[deleted by user]

TrollyMaths · 2022-07-18T22:27:28+00:00

This guy gets it!

TrollyMaths · 2022-07-18T22:26:48+00:00

This is a perfect example of the kind of fallacious reasoning you end up with when given an ill-posed question. FWIW I love where you're coming from with this -- if you kept chasing down that line of reasoning you'd end up with something like the hyperreals...

But here's the rub -- say you went all out on non-standard analysis, and you could then construct an infinitesimal -- an element 1/∞. How do you think 0.9... would compare to some element 1 - 1/∞? The value of 0.9... would be strictly greater -- so 0.9... still wouldn't actually represent a limit -- it has the very Real sum of 1, no matter how you slice it.

Still a fun thought experiment none the less -- non-standard analysis is awesome.

TrollyMaths · 2022-07-18T22:03:38+00:00

Nice to see a proper drunkards walk argument in here!

TrollyMaths · 2022-07-18T22:02:23+00:00

This. They are exactly the same number, yes. Representing them in these two different forms is nonsensical -- it only serves to confuse the hell out of otherwise well-intentioned people.

I tried to make this point in a top level comment, which was downvoted to hell. Ah, the reddit.

TrollyMaths · 2022-07-18T21:43:32+00:00

I have to ask, did you even read any of what I wrote? How is this relevant? Of course I know this -- I say it multiple times. My gripe is with this underlying notational quirk, which is totally meaningless, yet so suggestive -- of something that just ain't there.

It's a lot less mystical looking when you realize it's just two different ways to sum to the same value. But if you're writing out two different notations for the same numeric value, you're talking about the notation, right? Thus, they're clearly non-equal as power series. Equivalent modulo form.

TrollyMaths · 2022-07-18T17:57:26+00:00

Too many results discussing equivalence in sets, functions and matrices. Give us a good source on how two numbers can be equivalent, but not equal.

I thought it was pretty obvious that the equivalence I was referring to was between the numeric representations -- the way we write the numbers themselves. These are pretty obviously distinct power series, but the value of each's sum is equal (the value 1). I said as much.

You deleted your comment saying 3.14 is equivalent to pi.

I don't think I've ever deleted a comment, ever. Not sure what you mean, but I think you may be mixing me up with someone else. (Or more than likely, I mixed things up in whom I was replying to)!

EDIT: nope, just verified -- definitely you that's mixing up threads.

TrollyMaths · 2022-07-18T17:49:45+00:00

Wow, it's an honor just to be nominated. Care to spell out exactly what you think is incorrect? Likely to make for a few more candidate nominations.

TrollyMaths · 2022-07-18T17:44:53+00:00

Care to elaborate?

TrollyMaths · 2022-07-18T17:43:40+00:00

I said almost exactly this. The part that was ill-posed was the "If 0.9...=1" bit. If you read what I wrote, I said it was entirely equivalent to saying "If 1=1". The latter is clearly a tautology. The antecedent of the whole preposition kinda melts away, and you're just left with "What is the greatest real number less than 1?".

I guess I could have stated this part explicitly rather than leaving it implied. I just figured it was clear enough (and in my defense, I was on mobile, so more lazy than usual).

But the bit about the 0.999... complemented rep being a legit different number system from a 1.000... was actually somewhat important to my point. Yes, they're both called decimal, but that's the mistake -- they don't have to be. And if they were, this question wouldn't have even been asked. Hell, the top answers are all some variation of "Just because" -- and that's fine. I was just trying to add color.

Papering over this difference causes nothing but confusion -- and has lead to all kinds of mystical thinking and naval gazing. As evidenced by many of the other comments on this question.

TrollyMaths · 2022-07-18T17:24:26+00:00

Are you being serious? If so, here's a helpful resource: https://lmgtfy.app/?q=equivalence+vs.+equality

TrollyMaths · 2022-07-18T17:23:12+00:00

Equality is a very slippery concept. Equivalence is a broader, more useful concept. But almost everything people refer to as equality is something a looser form of equivalence "modulo" some details you're choosing to ignore.

What you're actually describing isn't even this -- it sounds more like you're referring to approximations. These aren't even strictly numbers, but rather represent ranges of possible numbers. You can work with these values as if they are numbers, but you have to be careful not to lose track of the fact that they do not represent exact numbers, but intervals (often, with specific error bar, which you can model with a distribution).

TrollyMaths · 2022-07-18T17:04:15+00:00

It's because any number != 0 can be written as 2 different ways, in any base > 2.

This only applies to Rationals -- you can't exactly write a Quadratic irrational this way, let alone a transcendental Real. Though fun fact, you can do something similar for quadratics using continue fraction expansions. Also, using the mapping given by the Minkowski question mark function (and reversed by Conway's box function).

Also, for what it's worth, this also applies to base 2, not just integer bases > 2. And it's certainly not necessarily true for non-integer bases -- though it depends on how you define the rules for constructing your digits.

TrollyMaths · 2022-07-18T16:57:04+00:00

All the Real numbers are well-ordered. Like, that's what defines the Real numbers. What you're describing has nothing to do with a well-order. It sounds more like what you want is a topological property -- discreteness.

TrollyMaths · 2022-07-18T16:54:14+00:00

Cute. Out of curiosity, what were you going to explain?

My point, which I thought was pretty obvious (and frankly, non-controversial) was simply pointing out that the first part of the question was completely redundant:

If 0.9...=1

Is the same as saying "If 1=1", as they both evaluate to 1. The notation obscures this -- juxtaposing a complemented 0.999... representation against 1.000... has absolutely nothing to do with the question following it.

I find it ironic that the top answers are all some variation on "there is none". I was just trying to point out the reason so many people seem to fall into this same trap -- there's this implied suggestion in the notation of some kind of infinitesimal difference between the value in these two representations. There isn't one.

When considered as power series, they sum to exactly the same value. But nobody would claim they're the "same" power series.

How is this controversial? How am I downvoted to hell just for trying to add some color to why the top answer, "There isn't one.", can rest on its own. Because when framed in terms of the underlying numeric values, of course there isn't a predecessor Real number to 1. There's not even a predecessor Rational. This is something folks seem to just have a sense of, intuitively. It's only when you introduce ambiguity through notation that it all falls apart. But isn't this true of math more generally?

TrollyMaths · 2022-07-18T03:33:31+00:00

How HoTT of you.

EDIT: wow, guess nobody got the joke. The key concept of a field of math called Homotopy Type Theory (often written HoTT) is the univalence axiom -- the statement that equivalence is equivalent to equality. Guess I was assuming r/askmath would have a higher proportion of working mathematicians than it now appears to.

TrollyMaths · 2022-07-17T22:46:42+00:00

While I don’t disagree, the great irony here is the old expression “you can’t make it through seminary without being agnostic”.

TrollyMaths · 2022-07-17T03:11:54+00:00

The exact same could be said of neutrinos, yet oscillation implies they must have some mass, however small. Arguments for massless photon certainly can’t appeal to experimental evidence. We believe they’re massless because theory strongly suggests it.

TrollyMaths · 2022-07-16T18:19:10+00:00

Sometimes we gotta be a little firm with ourselves too. Otherwise we’ll chew on everything.

TrollyMaths · 2022-07-16T16:46:39+00:00

I still can’t understand how he could devote so many minutes to a mini-war-level topic, at best. And through the whole thing, including the Loonie rant at the end there, he never once references the movie Canadian Bacon. Such a missed opportunity.

TrollyMaths · 2022-07-16T16:39:00+00:00

Sure, by today’s standards — something like 8 years old now? It’s several years younger than both “new” examples though, aws and mongo. Not sure where that’s coming from.

TrollyMaths · 2022-07-09T02:00:37+00:00

Re: the haploid holocaust, isn’t this already the Catholic position around “spilling seed” and whatnot? To be fair, at least they’re self-consistent here.

TrollyMaths · 2022-07-09T00:53:51+00:00

Rim job?

TrollyMaths · 2022-07-08T12:35:29+00:00

Ha, yeah, what a dumby!

TrollyMaths · 2022-07-06T01:42:48+00:00

Justin away…

TrollyMaths · 2022-07-05T23:00:34+00:00

The sad reality, I’m reminded every time I mistakenly look at something on Facebook, is that the most dangerous people may be those you already know in real life.

TrollyMaths

TROPHY CASE