# What is meant by the delta equivalent sign?

What is the meaning of the delta equivalent ($\overset{\Delta}{=}$) sign?

I met this in a communication theory text. It said,

signaling rate: $r\overset{\Delta}{=} 1/D$ symbols/s or also called ‘baud’.

• Can you show us where you encountered it?
– 5xum
May 19 '15 at 10:34
• I met it in communication theory text, says that "signaling rate: r≜ 1/D symbols/s or also called ‘baud’" May 19 '15 at 10:37

It is a definition. 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.

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

• Thanks for the explanation! Re: "equal by definition" -- I've seen the symbol used in that way, but isn't that a bit weird given that anything that's true in mathematics is true due to, eventually, a definition? Jun 21 at 10:44
• IMHO, this argument is not very strong as, in the end, everything just follows from the axioms :) The use of $\triangleq$ is meant to improve readability, and can be especially useful if the author does not use $:=$ for definitions. E.g. consider the following: "Let us define X=f(Y). ... Theorem: $X=f(Y)=g(Z)$." It may be useful for the reader if the author reminds him/her that $X=f(Y)$ carries no information (it is the definition of $X$), while the content of the theorem is entirely in the second equality. Jun 21 at 16:15
• Sometimes things get even worse, and the author expects the reader to be so familiar with the jargon of his/her subfield to not bother about specifying what is a definition and what is not, making the reading needlessly harder (personal experience). Now that I think about it, this was cleverly narrated by Serre in a youtube video :) youtu.be/ECQyFzzBHlo?t=1450 Jun 21 at 16:33

It is also used in physics, to indicate that forces are drawed on some scale, so for example $$1 N \overset{\Delta}{=} 0,1 \; \mathrm{m}$$

It is clear that a force isn't equal to some length, but they are corresponding with the same length. It is the same principle as Saphrosit says, i.e. they are by definition, and in this specific case, equal to each other.