Delta (letter)

"Δ" redirects here. For other uses, see ∆ (disambiguation).

Delta (/ˈdɛltə/;^[1] uppercase Δ, lowercase δ; Greek: δέλτα, délta, [ˈðelta])^[2] is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter dalet 𐤃.^[3] Letters that come from delta include Latin D and Cyrillic Д.

A river delta (originally, the delta of the Nile River) is so named 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.^[4]

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 operator^[5][6]); for example, in $${\displaystyle {\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}={\frac {\Delta y}{\Delta x}},}$$ 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:

  ${\displaystyle \Delta f=\sum _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}.}$

• The discriminant of a polynomial equation, especially the quadratic equation:^[7][8]

  ${\displaystyle \Delta =b^{2}-4ac.}$

• The area of a triangle

  ${\displaystyle \Delta ={\tfrac {1}{2}}ab\sin {C}.}$

• The symmetric difference of two sets.
• 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.^[9]
• In the financial markets, one of the Greeks, describing the rate of change of an option price for a given change in the underlying benchmark.
• A major seventh chord in jazz music notation.
• 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.^[10]
• The anonymous signature of James David Forbes.^[11]
• Determinacy (having a definite truth-value) in philosophical logic.
• In mathematics, the symbol ≜ (delta over equals) is occasionally used to define a new variable or function.^[12]

Lowercase

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 limits, in mathematics and more specifically in calculus
• The Kronecker delta in mathematics
• The degree of a vertex (graph theory)
• The Dirac delta function in mathematics
• 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
• A subunit of the F1 sector of the F-ATPase
• The declination of an object in the equatorial coordinate system of astronomy
• 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^[13]
• 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.^[14]
• An Old Irish voiced dental or alveolar fricative of uncertain articulation, the ancestor of the sound represented by Modern Irish dh

Computer encodings

• Greek Delta / Coptic Dalda

Character information

Preview	Δ		δ		ᵟ		Ⲇ		ⲇ
Unicode name	GREEK CAPITAL LETTER DELTA		GREEK SMALL LETTER DELTA		MODIFIER LETTER SMALL DELTA		COPTIC CAPITAL LETTER DALDA		COPTIC SMALL LETTER DALDA
Encodings	decimal	hex	dec	hex	dec	hex	dec	hex	dec	hex
Unicode	916	U+0394	948	U+03B4	7519	U+1D5F	11398	U+2C86	11399	U+2C87
UTF-8	206 148	CE 94	206 180	CE B4	225 181 159	E1 B5 9F	226 178 134	E2 B2 86	226 178 135	E2 B2 87
Numeric character reference	Δ	Δ	δ	δ	ᵟ	ᵟ	Ⲇ	Ⲇ	ⲇ	ⲇ
Named character reference	Δ		δ
DOS Greek	131	83	155	9B
DOS Greek-2	167	A7	221	DD
Windows 1253	196	C4	228	E4
TeX	\Delta		\delta

• Latin Delta

Character information

Preview	ẟ		ƍ
Unicode name	LATIN SMALL LETTER DELTA		LATIN SMALL LETTER TURNED DELTA
Encodings	decimal	hex	dec	hex
Unicode	7839	U+1E9F	397	U+018D
UTF-8	225 186 159	E1 BA 9F	198 141	C6 8D
Numeric character reference	ẟ	ẟ	ƍ	ƍ

• Technical and Mathematical symbols

Character information

Preview	∆		∇		≜		⍋		⍍		⍙
Unicode name	INCREMENT		NABLA		DELTA EQUAL TO		APL FUNCTIONAL SYMBOL DELTA STILE		APL FUNCTIONAL SYMBOL QUAD DELTA		APL FUNCTIONAL SYMBOL DELTA UNDERBAR
Encodings	decimal	hex	dec	hex	dec	hex	dec	hex	dec	hex	dec	hex
Unicode	8710	U+2206	8711	U+2207	8796	U+225C	9035	U+234B	9037	U+234D	9049	U+2359
UTF-8	226 136 134	E2 88 86	226 136 135	E2 88 87	226 137 156	E2 89 9C	226 141 139	E2 8D 8B	226 141 141	E2 8D 8D	226 141 153	E2 8D 99
Numeric character reference	∆	∆	∇	∇	≜	≜	⍋	⍋	⍍	⍍	⍙	⍙
Named character reference			∇, ∇		≜, ≜

• Mathematical Delta

Character information

Preview	𝚫		𝛅		𝛥		𝛿		𝜟		𝜹
Unicode name	MATHEMATICAL BOLD CAPITAL DELTA		MATHEMATICAL BOLD SMALL DELTA		MATHEMATICAL ITALIC CAPITAL DELTA		MATHEMATICAL ITALIC SMALL DELTA		MATHEMATICAL BOLD ITALIC CAPITAL DELTA		MATHEMATICAL BOLD ITALIC SMALL DELTA
Encodings	decimal	hex	dec	hex	dec	hex	dec	hex	dec	hex	dec	hex
Unicode	120491	U+1D6AB	120517	U+1D6C5	120549	U+1D6E5	120575	U+1D6FF	120607	U+1D71F	120633	U+1D739
UTF-8	240 157 154 171	F0 9D 9A AB	240 157 155 133	F0 9D 9B 85	240 157 155 165	F0 9D 9B A5	240 157 155 191	F0 9D 9B BF	240 157 156 159	F0 9D 9C 9F	240 157 156 185	F0 9D 9C B9
UTF-16	55349 57003	D835 DEAB	55349 57029	D835 DEC5	55349 57061	D835 DEE5	55349 57087	D835 DEFF	55349 57119	D835 DF1F	55349 57145	D835 DF39
Numeric character reference	𝚫	𝚫	𝛅	𝛅	𝛥	𝛥	𝛿	𝛿	𝜟	𝜟	𝜹	𝜹

Character information

Preview	𝝙		𝝳		𝞓		𝞭
Unicode name	MATHEMATICAL SANS-SERIF BOLD CAPITAL DELTA		MATHEMATICAL SANS-SERIF BOLD SMALL DELTA		MATHEMATICAL SANS-SERIF BOLD ITALIC CAPITAL DELTA		MATHEMATICAL SANS-SERIF BOLD ITALIC SMALL DELTA
Encodings	decimal	hex	dec	hex	dec	hex	dec	hex
Unicode	120665	U+1D759	120691	U+1D773	120723	U+1D793	120749	U+1D7AD
UTF-8	240 157 157 153	F0 9D 9D 99	240 157 157 179	F0 9D 9D B3	240 157 158 147	F0 9D 9E 93	240 157 158 173	F0 9D 9E AD
UTF-16	55349 57177	D835 DF59	55349 57203	D835 DF73	55349 57235	D835 DF93	55349 57261	D835 DFAD
Numeric character reference	𝝙	𝝙	𝝳	𝝳	𝞓	𝞓	𝞭	𝞭

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.

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.)
2. "Dictionary of Standard Modern greek". Centre for the Greek Language.
3. "Definition of DELTA". www.merriam-webster.com. Retrieved 26 October 2017.
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.
5. Clarence H. Richardson (1954). An Introduction to the Calculus of Finite Differences. Van Nostrand. Chapter 1, pp. 1—3.online copy
6. Michael Comenetz (2002). Calculus: The Elements. World Scientific. pp. 73–74. ISBN 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.
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.
9. Tepper, Pamela (2014). The Law of Contracts and the Uniform Commercial Code. Cengage Learning. p. 32. ISBN 978-1285448947. Retrieved 2018-04-30.
10. "Caduceus, the emblem of dentistry". American Dental Association. Archived from the original on 12 November 2012. Retrieved 26 October 2017.
11. Proceedings of the Royal Society, vol. XIX, p. ii.
12. "Who first defined the "equal-delta" or "delta over equal" symbol?". Archived from the original on 6 March 2022. Retrieved 2 October 2022.
13. "Faculty - Economics Department". econ.duke.edu. Retrieved 26 October 2017.
14. 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)