# Strange equal-sign

1. Dec 6, 2004

### danne89

Hi! Can anyone explain to me what this sign means. $\equiv$

2. Dec 6, 2004

### shmoe

It's often used in modular arithmetic, $x\equiv y (\text{mod }d)$ if and only if $d \text{ divides } x-y$, but this is by no means universal. Where did you see it used?

3. Dec 6, 2004

### arildno

It basically means "equal, by definition/identically equal"
If, for example, you want to introduce a new notation for something, many prefer to do it with this sign.
For example, a standard notation for the "normal derivative" of a function f is often introduced as follows:
$$\frac{\partial{f}}{\partial{n}}\equiv\vec{n}\cdot\nabla{f}$$
where the symbols on the right-hand side are assumed known.

4. Dec 6, 2004

### danne89

Okej. Great!!

5. Dec 6, 2004

### arildno

Note:
I was unaware of the formal use of this sign in modular arithmetic.
Without having seen your example, I suspect it means "equal, by definition" there..

6. Dec 6, 2004

### Galileo

In mathematics, it's used to denote a congruency. $x \equiv a \pmod{n}$

In physics, it's often used for defining new things (notations, substitutions and such).
$k \equiv \frac{\sqrt{2mE}}{\hbar}$.

In mathematics, they use " := "for such definitions.

7. Dec 6, 2004

### shmoe

I'm thinking of using it in place of the usual equality sign "=", not as a definition here. This notation is more commonplace in introductory texts in my experience. I think it's to draw attention to the fact that it's a congruence and not a usual integer equality, in case the student misses the flashing "mod d" that follows. It's tough to get the neon lights around that part in a text!

8. Dec 6, 2004

### arildno

Oh, I meant the example posted by OP, not yours, which he subsequently removed
(He'd encountered it when reading of the Uncertainty Principle)

9. Dec 6, 2004

### danne89

Yes it was in it's meaning as defination i encounter it.

10. Dec 6, 2004

### jcsd

Yep it's most common use is for 'equivalent to' which is a stronger statement than 'equal to'.

For example in x^2 - x - 2 = 0, the equals sign means 'equal for some x', but in x(x + 5) $\equiv$ x^2 + 5x the equivalent sign means 'equal for all x'.

11. Dec 6, 2004

### cogito²

tokiga svenskar

12. Dec 9, 2004

### cepheid

Staff Emeritus
I have a question about this symbol...it would seem to me that it would be appropriate in so many situations in which = is (carelessly?) used instead. For instance, physics notes are always full of "equality" statements in which the expression on the RHS is just the definition of the quantity on the LHS. So why not use "equivalent to/ identical to/ equal to always, by definition"?

$$K \equiv \frac{1}{2} mv^2$$

$$\vec{a} \equiv \frac{d\vec{v}}{dt}$$

$$S \equiv \int_{t_1}^{t_2}{L(Q, \dot{Q}, t) dt}$$

etc...

nobody ever does that. Is there an error in my reasoning? Because the symbol's use is so haphazard that it might crop up once or twice in a document in random places, leaving me wondering why the author chose those two instances to use it, and why he/she bothered using it at all.

13. Dec 9, 2004

### jcsd

14. Dec 9, 2004

### James R

In physics, I think it tends to be used mostly in definitions. Thus, a statement like:

$$\vec{a} \equiv \frac{d\vec{v}}{dt}$$

defines the symbol $\vec{a}$. Whereas, a statement like

$$K=\frac{1}{2} mv^2$$

is not a definition, since it is derived from more fundamental principles.

15. Dec 9, 2004

### NeutronStar

I've seen it used by authors in basically three ways:

"is formally defined as" - to express a formal definition.

"let this symbol or expression be defined as" - to create an informal definition within the context of a discussion.

"this expression is, by definitions, equivalent to" - showing that two expressions are equivalent by previous or formal definitions.

Generically it means, "equivalent by definition".

However, whenever things are equivalent by definition we can always use the regular equals sign to equate them as well because it's certainly also true. So author's usually only use the special "defined as" symbol to stress a definition. Many of them don't bother with this special symbol and simply use the regular equals symbol while just mentioning somewhere in the text that the equivalency is a definition.

16. Dec 9, 2004

### dextercioby

The "$\equiv$" should put in 2 places only:
1.A definiton.E.g.$\vec{v}\equiv\frac{d\vec{r}}{dt}$.
2.An identity. E.g.$(a+b)^{2}\equiv a^{2}+b^{2}+ab+ba$

The rest is just interpretation.Erroneous,sometimes.

Daniel.