Delta (letter)

Delta (uppercase Δ, lowercase δ or 𝛿; Δέλτα Délta; Modern Greek [ˈðelta]^[1]) 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 .^[2] Letters that come from delta include Latin D and Cyrillic Д.

A river delta (originally, the Nile River delta) is so named because its shape approximates the upper-case letter delta (the shape is a triangle). Despite 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 /d/. In Modern Greek, it represents a voiced dental fricative /ð/, like the "th" in "that" or "this". It is romanized as d or dh.

Upper case

The upper-case letter Δ can be used to denote:

• Change of any changeable quantity, in mathematics and the sciences (more specifically, the difference operator); for example, in:

${y_2-y_1\over x_2-x_1} = {\Delta y \over \Delta x} ,$

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.)

• The Laplace operator:

$\Delta f = \sum_{i=1}^n {\frac{\partial^2 f}{\partial x_i^2}}$

• The discriminant of a polynomial equation, especially the quadratic equation:

$\Delta = b^2 - 4ac\,\!$

• The area of a triangle

$\Delta = \tfrac{1}{2} a b \sin{C}$

• 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
• 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 of CCR5)
• The American Dental Association cites it (together with omicron for "odont") as the symbol of dentistry.^[4]
• The anonymous signature of James David Forbes.^[5]

Lower case

The alphabet on a black figure vessel, with a D-shaped delta.

The lower-case letter δ (or 𝛿) can be used to denote:

• A change in the value of a variable in calculus
• An auxiliary function in calculus, used to rigorously define the limit or continuity of a given function
• The Kronecker delta in mathematics
• 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^[6]
• In a system that exhibits electrical reactance, the angle between voltage and current

Keyboard Entry

On Microsoft Windows, a small delta can be generated by typing Alt+235. In certain programs (e.g. Microsoft Word), an uppercase delta can be typed using Alt+0916. However, in most other programs, Alt+0916 generates a closing quotation mark (”).

On OS X, option-j yields an upper-case delta (∆).

On Linux, Left Ctrl+Shift+u then 0394 yields an upper-case delta (Δ) and Left Ctrl+Shift+u then 03B4 yields a lower-case delta (δ).

Computer encodings

• Greek Delta / Coptic Dalda

Character	Δ		δ		Ὗ		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	decimal	hex	decimal	hex	decimal	hex	decimal	hex
Unicode	916	U+0394	948	U+03B4	8031	U+1F5F	11398	U+2C86	11399	U+2C87
UTF-8	206 148	CE 94	206 180	CE B4	225 189 159	E1 BD 9F	226 178 134	E2 B2 86	226 178 135	E2 B2 87
Numeric character reference	Δ	Δ	δ	δ	Ὗ	Ὗ	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	ẟ		ƍ
Unicode name	LATIN SMALL LETTER DELTA		LATIN SMALL LETTER TURNED DELTA
Encodings	decimal	hex	decimal	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	∆		∇		≜		⍋		⍍		⍙
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	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	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	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	decimal	hex	decimal	hex	decimal	hex	decimal	hex	decimal	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	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	decimal	hex	decimal	hex	decimal	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

• D, d
• Д, д
• ẟ - Latin delta
• ∂ - the partial derivative symbol, 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)
• Greek letters used in mathematics, science, and engineering
• ∇ - Nabla symbol

References

1. "Dictionary of Standard Modern greek". Centre for the Greek Language.
2. http://www.merriam-webster.com/dictionary/delta
3. Celoria, Francis (1966). "Delta as a geographical concept in Greek literature". Isis 57 (3): 385–388. doi:10.1086/350146.
4. Caduceus, the emblem of dentistry - American Dental Association - ADA.org
5. Proceedings of the Royal Society, vol. XIX, p ii
6. http://econ.duke.edu/people?subpage=unit&Gurl=%2Faas%2FEconomics&cname=Masters+Students