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0.999...= 1

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Re: 0.999...= 1

#361

Post by petters »

da_chicken wrote: post #298
thinkharderdev wrote: post #288

Earlier quoted context omitted.

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

Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly…

At the very least, don't write "Of course it does". It does not in the real number system.

Re: 0.999...= 1

#362

Post by brutt »

gspr wrote: post #352
brutt wrote: post #300

Earlier quoted context omitted.

0.00...1

> 0.00...1 And what does this mean? I will remind you that for an integer d between 0 and 9, 0.ddd… means the limit of \sum_{i=1}^N d/10^i as N tends to infinity.

  0.000...1 = 1/∞

Re: 0.999...= 1

#363

Post by brutt »

brlewis wrote: post #341
brutt wrote: post #322

Earlier quoted context omitted.

0.999... + 0.000...1 = 1 0.000...1 = 1/∞ 0.999... = 1 - 1/∞

You're repeating the same wrong thing you said earlier. It's 0.999... and not 0.999...0 In the same way, it's 0.000... and not 0.000...1.

  0.999... = 0.999...9
  0.999...9 + 0.000...1 = 1
  0.999...0 + 0.000..1 = 0.999..1
  0.000...1 = 1/∞
  0.999...9 = 1 - 1/∞
  0.999...0 = 1 - 1/∞ - 9/∞ = 1 - 10/∞
  If x/∞ = 0, then 0.999...x = 1.
  If x/∞ ≠ 0, then 0.999...x ≠ 1.

Re: 0.999...= 1

#364

Post by ubercow13 »

michaelcampbell wrote: post #338
CogitoCogito wrote: post #165

Earlier quoted context omitted.

> I think the point is not defining 0.9... to be 1, the point is that “...” means an infinite number of 9s. If you shift the decimal point by 1, then nothing changes, there are still an infinite number of 9s. If you shift the decimal point by 5 places, there are still an infinite number of 9s to the right. And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then t…

Don't conflate "I don't understand it" with "hand-wavy". The 2 are different things. Another proof by contradiction I've heard of this is that if 0.9... != 1.0... then there exists a number in between the 2. What is it?

It is hand-wavy, because it is using a vague conception of infinity which doesn't correspond to the real, useful definition of the infinity labelled "..."

Re: 0.999...= 1

#365

Post by raattgift »

solotronics wrote: post #202
Okx wrote: post #182

Earlier quoted context omitted.

I usually say "If and only if two numbers are different, then you can find a number between them". People often accept this axiom. Then, I offer them to find a number between 0.999... and 1.

is there any difference between a black hole and nothing? (somewhat joking but I was thinking of a physical analogy of the limit approaching zero)

Yes, there's a difference.

Firstly, though, there are multiple different types of black hole, from the theoretical to the astrophysical. We must narrow your question down to have any hope of a good answer.

The simplest theoretical black hole, the Schwarzschild black hole, has one variable -- the central mass -- which must be positive.

If we set the central mass in a Schwarzschild spacetime to zero, then we have Minkowski spacetime: no curvature, no horizon, no black hole.

The Schwarzschild spacetime is completely empty except for the central mass, which is constant and located at an infinitesimally small point at all times. The Minkowski spacetime is completely empty everywhere and at all times.

The symmetries of Schwarzschild and Minkowski spacetime are different, and if one were to probe the spacetimes in question with a Synge curvature detector [1], we would quickly discover which we were probing if our probes happened to be placed close to the central mass, and eventually if they were placed far from the central mass.

If one placed the probes infinitely far from the central mass, it would take an infinitely long time to distinguish the presence of the central mass (which makes spacetime non-Minkowski); but these spacetimes are eternal anyway, so that's OK. So that's almost a "yes" to there being a theoretical black hole analogy between (1-) 0.999... and (1-) 1.

I would not call this a physical analogy since neither Minkowski spacetime nor Schwarzschild spacetime is at all physical. Nature is full of stress-energy (gas, dust, ...) any of which breaks the vacuum condition of these spacetimes, there seem to be a lot of astrophysical black holes at the centres of galaxies and individual/binary stars that have become black holes, and even a two-black-hole universe is markedly different than a Schwarzschild spacetime. Additionally, these astrophysical black holes are not eternal, unlike Schwarzschild. In particular, the stellar mass ones were once stars, and the galaxy-centre ones at least had less mass in the past. These last conditions alone are substantial deviations from Schwarzschild that are even more obviously not Minkowski (e.g. if you put probe finitely but sufficiently far away, you could see an image of the radiant precursor star rather than the black hole!).

Finally, in our physical galaxy the answer to your question is a big "yes!". The observed orbits of these stars [2] would be noticeably different if the central mass in the Milky Way's central parsec were anything but a black hole, and would be even more different if that central mass were not there at all.

- --

[1] Synge, J.L., _Gravitation. The General Theory_, ch. XI §8, "A five-point curvature detector".

[2] http://www.astro.ucla.edu/~ghezgroup/gc/animations.html and http://www.astro.ucla.edu/~ghezgroup/gc/blackhole.html

Re: 0.999...= 1

#366

Post by RHSeeger »

dacohenii wrote: post #207
hilbertseries wrote: post #177

Earlier quoted context omitted.

Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?

I'm pretty sure it's a real number in that it falls within ℝ (the set of all real numbers).

I doesn't, because 0.666....7 is not a valid representation for a number. The ... means goes on forever, and you cannot put something "after forever". It's not different than saying "0.0j" is a real number in base 10; it's not, that string does not represent a number in our number base 10 number system.

Re: 0.999...= 1

#367

Post by hilbertseries »

dacohenii wrote: post #207
hilbertseries wrote: post #177

Earlier quoted context omitted.

Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?

I'm pretty sure it's a real number in that it falls within ℝ (the set of all real numbers).

There’s not a way to define it, typically you’d define .666... as the sum of 6/10^n from 1 to infinity. This decimal representation does not terminate, so you can’t put a 7 at the “end of it” because there is no end.

Re: 0.999...= 1

#368

Post by shawnz »

_Donny wrote: post #264
317070 wrote: post #192

Earlier quoted context omitted.

It depends on more than just ZFC, also on the definitions of the real/complex numbers. The crux of the proof is that 0.99999... is being constructed within the real/complex numbers, and in that system it is equal to 1. And at the point where students see this, the whole concept of real numbers and infinity is usually ill-defined. I actually understand the scepsis for this theorem and where it comes from. The proof re…

I think this is spot on, at least for me personally. I am not very good at mathematics, so I never questioned my professors when they said that "You cannot treat infinites as regular numbers". Perhaps due to that statement, I did not really pursue these kinds of equations. For instance, I do not really see how the algebraic argument on the Wiki is any different from: 2 * inf = inf inf + inf = inf (subtract inf from b…

That causes a contradiction which is why infinity can't be used that way. But what is the contradiction with 0.999... = 1?

Re: 0.999...= 1

#369

Post by jordigh »

blauditore wrote: post #311
mjd wrote: post #280

Earlier quoted context omitted.

I think this is a very insightful remark. People think that numerals _are_ numbers, and it's hard to explain why this is not the case, because we have no way to talk about specific numbers _except_ by using numerals. But many frequently-asked questions are based in a confusion between numbers and numerals. For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that a…

>the mistaken belief that an irrational number is one whose decimal representation doesn't repeat ...which is true for any base-n representation where n is a natural (even rational) number. And that's kind of implied most of the time, so it seems like a useful definition. Where would this lead to problems?

I think the problem is that the definition, while valid, makes you hyperfocus on irrational numbers this way.

Seldom do we prove that a number is irrational by inspecting its decimal expansion. This would be in most cases a very unnatural proof. Since irrationality is a negative property (meaning, one arising out of a negation: the number is not a ratio), most of the time you prove it by contradiction. But people who just know the "digits don't repeat" definition expect us to somehow be able to list all of the digits of an irrational number and show that this infinite list doesn't repeat, which is, of course, an impossible task.

Re: 0.999...= 1

#370

Post by brlewis »

gspr wrote: post #342
da_chicken wrote: post #298

Earlier quoted context omitted.

Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly…

> If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. Not at all. Standard calculus uses standard real numbers, for which there is no infinitesimal. One may well speak of infinitesimals as a mental tool when building a mental model for calculus, but those infinitesimals are not actual real numbers (or a well-defined mathematical object at all - in standard calculus).

Correct. This is covered in the article. https://en.wikipedia.org/wiki/0.999...#Infinitesimals
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