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?
Best Answer
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. $\Bbb R$).