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I was recently thinking about some of my past math classes, and depending on the context I recall my professors would sometimes use the "$\equiv$" symbol in places where I'd feel "$=$" to be more appropriate. For example, since this would often be the case in my classes on differential equations and Fourier series, we would have (for $n \in \Bbb N, k \in \Bbb Z$)

$$(-1)^{2n+1} \equiv -1$$ $$\sin(k\pi) \equiv 0$$

Is there a particular reason in this context why we would say "$\equiv$" instead of "$=$"? The latter feels more natural in this context, which makes me think that there's some reason my professors would use the former.

I'm familiar with the notion of the "$\equiv$" symbol in the context of, say, elementary number theory (specifically modular arithmetic) where we might say

$$10 \equiv 1 \pmod 3$$

which isn't saying "$10$ equals $1$", just that "$10$ is like $1$ in this context." But that doesn't seem to fit the case as with the first two statements - because I don't believe it is that $(-1)^{2n+1}$ is like $-1$, or that $\sin(k \pi)$ is like $0$, they are $-1$ and $0$ respectively.

Am I just mistaken on this latter fact? Is there something I'm missing? What, precisely, is the difference between the two notations?

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    $\begingroup$ I think people usually use $\equiv$ for equality of functions. For your examples, the functions $f: \mathbb{Z} \to \mathbb{R}$ and $g: \mathbb{Z} \to \mathbb{R}$ given by $f(n) = (-1)^{2n+1}$ and $g(k) = \sin(k \pi)$ are identically the functions $-1$ and $0$, so we write $f \equiv -1$ and $g \equiv 0$. $\endgroup$ – mathworker21 Jan 10 at 3:39
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    $\begingroup$ when speaking about functions, we would interpret $f(x) = 0$ as the statement that there exists an $x$ where $f(x) = 0$, and interpret $f(x) \equiv 0 $ as the statement that for all $x$, $f(x) = 0$. The first statement read aloud says $f$ is equal to 0 at $x$, the second is $f$ is identically 0, that is, it is equal to the 0 function. $\endgroup$ – staedtlerr Jan 10 at 3:40
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    $\begingroup$ @staedtlerr nobody interprets "$f(x) = 0$" as "there exists an $x$ where $f(x) = 0$". people would assume the speaker is referring to some specific $x$. $\endgroup$ – mathworker21 Jan 10 at 3:41
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    $\begingroup$ It is usually used as denoting “identically equal to”. For example, $\sin(k \pi)=0$ could be read as an equation in $k$, but $\sin(k \pi) \equiv 0$ means that this is always 0 (for integer k). Just like $f(x) \equiv 0$ would mean the function is identically 0 $\endgroup$ – Fede Poncio Jan 10 at 3:41
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    $\begingroup$ To piggy-back on mathworker21's comment, $(-1)^{2n+1}$ is a number, depending on a variable $n$, whereas $\mathbb{N} \to \mathbb{Z} : n \mapsto (-1)^{2n+1}$ is a function. To say $(-1)^{2n+1} \equiv -1$ is shorthand for saying the above function is equal to the constant function $1$, or in other words,$$n \mapsto (-1)^{2n+1} = n \mapsto -1.$$ $\endgroup$ – Theo Bendit Jan 10 at 3:42
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I'll give an example of each.

$$2x=x+1$$

This holds when $x=1$ only, and so the equality symbol is appropriate. In short, we use an $=$ when specific values solve the expression.

On the contrary, we have:

$$2x\equiv x +x$$ Whatever the value of $x$, this holds. This is an algebraically obvious one, but another might be $$\sin^2 x + \cos^2 x \equiv 1$$

The identity symbol $\equiv$ is used when an equality holds for all values in the domain specified (e.g. $\Bbb R$).

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    $\begingroup$ +1. I like to think of "$\equiv$" as "$=$" with an emphatic underscore. :) $\endgroup$ – Blue Jan 10 at 4:11
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    $\begingroup$ Oooh, nice analogy, thanks for that Blue. $\endgroup$ – Eevee Trainer Jan 10 at 4:17
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The equal sign "$=$" is used for equalities ,equations, identities and definitions of functions.

Examples of equalities are $3=2+1$ or $12^2 = 144$ where two numbers are the same.

Examples of equations are $ 3x+1=10$ or $x^2-4=0$ where for some values of the variable both sides result in the same value.

Examples of identities are $\sin ^2 x + \cos ^2 x =1$ or $(x+y)^2 = x^2+y^2+2xy$ where both sides are identical for every value of variables.

For identities sometimes $\equiv $ is used instead of $=$ for example we may use $ e^{i\theta} \equiv \cos \theta + i \sin \theta$ or $\sin ^2 x + \cos ^2 x \equiv 1$ or $(x+y)^2 \equiv x^2+y^2+2xy$ to emphasize that this is an identity not an equation.

For functions we sometimes use $ f(x) \equiv c$ to emphasize that the given value $c$ is for all values of $x$

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