Delta

How to read it

Delta is said DEL-tuh (Cambridge Dictionary: /ˈdel.tə/, American /ˈdel.t̬ə/), and Delta, said DELL-TAH, is the word for D in the ICAO spelling alphabet the US Federal Aviation Administration uses. Modern Greek δέλτα starts with a different sound: the University of Arizona’s guide writes “Dhelta”, with the th of “this”, “NOT D”.

Also searched as: delta letter, delta symbol, triangle symbol delta, δ, Δ, Δ meaning.

Capital, small and look-alike forms

The increment sign ∆ looks like the capital delta but is a separate maths character. Unicode lists it for the Laplace operator, the forward difference and the symmetric difference of two sets, and cross-refers it to the Greek Δ and to the plain white triangle △ (U+25B3). The upside-down triangle ∇ is Nabla (∇).

Delta in maths

The capital Δ is a change or a difference: the American Heritage Dictionary defines delta in maths as “a finite increment in a variable”, and Cambridge Dictionary as “a change in a figure or amount”. Jeff Miller’s history of mathematical symbols quotes Euler in 1755: “Just as we used the symbol Δ to signify a difference, so we use the symbol Σ to indicate a sum”, the first use of the summation sign Σ. NIST’s Digital Library of Mathematical Functions writes the forward difference as Δf(x) = f(x + 1) − f(x). Δ is also the Laplacian, a use Miller credits to Robert Murphy in 1833, also written ∇² (Nabla (∇)).

The small δ is a small quantity. In the ε-δ definition of a limit, for every ε > 0 there is a δ > 0 that keeps f(x) within ε of the limit whenever x is within δ of a (OpenStax, Calculus Volume 1). Augustin-Louis Cauchy used both letters in the 1820s, and Miller cites Finney and Thomas that δ stood for différence and ε for erreur. δ also names two standard functions: the Kronecker delta δⱼₖ, which is 1 when j = k and 0 otherwise, and the Dirac delta (NIST Digital Library of Mathematical Functions).

In quantum physics

Δ is uncertainty. Heisenberg’s uncertainty principle is written Δx Δp ≥ ħ/2: the uncertainty in a particle’s position times the uncertainty in its momentum is at least half the reduced Planck constant (OpenStax, University Physics).

Other uses of delta

A river delta is low, flat land, sometimes shaped like a triangle, where a river divides into smaller rivers before reaching the sea (Cambridge Dictionary); the American Heritage Dictionary gives “an object shaped like a triangle” as another meaning of the word. In finance, an option’s delta measures how its price changes compared with the price of the shares it is based on (Cambridge).