Does .999... equal 1?

By stevea68 | SteveA | 15 Dec 2020

I'll admit ahead of time that this is really just a fun rhetorical question to me with the answer being basically that - it's a cool idea with all sorts of interesting "mathemagical" tricks that can be used but truly it sort of depends upon what you desire the answer to be.

If you really want a precise answer, then you need the question to be AT LEAST as precise - if the question or its context can vary, then the answer can similarly vary, and the reason why I said "AT LEAST" is because there can be multiple ways of coming up with assumed solutions.

To delve in a bit deeper, is there a fixed intent/desire behind it that remains constant throughout all?  If that is so, then we might actually be able to come up with a similarly fixed/constant answer.

Overall, though the two values can be made potentially as closely equal as one may desire, at a minimum they're written differently but it can be fun exploring different ideas with mathematics and infinities ... so maybe there's a bit of fun story telling possible?  It's not a completely useless subject in life though, as the ideas can be extrapolated upon and applied for real utility (oh, did I mention it can be fun too? ;) )

As an interesting sidenote, here's a funky relationship I bumped into along the way:

0.9^9 ~= 0.38742 (That's taking 0.9 and multiplying it by itself 9 times)

0.99^99 ~= 0.36973

0.999^999 ~= 0.36806

0.999... ^ ...999 = 1/e (where e is Napier's/Euler's Constant ~=2.718281828459... It comes up a lot in relationship to exponential functions)

Just tossing that out there in case anyone else enjoys this stuff (forewarning, I've already gone through about 7 of the most significant "proofs" of this "identity" and found the weakness in all of them, so this is intended just for fun exploring ideas and not attempting to make hard proofs).

Enjoy what you want/can

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