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0.999...= 1

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Re: 0.999...= 1

#551

Post by gowld »

SkyBelow wrote: post #506
thinkharderdev wrote: post #288

Earlier quoted context omitted.

That doesn't exist. An open interval doesn't have a smallest number.

It does exist. The other poster just clearly showed that it exists by referring to it. The problem is that if we include such a number in our formal system of math, we quickly find contradictions and the whole system falls apart. So such a number is incompatible with any formal system of math (though I guess you could start building one which does include such a number and see what properties it has). Herein lies the…

In math, when assuming the existence of something proved a contradiction, we conclude that the thing does not exist. The description may exist "integer between 3 and 4", but there is not described object. A description names a set or a class, and that class can have 0,1, or more numbers.

Re: 0.999...= 1

#552

Post by dvt »

jturpin wrote: post #528
dvt wrote: post #429

Earlier quoted context omitted.

Did you read what @jl2718 posted? Namely: > the supremum of an increasing sequence is equal to the limit -- this is not misinformation (and to anyone familiar with some introductory analysis, correct[1]). Of course, calling it "dogma" is a bit inflammatory, but not technically wrong. It's kind of of a made-up rule to help us work with infinities (particularly in ℝ -- but it happens all the time in set theory, as well…

I do think it is technically wrong to call it dogma - the decimal is a geometric series with a limit, right? And limits have an unambiguous definition, it's the smallest value that the series approaches but never exceeds as it tends to infinity. I think the part that is admittedly weird is that the notation "0.999..." refers to the limit as the series tends to infinity, and it kind of hides that fact from you. Even j…

> the decimal is a geometric series with a limit, right..

Right, but that's not really the crux of the matter. Hint: look at how the supremum is defined[1]. The definition of the supremum is how we end up with 0.999... = 1.

[1] https://math.stackexchange.com/questions/1977204/limit-of-mo...

Re: 0.999...= 1

#553

Post by gowld »

gspr wrote: post #496
crispyambulance wrote: post #354

Earlier quoted context omitted.

> It is hard to have the patience to chase down the consequences of their ill-fated definitions, though. Of course it's hard because in day to day life, even for the vast majority of STEM practitioners, the nuance of the proof that 0.9999... is 1 is not of much utility. Whenever one sees a 0.999[... to however many digits] one can safely assume it's less than one or perhaps more realistically "almost 1". To say 0.999…

> Of course it's hard because in day to day life, even for the vast majority of STEM practitioners, the nuance of the proof that 0.9999... is 1 is not of much utility. Oh absolutely. I'm not expecting STE(no M this time!) practitioners to necessarily be aware of why 0.999…=1 in their daily lives, but I do expect them to have encountered enough situations in their field of expertise where scraping the surface using sh…

> start arguing Anatomy 101 facts with a heart surgeon at a bar

Well, there's this:

wikipedia.org/wiki/Vaccine_hesitancy

wikipedia.org/wiki/Homeopathy

Re: 0.999...= 1

#554

Post by gigatexal »

ben509 wrote: post #518
gigatexal wrote: post #498

I hate to say it but I still don't believe this, it just goes against all intuition that I have, but people much smarter than I have proven it so I take it on faith for doing things like calculus etc just my lizard brain won't let me accept something that looks like less than 1 being 1 the same way that the limit of 1/x as x goes to infinity is zero but it doesn't seem like ti should be. The number gets infinitesimal…

Maybe try this: do longhand division of 7 by 7 or 9 by 9 or whatever, but instead of putting 1 in the first place, start with 0. 0.99 7 | 7.00000 0 7 0 6 3 70

I follow that but then I could see it just continues to .9999 forever never converting to 1 it’s always that tiny fraction less than 1...

Re: 0.999...= 1

#555

Post by cthalupa »

ubercow13 wrote: post #521
cthalupa wrote: post #513

Earlier quoted context omitted.

Are you saying that Cantor showed you can have 0.9..4? I'm not even remotely an expert here, so I might certainly be wrong, but I don't understand how Cantor's theorems show an ability to stick a finite number and stop on the end of an infinite number.

Yes, one thing Cantor showed is that it is somehow meaningful to define numbers like infinity+1, where there are an ordered infinity of elements followed by one more element. Sets like this are called ordinals https://en.wikipedia.org/wiki/Ordinal_number So you could, if you like, define 0.9...4 to be a bunch of digits indexed by the ordinal ω+1. However the thing you have now defined isn't really a representation of…

Huh, interesting. Looks like I've got some more reading to do.

Thank you!

Re: 0.999...= 1

#556

Post by gspr »

brlewis wrote: post #517
gspr wrote: post #492

Earlier quoted context omitted.

You're asking them to perform subtraction. They probably know how to do that with real numbers, but problably not with much else. So they'll have to know that they're real numbers (or whatever numbers you are demanding that they be – you're still unclear on this point if it's not actually the reals).

Internally I'm saying they're real numbers. In what I say to the person trying to intuit that 0.999... = 1, I'm deliberately avoiding talking about number systems. I'm assuming this person thinks of numbers as sequences of digits, possibly with a decimal point.

And that is where it all goes wrong.

Re: 0.999...= 1

#557

Post by Pudini »

andrewprock wrote: post #500
brlewis wrote: post #471

Earlier quoted context omitted.

Apples and oranges. For any two different real numbers, there's a number between them. Integers work differently.

This is a bold assertion, and one that is not obviously true, especially in cases like 0.999... and 1.0

Obviously saying it is not obviously true is false if 0.999... == 1.0

Re: 0.999...= 1

#558

Post by empath75 »

andrewprock wrote: post #500
brlewis wrote: post #471

Earlier quoted context omitted.

Apples and oranges. For any two different real numbers, there's a number between them. Integers work differently.

This is a bold assertion, and one that is not obviously true, especially in cases like 0.999... and 1.0

Those aren’t different real numbers. That’s the whole point of the conversation.

Re: 0.999...= 1

#559

Post by esmi »

tom-thistime wrote: post #468
0xff00ffee wrote: post #444

Earlier quoted context omitted.

You asked for a "law" of equality (whatever that is) and provided an answer that proves it converges to 1. What more could you possibly want?

We seem to agree on this: you don't think there's any need for a way to determine whether two real numbers are equal. For ordinary math, though, using some criterion for equality (for example x>=y and y>=x) is basic and not controversial. So it seems unconvincing (to me) when you seem to imply the opposite.

There is an easy way to prove two numbers are equal. Typically in the reals there are three possibilities: a > b, a b and a < b then you are left to conclude a = b. And this is exactly what is done in Apostol's Calculus Vol 1 (IMHO the greatest calc book ever written) chapter 1 when he proves that the area under n^2 is EXACTLY (n^3)/3, with no "calculus". You would be shocked how far into calculus the author gets with just that theorem. Can't recommend that book enough.

Re: 0.999...= 1

#560

Post by esmi »

tom-thistime wrote: post #169
Igelau wrote: post #164

Earlier quoted context omitted.

What's interesting is that people pretty quickly become comfortable with the idea that 1/3 = 0.333… So using that as a foothold, we can express 1/3 + 1/3 + 1/3 as 0.333… + 0.333… + 0.333… and it should be pretty easy to digest. At once we can see that in this little zone we've defined, 1 and 0.999… mean the same thing. Not a rigorous proof, and one or two people will probably bring up whataboutisms like "that's just…

This is a really good point. Maybe the problem is how we define equality. What's the test for when two numbers are equal? People accept that 1/3 = 0.333333... The same people don't always seem to accept that 3*0.33333... = 1. Well, how are we defining "equals"? If we can give that definition in black and white, I think that may help.

> What's the test for when two numbers are equal?

I put this in another comment but: For the reals: eliminate a b then conclude a = b.

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