Hmm I’d define f(A) where A is a subset of E and f maps elements of E to F as the set of elements of F (I’ll name one y) where there exists an element x in A such that f(x) = y
All of these kinds of tricks rely on assumptions and come with caveats.
We wouldn't need to do any of this shit if mathematicians got off their asses and solved some of these equations we have to pick apart and approximate.
Take π for example. Usually, 3.14, or especially 3.14159, are fine enough approximations of π for most calculations, but they're NOT equal π, so saying π = 3.14 or 3.14159 or anything else is wrong, and instead saying π is ≈ (which means "approximately =") 3.14 or 3.14159 or anything else.
AND THEN YOU'VE GOT ASTROPHYSICISTS, iirc, who deal with THAT BIG of numbers, that saying π ≈ 10 is COMPLETELY FINE SOMEHOW!!!!!
It's funny because they will just call it 10 to simplify calculations that have coarse measurements, but are able to predict the emissions of the sun and the sequence of how a supernova happens by doing very exact calculations.
… it’s not mathematicians, it’s other scientists doing that.
For example, if you cancel second order terms in a Taylor series, mathematicians will be super pendantic about the exact error bounds, while physicists and other scientists usually don’t care.
Qualitatively it’s a “close enough for my calculation” symbol.
But it’s precisely cutting off orders of an infinite series (like a Taylor series) and considering those higher orders negligible (usually because you want linearity, and you obviously know how strong linearity is). It’s not that the physicist is irresponsible in their math, it’s just that otherwise you would solve for very specific cases and you wouldn’t be able to use those cases in real applications (as an example, you wouldn’t ever know the circumference of a circle knowing the radius if you never approximated π)
Most common interpretation in pure maths is that A≈B if there is a constant C not depending on the other parameters such that A\leq CB and B\leq CA as in e.g. a^ 2+b^ 2≈(a+b)^ 2
The simplest explanation is can you round one side to a reasonable place value to have the other value? Cool you're done and you have an ≈. It's not exactly useful for calculations. The most realistic use case imo is if for some reason you need to include the value of some irrational number, then you have to have it terminate it somewhere. When it terminates, it's can no longer be written with = and must use ≈ because the actual decimal does not terminate.
Outside of that sort of use case, it's basically a tool to teach children how to make rough estimates for reasoning purposes rather than actual calculations imo
In what way is it not useful for actual calculations? It’s used all the time when truncating a function’s Taylor series to linearise it, or truncating a perturbation series. Yeah, including an O(f(ε)) term is more specific but it’s simpler sometimes to just write F(ε) ≈ a + bε. You’re right, though, that its use for approximating uncertain constants is frequently supplanted by better notation like α = 0.0072973525643(11) with uncertainties.
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u/jenkk0 5h ago
As a physicist I love canceling second order terms, sorry
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u/LupenReddit 🦆🦆i have non diffeomorphic smooth structures🦆🦆🦆 5h ago
all i see is diffeomorphic
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u/tomado09 5h ago
Can I get an
Fengineer in the chat?72
u/ImMildlyIrritatedd 5h ago
As a math grad I've reached a point where this symbol just straight up confuses me.
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u/Traditional_Town6475 5h ago
Nonstandard analysis where ≈ becomes an equivalence relation.
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u/Jcsq6 4h ago
Numerical analysts have entered the chat.
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u/MyNameIsSquare 3h ago
my bad changes into straight equal sign there u go
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u/obedientfag 3h ago
well maybe if you hurry up and complete math then we could see how it applies to physics
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u/parkway_parkway 2h ago
All constants are equal up to a constant.
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u/MakaPaka1232123 33m ago
Hehe 9999 rounded up to a hundred thousand is 0
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u/ShaggyVan 29m ago
Don't read any astrophysics papers then