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0.999...= 1

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Re: 0.999...= 1

#441

Post by rrmm »

danharaj wrote: post #438
rrmm wrote: post #414

Earlier quoted context omitted.

Certainly not non-computable ones, but presumably they lie somewhere regardless of my inability to do it on a TM. Which presumably gives rise to all the weirdness uncountable infinities give you. I guess I shouldn't phrase it as "you can fully order it". :D Zermelo's theorem at that point right?

Even computable reals do not have computable ordering. > but presumably they lie somewhere regardless of my inability to do it on a TM. Why? > Zermelo's theorem at that point right? It is declared by fiat in standard set theory that infinite sets can be well ordered. This is no real mathematical justification. The real justification is social: that it is convenient for mathematicians to not care about the ontology of…

I mean that's Wildberger's whole point isn't it?

Re: 0.999...= 1

#442

Post by burnte »

2OEH8eoCRo0 wrote: post #221
gspr wrote: post #162

Earlier quoted context omitted.

> There is no proof that will ever satisfy a person dead-set against this. Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration. I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thi…

Ask for a number between .9 repeated and 1

Ask for a letter between G and H.

Re: 0.999...= 1

#443

Post by brutt »

aliceryhl wrote: post #439
brabel wrote: post #422

Earlier quoted context omitted.

If you accept that 1/∞ = 0 Then you accept that ∞ * 0 = 1 But the definition of 0 is exactly that anything multiplied by it must be 0. So this cannot be true. To take a more verbal route: you cannot take nothingness and repeat it. Repeating (or multiplying) nothingness (or 0) is fundamentally nonsense. Programmer explanation: one cannot loop through `null` even once, let alone a large number.

No, you don't have to accept that. In my Analysis 2 course we worked a whole bunch with [0, ∞], i.e. the positive real numbers together with infinity, and we defined 1/∞ = 0 and ∞ * 0 = 0. You lose some of your usual rules of arithmetic, but it gets a lot easier to talk about integrals.

https://en.wikipedia.org/wiki/Surreal_number

Quote:

There are also representations like

{ 0, 1, 2, 3, … | } = ω

{ 0 | 1, 1/2, 1/4, 1/8, … } = ε

where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about 2ω or ω − 1 and so forth.

Re: 0.999...= 1

#444

Post by 0xff00ffee »

tom-thistime wrote: post #423
tom-thistime wrote: post #224

Earlier quoted context omitted.

Fine. Define 0.999... as the limit of the series sum(n=1 ... N)(10^-n), as N-> infinity. This is standard high school calculus. "Number" and "series" and "limit" and "convergence" don't all mean the same thing. However this number is defined as the limit of a convergent series. So the question really is meaningful. (One clue that this question is meaningful is the amount of space introductory calculus textbooks use t…

Thanks to a commenter who pointed out that my sum above should be sum(n=1 ... N)(9*10^-n). I can't, uh ... fully endorse that comment, which is not entirely accurate and doesn't answer my question. But I sure did miss the '9'.

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?

Re: 0.999...= 1

#445

Post by brlewis »

brutt wrote: post #420
brlewis wrote: post #380

Earlier quoted context omitted.

Ok, now you're saying that infinite decimals have final digits.

If Universe is infinite, then if we compare you to size of Universe, you are infinitely small, so you don't exists at all. Why I should waste my time? If Universe is finite, then finite number of elements can make only finite number of combinations, thus this discussion is repeated infinite number of times again. Why I should waste my time again?

Yes, you're wasting your time if you compare my size to the size of an infinite universe. If you really want to waste your time that way, you don't want to use real numbers. On the real number line my size is exactly zero. You need to go into infinitesimals, which are out of place when you're looking at decimal notation, which is only for real numbers.

Re: 0.999...= 1

#446

Post by danharaj »

rrmm wrote: post #441
danharaj wrote: post #438

Earlier quoted context omitted.

Even computable reals do not have computable ordering. > but presumably they lie somewhere regardless of my inability to do it on a TM. Why? > Zermelo's theorem at that point right? It is declared by fiat in standard set theory that infinite sets can be well ordered. This is no real mathematical justification. The real justification is social: that it is convenient for mathematicians to not care about the ontology of…

I mean that's Wildberger's whole point isn't it?

Yes, I'm just trying to emphasize that there is real serious mathematics behind his point, it's not just a matter of philosophical taste.

Re: 0.999...= 1

#447

Post by WORLD_ENDS_SOON »

black_knight wrote: post #301
WORLD_ENDS_SOON wrote: post #278

Earlier quoted context omitted.

I wonder if this is related to "intuitionist" math. This is an alternative formulation of math which doesn't have the law of excluded middle, recently discussed on Hacker News relating to this physics research: https://www.quantamagazine.org/does-time-really-flow-new-clu...

If you want to work with the real numbers intuitionistically (or constructively), you quickly find out that infinite decimal expansions are not what you want. In classical mathematics all the usual definitions of real numbers (decimal, Cauchy sequences and Dedekin cuts) are equivalent. If you overthrow the Law of Excluded middle, these are all different. Infinite decimal expansions are bad intuitionistilcally for sev…

Thank you for the reference! Your explanation makes sense to me. I'm curious if students that struggle with learning decimal expansions and limits are just subconsciously rejecting the non-constructive foundations of the math they're learning. Probably not, but it's fascinating regardless.

Re: 0.999...= 1

#448

Post by aliceryhl »

brutt wrote: post #443
aliceryhl wrote: post #439

Earlier quoted context omitted.

No, you don't have to accept that. In my Analysis 2 course we worked a whole bunch with [0, ∞], i.e. the positive real numbers together with infinity, and we defined 1/∞ = 0 and ∞ * 0 = 0. You lose some of your usual rules of arithmetic, but it gets a lot easier to talk about integrals.

https://en.wikipedia.org/wiki/Surreal_number Quote: There are also representations like { 0, 1, 2, 3, … | } = ω { 0 | 1, 1/2, 1/4, 1/8, … } = ε where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numb…

I'm not quite sure I understand what you're trying to tell me? The example I mentioned is not about surreal numbers.

Re: 0.999...= 1

#449

Post by cthalupa »

sebringj wrote: post #233

The thing that helps me "understand" it is that the universe has finite sizes of things like the Planck length for example being a theoretical thing at the smallest distance I would imagine. Now imagine it going smaller than the Planck length (finite) in terms of the difference of .9 repeating and 1 since infinitely small differences can do that. Essentially there is no way to tell the difference between .9 repeating…

>The universe has finite sizes of things like the Planck length for example being a theoretical thing at the smallest distance I would imagine

The Planck length might or might not be a physical limit of the universe. We don't have any specific proof that it's the smallest, just that we will not be able to observe any of that size or smaller. To look at something, we need to use light, and we must use a wavelength smaller than the details we wish to resolve. For something of the Planck length or smaller, this ultimately results in a photon that would have more energy in that area than can exist without a black hole forming... so one does, which then prevents us from measuring it, much less anything smaller.

Space and time might very well be discrete and not continuous - certainly the Loop Quantum Gravity folks would agree there. But there are widely supported theories that take both sides.

(I tend to lean towards them being discrete, but I would hesitate to call myself even an amateur hobbyist when it comes to theoretical physics...)

Re: 0.999...= 1

#450

Post by burmer »

2OEH8eoCRo0 wrote: post #221
gspr wrote: post #162

Earlier quoted context omitted.

> There is no proof that will ever satisfy a person dead-set against this. Indeed. I've torn my hair out trying to convince smart people with PhDs in hard sciences and had to give up in frustration. I usually find that the most success can be had by kicking the ball to them immediately and having them define what they actually mean when they say "0.999…". If we're going to debate whether that thing equals another thi…

Ask for a number between .9 repeated and 1

How about, ask for an integer between 1 and 2. Can't think of one? Guess they're the same number then.
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