There are few more examples that come to mind, like statements about intervals (π = 3.14 ± 0.01) and the usual notation for modular arithmetic; 3 * 3 = 1 (mod 4).
Oh, and the wonderful notation for integrals, ∫ 2x dx = x² + C
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The integral notation is a set-builder in disguise:
∫ f(x) dx = set of solutions of diff. eq. g'(x) = f(x)
Of course, writing
∫ 2x dx = { x² + C | C ∊ R }
grows old pretty fast, so we drop a couple of characters here and there, but it's pretty consistent with set-builder.
Of course, this only goes to show the article's point that what "=" means here requires a lot of context.
Well the (mod n) notation is just an implicit homomorphism, phi(3*3) = phi(1) where phi : I -> Z4. But during lectures it was pretty common for professors to just drop all the greek and squiggly lines and commonly write things like 9 = 1, and use '=' with whatever arcane RST relation which happened to be relevant at the time.
Isn't the integral notation just saying that two sets (or equivalence classes or similar) are equal?
The usual notation for modular arithmetic uses three dashes, not two, to denote congruence.
At least in my university, in one of the first Algebra courses about integers, factorization and congruence the usual notation is
3 * 3 ≡ 1 (mod 4)
3 * 3 = 1
I don't think "=" is being overloaded in this example. The symbols "1", "3" and "*" are, since they're working in Z_4 rather than Z, but equality is just equality.
Sure; I think we’re in violent agreement here, it’s absolutely the case that people write the simpler version when the meaning is clear from context. I’ve definitely done that a bunch.
Piling on with a bit more pedantry, my experience is a bit different.
In my current ring theory course, we have indeed written things like 3 * 3 = 1 when working in |F_5 (not sure that notation is going to work as well as I hope, looks alright in the app I use), but it's not the equality symbol is overloaded, but the numbers themselves. Rather than using = to mean numeric equality and equality w.r.t. equivalence classes, we just use the numbers themselves as shorthand for their equivalence classes.
That seems odd to me. I don't think I've read any ring/algebra/module theory text that doesn't explicitly denote equivalence classes with, for example, square brackets.
There's a canonical ring homomorphism from the integers into any commutative ring with 1. When such a homomorphism is unique, you often omit it, hence mathematicians sometimes just write numbers without equivalence class brackets.
This is not limited to rings of the form Z/nZ.
In my experience, mathematicians tend to drop the third dash.
I should also add that, as generally presented in higher level math, modular arithmetic actually does use literal equality. The abuse is in the numbers.
That is to say, when we write "9" in modular arithmetic, we mean the set of all numbers congruent to 9. In this case in "9=1 mod 4" the equality is literal because the set of numbers congruent to 9 is literally the same set as the set of numbers congruent to 1.
I probably should have said "common" notation; it's certainly what I learned. I think you're right in that many authors prefer ≡ to emphasize that it is an equivalence rather than equality relation.
Source that both are in use: https://math.stackexchange.com/questions/196081/the-right-wa...
I don't understand the schism. For example, 11=1 gives the equivalence of two different programs. Whereas 33 mod 4 and 1 are equal in Z_4 ...
It bothered me a lot when I took logics lectures that equality wasn't treated as operator or even just anything within a theory (whereas turnstyle was described, at least).
Yeah, sorry, I didn’t mean to split hairs. They’re definitely both in use. I do plenty of work in Zp and I feel bad every time I don’t write \equiv or the \pmod for that matter :)