I've always been taught to write, for example, $$\sin^2(\theta)+\cos^2\theta \color{red}{\equiv} 1,$$ rather than $$\sin^2(\theta)+\cos^2\theta \color{red}{=} 1,$$ and $$x(x+2)\color{red}{\equiv} x^2+2x$$ rather than $$x(x+2)\color{red}{=}x^2+2x, $$as $\equiv$ denotes an identity, so the equation is true for all values of the variable(s), whereas $=$ is an equation that only holds for some specific values of the variable.

I've happily taken this on board, but I've seen so few people write it.

In this context only (i.e. with regards to identities), are $=$ and $\equiv$ interchangeable, or have I been lied to/deceived?

Are both correct in this situation?


  • $\begingroup$ Because, in most of the cases it doesn't cause any confusion. $\endgroup$ – Kaster Jun 1 '14 at 0:27
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    $\begingroup$ I was never taught $\equiv$ for identities. I've always considered identities to be bona fide equalities of functions or elements of polynomial / power series rings etc. or at least logical $\forall$ statements with the symbols suppressed for efficiency. $\endgroup$ – blue Jun 1 '14 at 0:27
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    $\begingroup$ The person who taught you is more careful than many of us are. One usually lets context determine the meaning of $=$. $\endgroup$ – André Nicolas Jun 1 '14 at 0:28
  • $\begingroup$ I've only ever seen $\equiv$ used when defining a new quantity. $\endgroup$ – hasnohat Jun 1 '14 at 0:30
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    $\begingroup$ The problem is, for example, if I'm doing a mechanics problem and there are lots of trig. functions, and I'm trying to find the values of $\theta$, say, for which $f(\theta)=0$ (where $f$ is some complicated function of $\theta$), only to find out that it should be an identity, and it's, in fact, true for all $\theta$, so I spend 20-odd minutes deducing that $0=0.$ Is there a way I could avoid this and immediately identify whether the context means $\equiv$ rather than $=$? $\endgroup$ – beep-boop Jun 1 '14 at 0:37

The $\equiv$ symbol indeed is used when the values are identical, independent of the variable. For example, if $f(x) = 0 \quad \forall x$, then we write $f \equiv 0$. Write $x(x+2) \equiv x²+2x$ is correct also, because it is true for all x. However, people really don't give it so much importance, it serves just as a reminder.


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