# When do you use this symbol ≡

1. Mar 23, 2012

### autodidude

How do you know when to use ≡ or =? I've seen the the formula for density written with that = and ≡.

Also, whilst we're on the topic of symbols, what does an inequality sign with two lines beneath it mean? Is it the same as with one line (greater/less than or equal to)

Thanks

2. Mar 23, 2012

### cepheid

Staff Emeritus
I think you use the special three-line equality symbol:

1. If it's a definition
2. If it's an identity

3. Mar 23, 2012

### Dr. Philgood

When you see a three bar equal sign it means by definition it is equal to something. For instance you could say x=3 and that would be for a particular circumstance.

But when you say for instance, Ampere's law, ∫B dot ds, that is by definition(three bar equal sign) equal to current enclosed multiplied by η naught.

Likewise greater than or less than signs with two bars underneath mean it is greater or less than by definition.

I apologize I still haven't quite figured out how to use symbols properly.

4. Mar 23, 2012

### scurty

Here's the code for you if you are wondering:

Code (Text):

$\equiv \\ \geqq \\ \leqq$

$\equiv \\ \geqq \\ \leqq$

5. Mar 23, 2012

### nonequilibrium

The three line symbol is also (sometimes) used in modular arithmetic.

6. Mar 23, 2012

### cocopops12

that symbol is read "is defined as"
it's simply used to save space in text. instead of keep writing "is defined as" in words...

get what i mean? lol

7. Mar 23, 2012

### Robert1986

Additionally, sometimes you might see something like $:=$ or $=:$ where the thing closest to the colon is defined as the thing on the other side. For example, take the Euclidean Norm in 2 space, where $x=(x_1,x_2)$:

$$|x| := \sqrt{x_1^2 + x_2^2}$$

or

$$\sqrt{x_1^2 + x_2^2} =: |x|$$

Last edited by a moderator: Apr 9, 2012
8. Mar 24, 2012

### Staff: Mentor

≡ is used for congruency, e.g., ΔABC ≡ ΔDEF

To me, it seems little different from = in this usage. I'm not sure why = isn't sufficient.

9. Mar 24, 2012

### nonequilibrium

Because it isn't the same triangle, it can be rotated for example.

10. Mar 24, 2012

### Alpha Floor

I have also seen $\equiv$ used a lot as "equivalent to".

For example, if you have two systems of equations S1 and S2 that have the same set of solutions, you can write $S1 \equiv S2$, but not $S1=S2$ because strictly speaking they are different systems.

Also I've seen it used for combining written expressions with mathematical expressions, for example:

Integral of f over an interval [a,b] $\equiv \int^{b}_{a} f(x)dx$

For "is defined as" I've always used $:=$ or $=:$ as someone mentioned above.

Another notation I've seen and doesn't seem to be a global convention is puting an exclamation mark "!" right above an equal sign meaning "imposed to be equal to". Like when you have a physical problem, get two separate expressions, and because of some given conditions they must be equal to each other.

11. Mar 24, 2012

### Staff: Mentor

Perfect.

The neatest "is defined as" symbol I've seen (IMHO) is in elec eng journals, and comprises an equals sign with a triangle atop it. Picture = capped with a small ∆ of base slightly shorter in length than the equals sign.

12. Mar 24, 2012

### I like Serena

I like $\overset{def}{=}$ for definitions, since it eliminates the ambiguity (and obscurity) of what the symbol means.
There's even a unicode symbol () for it.

13. Mar 24, 2012

### I like Serena

In this case it means "is defined as".

14. Mar 24, 2012

### Staff: Mentor

I share your sentiment about eschewing ambiguity, but I don't like the "def" idea because it's (prone to be) small indecipherable print. (Absolute certainty by the stage it reaches a second generation photocopy.)

Worse, it doesn't lend itself to being handwritten (legibly and without deliberate exaggeration). https://www.physicsforums.com/images/icons/icon9.gif [Broken]

Last edited by a moderator: May 5, 2017
15. Mar 24, 2012

$\equiv$ is used when you want to mention that two things are equivalent/congruent. I think the notation is due to Gauss in Disquisitiones Arithmeticae and maybe Legendre before him. In the first page of Disquisitiones Arithmeticae Gauss says that he has adopted this symbol from Legendre but Legendre used this symbol to show both equality and congruence while Gauss uses this symbol only for congruence to distinguish it from equality.
Anyway, I don't think that this symbol is meant to be used for 'is defined as' naturally. If two propositions are logically equivalent, then they are congruent, but not equal because they are not the same propositions. For example you can define something in mathematics in one way and then prove other logically equivalent propositions in theorems after it. Since logically equivalent propositions are congruent you can use this symbol to define something as well I guess.

16. Mar 24, 2012

### arildno

That is also my experience with the triple-rod, as introducing&defining a handy, short notation for some complex object that can be defined in a more langorous fashion.

17. Mar 25, 2012

### autodidude

So if I understand correctly...in physics, for density, you can write p ≡ m/v because the definition of density is just that but you can't write F≡ma because force isn't defined as mass times acceleration?

I'm not familiar with the usage of the symbol in geometry as I never took the subject at that level but it seems like that's where most people first encounter it

18. Mar 25, 2012

### Staff: Mentor

Well, I suppose you could, but I can't recall seeing that. I'd be happier with it being made more explicit, e.g.,
density ρ ≡ mass ÷ volume​

(Besides, I think v in a physics formula is more commonly velocity.)

That makes sense.

19. Mar 29, 2012

### autodidude

^ Thanks

Yeah, I'm sure I saw it in Serway's text but have seen it with an equal sign elsewhere

20. Apr 7, 2012

### Staff: Mentor

I just noticed that latex has it as \ triangleq , e.g., $A \triangleq length \times width$

There is also a unicode symbol ≜

21. Apr 9, 2012

### jing2178

Way back in time when I was doing my 'A' levels (last centuary) I was taught it was an equivalence sign as in

(x+2)(x+1) ≡ x2 + 3x +2, ie true for all values of x

wheres as = was for an equation, true for some x but not all, as in x2 + 3x +2 = 12

Last edited: Apr 9, 2012
22. Apr 9, 2012

### Staff: Mentor

x2 + 3x +2 = 12 is an example of a conditional equation, as opposed to an equation that is identically true (your first example).