The symbol $\triangleq$ is sometimes used in mathematics (and physics) for a definition. It is instantiated for instance in the Unicode Character 'DELTA EQUAL TO' (U+225C). The notation $t \triangleq m$ (generally) means: "$t$ is defined to be $m$" or "$t$ is equal by definition to $m$" (often under certain conditions).

In a similar sense, some use $:=$ or $=:$ (see for instance Symbols based on equality). Yet, this Delta variant is more important to me.

The SE. Maths post What is meant by the delta equivalent sign? proposes a slight distinction (not crystal-clear to me) between the above similar senses:

Sometimes it is used with the slightly different meaning of "equal by definition", to underline the difference w.r.t. "$:=$ " which is the definition itself. i.e.

$$ a:=3;\\ 5+a \triangleq 5 + 3 = 8 $$

I always took for granted that the $\Delta$ stood for letter "D", i.e. for the initial of "definition". Indeed, one sometimes finds $\overset{\mathrm{def}}{=}$ too. In German apparently, one also uses $≙$ (Entspricht-Zeichen, with Unicode U+2259).

Based on these prior hints, my questions are as follows:

  • Who introduced this dual symbol first in science, and where (which source)?
  • What motivated the Greek $\Delta$ notation? The abbreviation of some word, a symbol (why not a latin notation)?
  • Why not merge the lower bar of the Delta with the upper bar of the equal sign, to save some ink, and create a lighter symbol?

References: the symbol itself was already discussed in StackExchange:

  • 8
    $\begingroup$ This notation seems very strange, as it suggests symmetry between the definiens and the definiendum. I am more used to := or =:, with the colon on the side of the definiendum. $\endgroup$ Commented Sep 5, 2016 at 1:45
  • 3
    $\begingroup$ Mathsym jeff560.tripod.com/mathsym.html has only $=_{\mathrm{Def}}$ (from 1894) but not the Delta version. $\endgroup$ Commented Sep 5, 2016 at 13:44
  • 1
    $\begingroup$ @MargaretFriedland The asymmetry is given by what is before the symbol and what is after the symbol. Thus $A \triangleq B$ always means "A is defined to be B". $\endgroup$
    – juanrga
    Commented Mar 19, 2018 at 19:42
  • 3
    $\begingroup$ @MargaretFriedland, I am sympathetic to your objection, and, in fact, would go so far as to object somewhat to the $:=$ usage, since it lends itself to typos so easily. Rather, I'd strongly prefer that the context clarify any asymmetry (such as a definition or notational abbreviation) in a mathematical equality. (Mathematicians seem unlikely to take these asymmetries as seriously as programmers may, so I don't necessarily trust them!) $\endgroup$ Commented Jun 28, 2018 at 23:43
  • 2
    $\begingroup$ The symmetric way understand it is to read it as "this equality holds by definition". From that perspective there is no "definiens" or "definiendum". $\endgroup$ Commented Jan 7, 2021 at 16:20

1 Answer 1


This is really a comment rather than a full answer.

Florian Cajori's encyclopedic tome A History of Mathematical Notations does not seem to discuss this symbol specifically; the closest thing is the following remark in Section 269:

L. Gustave du Pasquier (Comptes Rendus du Congrès International des Mathématicians (Strasbourg, 22–30 Septembre 1920), p. 164) in discussing general complex numbers employs the sign of double equality $\overset{\displaystyle =}{=}$ to signify “equal by definition.”

I checked MathSciNet, and the review of the 1953 paper "Zwei Klassen von Flächen, deren Bestimmung von einem Integral der Telegraphengleichung abhängt" by Hans Jonas (MR0058259) uses the notation $=_{\text{def}}$ (but note that Jonas's paper itself does not use that notation). This seems to be the earliest occurrence of $=_{\text{def}}$ or $\overset{\text{def}}=$ in MathSciNet. But I do not have access to a print version to verify that the notation was actually used back in the 1953 edition of Mathematical Reviews.

  • $\begingroup$ I have been waiting for a long time, let me validate yours, until... $\endgroup$ Commented Aug 29, 2021 at 16:47
  • 1
    $\begingroup$ After I posted this, I noticed Gerald Edgar's comment which points to an 1894 use of $=_{\rm Def}$ to mean equal by definition. $\endgroup$ Commented Aug 29, 2021 at 20:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.