Is $2+2=4$ an identity? I know this seems like a silly question, but someone was trying to debate with me about how $2+2=4$ should be called an identity and not an equation. I mentioned how it has no variables and isn't true for all numbers, but they claimed you could plug in any value for its variables(which don't exist) and it would be true and so is an identity. Which is kind of true in a round about way, but I still don't think anyone would call it an identity in a casual situation.

I consider an equation to be a statement that two things (the two sides of the equation, in particular) are equal. In this sense, $2 + 2 = 4$. But perhaps I also know that the set of $b$ in $B$ with some property satisfy the equation $b^4 + b^3 + b = 0$.
I consider an identity to be a relation that's tautologically true. Thus $2 + 2 = 4$ or $\sin^2 + \cos^2 = 1$ are identities. But $b^4 + b^3 + b = 0$ is different from this, and so isn't.
So yes, I consider it an equation and an identity. And as Alex said, it's not particularly interesting as either. 
