What is ≡ operator used for in math? [duplicate]

I heard about this in our calculus class years ago. I was actually not in that class when the professor explained this. 95% of engineering student do not know about this operator. I'm trying to recall what it meant. Is it used standard in math classes? I think it means approximately equal to.

I am not 100% sure about the syntax.

Edit: Originally, I asked for === operator.

• "Approximately equal to" is $\approx$. ===, I've seen in some programming languages... Sep 13, 2011 at 14:13
• Maybe it means "is defined to be equal" like "≡", except they didn't have "≡" on a typewriter.
– xpda
Sep 13, 2011 at 14:23
• "I was actually not in that class when the processor explained this." That's a curious typo. :-) Sep 13, 2011 at 14:39
• It is often used in the form $f(x) \equiv g(x)$ to say $f(x) = g(x)$ for all $x$, as opposed to $f(x) = g(x)$ for some specific $x$.
– t.b.
Sep 13, 2011 at 14:40
• When used in the sense of @Theo's comment, I think it is common to read it like: "$f(x)$ is identically equal to $g(x)$". Sep 13, 2011 at 14:42

Since your professor was referring to engineering students, then it's likely they were referring to the identity symbol, which is used in an expression to mean the left and right hand sides are true for all values. So $\cos^2\theta +\sin^2\theta \equiv 1$ since it's true for all $\theta$ whereas $\cos\theta = 1$ since it's true only for some.

• All algebraic and trigonometric identities can be written using the $\equiv$ symbol, e.g. $(a+b)^2\equiv a^2+2ab+b^2$. Sep 13, 2011 at 15:40
• A special case of that, function identically equal to a value (zero here) can be written as $f(x)\equiv 0$. Sep 13, 2011 at 16:24

There's the obvious meaning: Congruence modulo an integer, i.e. "$a \equiv b \pmod c$" (read: "$a$ is congruent to $b$ modulo $c$") which means $c \mid(b-a)$ ("$c$ divides $b-a$").

As someone else has mentioned it's also occasionally used to indicate equality of functions by writing $f(x) \equiv g(x)$, but why not just write $f=g$ then?!

• I've seen $\equiv$ used for definitions... Sep 13, 2011 at 14:50
• I think the "identically equal" usage is to really draw the reader's attention to whether you mean a function has a root ($f(x) = 0$) or whether it is identically zero ($f \equiv 0$). Sep 13, 2011 at 14:59
• As for why you'd want a separate symbol for modular congruence, instead of just using "$=$", it's convenient when writing something like "$-1234 \equiv 9999 - 1234 = 8765 \pmod{9999}$" or "$(a+b)(a-b) = a^2-b^2 \equiv a^2 \pmod{b^2}$". Here, $\equiv$ marks the places where we add or subtract multiples of the modulus, while ordinary $=$ signs denote equivalences that hold also in ordinary arithmetic. Sep 13, 2011 at 15:01
• @Theo: I know you're being facetious, but it's a good question nonetheless. First of all there's the principle that we should reserve "$=$" for actual equality (mostly, anyway...). Then there's the idea that "$a \equiv b \pmod c$" is a single "symbol" that makes a statement about the three integers, $a$, $b$ and $c$. And finally: It's what Gauss used in Disquisitiones Sep 13, 2011 at 15:04
• This is correct but from what I remember it was not used in this context. Could it mean something else , something simpler? Or may be our professor could be wrong :( Sep 13, 2011 at 15:10

It's used for various things in various contexts. The one about "defined to be equal" is often rendered as ":=". I haven't seen "$\equiv$" used for that. It's certainly used for congruence with respect to a modulus; e.g. $44\equiv62 \pmod 6$, etc. It's used for identities like $(x+1)^2 = x^2+2x+1$ when one wants to say that that is true for all values of $x$.

However, the variety of different uses that this symbol temporarily has in more advanced work has probably never been tabulated.

The "≡" operator often used to mean "is defined to be equal."