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0.999...= 1

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Re: 0.999...= 1

#411

Post by sebringj »

enedil wrote: post #387
sebringj wrote: post #258

Earlier quoted context omitted.

Is this problem simpler than we want it to be? Meaning 1/3 is a concept stating there is 1 part of 3 total. If you have 3 total parts, added then it is a whole. Trying to shoe-horn it into the decimal system, similarly to try to represent pie as a clean number into the decimal system etc. Isn't the issue representing the number in one for and another, not the actual logic of the issue? idk

Why this is an issue, when 0.9999... is exactly 1?

I don't think it is directly. I was referring to the 1/3 comment but possibly it is related in how we are representing decimal numbers as irrational numbers. I had made another comment directly on that somewhere in this tree but it was more an intuitive one rather than a proof.

Re: 0.999...= 1

#412

Post by jacobparker »

scrollaway wrote: post #333
theincredulousk wrote: post #308

Earlier quoted context omitted.

great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. There is nothing magic or unknowable about the quantity 1/3. I've often wondered if there is some alternate base or mathematical system entirely that would be "better" in these respects. The thought usually comes up thinking about why pi is such an "ugly" number in…

Base 12 is a better base overall. It is divisible by more numbers. That is why it is used in various monetary, time and measurement systems. It's the one issue I have with the metric system... But that ship has sailed :) look up the dozenal society if you're curious how fervent some supporters might be.

Any other argument for a different base aside, in base 12 you have the analogous “problem” that 0.BBB... = 1. None of the usual bases, equipped with this “...” power, avoid the “problem” (non-unique representation).

Re: 0.999...= 1

#414

Post by rrmm »

danharaj wrote: post #171
rrmm wrote: post #48

Earlier quoted context omitted.

Wildberger is great. His lectures that he teaches at UNSW (i think) are interesting, and he usually keeps a clear dividing line between std math and his own predilections. It threads the line between being a kook and legitimate published mathematician very finely. I actually have some sympathies with his contention that real numbers (limit points of infinite series) are somehow a different animal than rational number…

If I give you two representatives of real numbers, say turing machines that write out on their tape the binary digits of those real numbers, in general you will not be able to order them.

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?

Re: 0.999...= 1

#415

Post by camelite »

pwdisswordfish2 wrote: post #306
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…

There is no smallest positive infinitesimal either. At least in theories that manage to define those rigorously. And it’s mostly a formal trick anyway; standard epsilon-delta calculus avoids them entirely. Had you actually meaningfully studied this subject, or did you just link to a Wikipedia article you half-heartedly skimmed one day?

> There is no smallest positive infinitesimal either.

There is in the surreal and hyperreal number systems. I got that from skimming wikipedia though....

Re: 0.999...= 1

#416

Post by jl2718 »

shawnz wrote: post #368
_Donny wrote: post #264

Earlier quoted context omitted.

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?

Multiply both sides by ‘x’, then subtract one side from the other, then take the limit as x -> inf. This is obviously undefined. To get to zero, you have to make a new rule that one form of infinity is bigger than another form of infinity.

Infinity is very slippery, and there are several divergent fields of math that depend on particular definitions of it.

Re: 0.999...= 1

#418

Post by brutt »

gspr wrote: post #385
brutt wrote: post #363

Earlier quoted context omitted.

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.

None of these, except 0.999… and 1 are well-known standard objects in this setting. You have to define what you mean.

I defined it:

    0.000...1 = 1/10^∞ = 1/∞

Re: 0.999...= 1

#419

Post by supercasio »

jl2718 wrote: post #402

The proof relies on the assertion that the supremum of an increasing sequence is equal to the limit. This is mathematical dogma, and should be introduced as such. Once that is accepted, it becomes obvious. This is illustrative of what I see as a fundamental problem in mathematics education: nobody ever teaches the rules. In this case, the rules of simple arithmetic hit a dead end for mathematicians, so they invented…

What? 0.999... = 1 is not dogma. Please don't spread misinformation. And at least read the link before commenting on something.

Re: 0.999...= 1

#420

Post by brutt »

brlewis wrote: post #380
brutt wrote: post #363

Earlier quoted context omitted.

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.

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?

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