Delta (letter)

Delta (/ˈdɛltə/ ^ⓘ DEL-tə;^[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 four. It was derived from the Phoenician letter dalet 𐤃.^[3] 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.^[4]

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", "this", or "though" (while IPA: [d] in foreign words is instead commonly transcribed as ντ). Delta is romanized as d or dh.

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:

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

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 ratio^[27]

• 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 DELTA^[a]
• 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

Look up Δ or δ in Wiktionary, the free dictionary.

• 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