I recall an anecdote about a mathematician being asked to clarify precisely what he meant by “a close approximation to three”. After thinking for a moment, he replied “any real number other than three”.
Yes, informally in the sense that the error between the two numbers is “arbitrarily small”. Sometimes in introductory real analysis courses you see an exercise like: “prove if x, y are real numbers such that x=y, then for any real epsilon > 0 we have |x - y| < epsilon.” Which is a more rigorous way to say roughly the same thing. Going back to informality, if you give any required degree of accuracy (epsilon), then the error between x and y (which are the same number), is less than that required degree of accuracy
It depends on the convention that you use, but in my experience yes; for any equivalence relation, and any metric of “approximate” within the context of that relation, A=B implies A≈B.
pruwybn@discuss.tchncs.de 2 years ago
Is it true to say that two numbers that are equal are also approximately equal?
SpeakerToLampposts@lemmy.world 2 years ago
I recall an anecdote about a mathematician being asked to clarify precisely what he meant by “a close approximation to three”. After thinking for a moment, he replied “any real number other than three”.
mpa92643@lemmy.world 2 years ago
“Approximately equal” is just a superset of “equal” that also includes values “acceptably close” (using whatever definition you set for acceptable).
Unless you say something like:
a ≈ b ∧ a ≠ b
which implies a is close to b but not exactly equal to b, it’s safe to presume that a ≈ b includes the possibility that a = b.
EatATaco@lemm.ee 2 years ago
Can I get a citation on this? Because it doesn’t pass the sniff test for me. en.wikipedia.org/wiki/Approximation
mpa92643@lemmy.world 2 years ago
Sure! See ISO-80000-2
Here’s a link: cdn.standards.iteh.ai/…/ISO-80000-2-2019.pdf
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match@pawb.social 2 years ago
np.isClose(3, 3) is TRUE
myslsl@lemmy.world 2 years ago
Yes, informally in the sense that the error between the two numbers is “arbitrarily small”. Sometimes in introductory real analysis courses you see an exercise like: “prove if x, y are real numbers such that x=y, then for any real epsilon > 0 we have |x - y| < epsilon.” Which is a more rigorous way to say roughly the same thing. Going back to informality, if you give any required degree of accuracy (epsilon), then the error between x and y (which are the same number), is less than that required degree of accuracy
Leate_Wonceslace@lemmy.dbzer0.com 2 years ago
It depends on the convention that you use, but in my experience yes; for any equivalence relation, and any metric of “approximate” within the context of that relation, A=B implies A≈B.