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0.999...= 1

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Re: 0.999...= 1

#501

Post by kosievdmerwe »

vfinn wrote: post #433
kosievdmerwe wrote: post #303

Earlier quoted context omitted.

Other people have pointed out that induction never makes the jump from a finite number of 9s to an infinite number of 9s. I feel the easiest "proof" is a proof by contradiction. First hopefully we can agree that if we have two real numbers x and y that are not equal then we have a number z, such that x If 0.999...!= 1 then there must exist a number A, such that 0.999... Now since 0 For instance, we might have A = 0.9…

"If 0.999...!= 1 then there must exist a number A, such that 0.999... Thanks for the reply, but I don't know how you can make the above claim, because to me the question of what we mean by 0.999... is intertwined with the claim itself. I mean to me it seems to be a matter of how you interpret the approach to infinity. I don't see why there has to be A in between, if you interpret 0.999... as the "biggest possible num…

That statement is based on:

> First hopefully we can agree that if we have two real numbers x and y that are not equal then we have a number z, such that x It's one of the properties of the reals and rationals that if two numbers aren't equal, then there are infinitely many numbers between them. It doesn't work with the integers, 2 and 3 aren't equal, but there's no integer x, such that 2 So if we say 0.999... != 1 then that means 0.999... But this is modern mathematics. In the past mathematicians dealt with "infinitesimals", especially in the early days of calculus, but I think they were discarded because they were confusing and also not necessary in favor of limits. This is where I think where some of your confusion is coming from. Infinitesimals don't exist in the real (and therefore rational) number space, but the concept exists for other "weirder" number systems.

According to the wikipedia page "This repeating decimal represents the smallest number no less than every decimal number in the sequence (0.9, 0.99, 0.999, ...)"

The clearest resolution to "if you interpret 0.999... as the "biggest possible number below 1"" is that in the reals and rational this concept doesn't exist. There's no biggest number smaller than x and ditto for smallest number bigger than y. (The distinction from the wikipedia definition is the difference between It's similar to saying you can't divide by 0. Sure in some cases you can define it, but doing so causes many issues and costs you so much that it's not worth it.[1] Another example is 1 not being prime, there's no reason for it not to be prime, but it's just much more convenient to say arbitrarily that it's not prime.[2]

The other thing that might confuse you is a proof by contradiction, I know it certainly confused me the first few times I saw it. I'm happy to help you if this is tripping you up too.

> I'm slightly embarrassed I don't know more mathematics, but I'm trying to learn some more...

No problem, we all start knowing nothing :)

[1] https://www.youtube.com/watch?v=BRRolKTlF6Q [2] https://www.youtube.com/watch?v=IQofiPqhJ_s

Re: 0.999...= 1

#502

Post by marcosdumay »

CogitoCogito wrote: post #186
marcosdumay wrote: post #166

Earlier quoted context omitted.

The GP's definition is the fundamental definition of the decimal notation. It's exactly what the "0.9..." symbol means. You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.

> The GP's definition is the fundamental definition of the decimal notation. It's exactly what the "0.9..." symbol means. > You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way. I _know_ that. I think I maybe should just bow out of this conversation. I'm apparently incapable of expl…

Oh, ok. If I understand correctly, you mean that there isn't any standard algorithm for handling a sum with infinite terms if you don't include limits.

Well, I do disagree, not with the above statement, but the meaning of "0.9..." itself requires limits, so the discussion can never go anywhere if your assumptions do not include limits.

Re: 0.999...= 1

#503

Post by samatman »

barrkel wrote: post #344
tom-thistime wrote: post #326

Earlier quoted context omitted.

Yeah, can we have an example of an irrational number whose decimal representation repeats or terminates? Or a rational number whose decimal representation doesn't repeat?

I expect mjd is thinking of irrational bases. The number might still be written as 10, in digits that look decimal.

Thank you! This thread is full of people insisting on something wrong because they were taught incorrectly; an irrational number is defined in terms of integer ratios for a reason.

It's not like those people haven't worked with an irrational base before, either! Radians have an irrational base. When we talk about 2π radians, or 1/4π radians, that's exactly what we're doing.

Re: 0.999...= 1

#504

Post by throwaway894345 »

bcrosby95 wrote: post #457
throwaway894345 wrote: post #325

Earlier quoted context omitted.

Maybe I'm misunderstanding, but I think the issue with dates is strictly different. Dates are hard not because time is fundamentally hard, but because there is lots of complexity in human representation of time (different places at different times have had similar but different representations of time). But that's not inherent to time. Ignoring relativity, if everyone throughout time used something like seconds since…

Yet the way we represent numbers is also a human construct. 1 = .999... is hard for people to understand because we think and write in base 10 rather than base 3. There's nothing that is fundamentally hard to reason about here.

Oof, I meant to write "numbers are fundamentally hard to reason about", not "numerals". Missed the edit window. The point I was trying to make is that numbers are very often fundamentally counterintuitive irrespective of notation. E.g., is the set of natural numbers larger than the set of decimal numbers? Or the other way around? Or equal? The complexities of time are almost exclusively inferring and converting between (often ambiguous) representations.

Re: 0.999...= 1

#505

Post by DavidVoid »

tom-thistime wrote: post #485
DavidVoid wrote: post #463

Earlier quoted context omitted.

It is not. The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.

Ok. (a) I think you're saying that 0.00 ... 1 is not a real number. (b) Do we agree that lim n->inf 10^-n is a real number? (I.e. zero?) (c) Then you're saying that limit in "(b)" is not a reasonable definition of the string "0.00 ... 1". Is that correct?

That is correct.

The limit of lim n->inf 10^-n is exactly 0, it is not 0.00...1.

Re: 0.999...= 1

#506

Post by SkyBelow »

thinkharderdev wrote: post #288
traderjane wrote: post #259

Earlier quoted context omitted.

It's the smallest number bigger than 0.

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 problem, the people you are talking with do not use a form system. There system of math has something similar to the same flaw of their system of grouping of things, which would include the whole grouping that contains every grouping that doesn't contain itself. People rarely deal in formal systems and thus they can handle completely illogical statements fine as long they are protected from seeing the consequence of it.

Re: 0.999...= 1

#507

Post by qppo »

ristos wrote: post #459
qppo wrote: post #122

Earlier quoted context omitted.

I think the mistake in your thinking is that infinity is a value or reachable destination. It's a concept and it behaves differently than a number. Also go read Generatingfunctionology and Concrete Mathematics

That's the point I'm trying to make actually. 1/infinity is an infinite series, which would be a computation that takes infinite time to compute. Saying that 0.9999...=1 is saying that the infinite series is the same as that concrete value, which don't actually know and can't prove.

I think the hiccup here is the notion that an open form expression doesn't have a closed form representation, which is not always the case. Closed forms exist for infinite series and vice versa.

1 - sum{k=1}^{\ifnty} 10^-k is one example.

Re: 0.999...= 1

#508

Post by ben509 »

CogitoCogito wrote: post #94
knzhou wrote: post #44

Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. The point is that in elementary school arithmetic, you define addition, multiplication, subtraction, division, decimals, and equality, but you never define "...". Until you've defined "...", it's just a meaningless sequence of marks on paper. You can't prove anything about it using arithmetic, or otherwise. What the "a…

> Personally I've always thought "proofs" using "arithmetic" are right, but kind of stated backwards. I've never considered them right at all. By saying something like 0.9... x 10 = 9.9... and then saying that 9.9... - 0.9... = 9 you're basically just a priori defining 0.9... to be 1. In other words you're basically just defining 0.9... as a symbol to be some number x which has the property that 10x - x = 9. So you'r…

Here's one stupid, simple proof that doesn't require any predefinition.

We do longhand division of 9 by 9, but start by putting a 0 in the first place.

        0.99
    9 | 9.0000
        0
        9 0
        8 1
          90
Hence, 9 / 9 = 0.999 = 1.

Re: 0.999...= 1

#509

Post by Dylan16807 »

chottocharaii wrote: post #454
twiceaday wrote: post #440

Earlier quoted context omitted.

.999... and 1 exist on a continuous line. If they are different numbers, name a number between them.

? Just because there is nothing between two numbers does not mean the two numbers are equivalent. What nonsense is this

Without using 9999..., please name any two real numbers that don't have a number in between.

Re: 0.999...= 1

#510

Post by esmi »

tom-thistime wrote: post #314
CogitoCogito wrote: post #293

Earlier quoted context omitted.

Then I guess you're the smart one. My apologies.

Sorry, sorry. I've been trying to bring up nonstandard analysis, and repeatedly getting poked by people saying "what do you think 'calculus' is?" "have you ever heard of a limit?" and so forth. In the process I have apparently become even more ornery than usual.

As we have no idea who you are don’t you think it makes sense we try to figure it out? How one answers the question will vary based on background.
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