en.