Is 1 = 0.999... ? Really?

Philosophy Forums Science, Technology, and Math
1961

My apologies that I’ve not addressed this when apparently you wanted me to.

So your point is that for all the finite fractions of the form (n\times\frac1{n}) equalling (1) without exception as (n) tends towards infinity, somehow “at infinity” it’s (0) because (\frac1\infty) disappears?

The conclusion you should be making, mathematically, is that (\frac1\infty) is undefined, not that it definitively equals (0) even when multiplied by its reciprocal.

Limits are the only thing we can be definitively talking about here, as I covered in my last post.

I don’t laugh at mathematical incompetence relative to me - this isn’t about me, and I’m only laughing at all the mathematical posturing going on from self-confessed non-mathematicians unsurprisingly falling so short.
I’m just another mathematician, and we all understand why (1=0.\dot9)
I’m just trying to let you guys in on why the correct answer is correct. I can’t make you understand, nor would it venerate me in any way if I did. I’m just a messenger. This is all for your respective benefits, not for mine in any way. I’m no god or anything, I’m more like a janitor cleaning up incorrectness merely for aesthetic purposes.

1962

Silhouette,

“Limits” is a code word for “convergence”. Let’s not play word games here.

I didn’t say 1. I said “every real and imaginary number” — if and only if convergence is TRUE, they all equal zero.

When I was a mathematician isolating consciousness signatures, a number like 0.999… was MUCH more valuable to me than the number one. If they were equalities, they couldn’t have separate utilities.

1963

Well not really.

1964

I’ve always understood that limit means something different than “approaches”.

What’s ironic to me about that link is that even in the definition, “approaches” is in scare quotes!

1965

Of course, you might be convinced of this, but I’d say you’re more of an evangelist than a teacher. Your approach is best suited for producing people who mindlessly agree with you.

Correct. The only question remaining is who’s the teacher and who’s the student when it comes to this subject. It is possible that you know everything on every subject in the world except for this one.

Everything you say about me might in fact apply to you.

I am not sure you’re comparing numbers. It seems like you’re saying that (2\pi rad) represents the same angle as (0 rad). Which is true but . . . .completely irrelevant.

Certainly, (100cm) is the same as (1m). But that’s not what we’re talking about here. We’re talking about decimal numbers.

The claim is that two decimal numbers are equal if and only they have the same exact digits.

Of course it is. (1), (\text{one}), (\frac{10}{10}) and (1.000\dotso) are four different symbols representing one and the same number. I wish you could stop misinterpreting your interlocutors (:

On the other hand, (1) and (0.999\dotso) are two different symbols representing two different things.

Well, if I absolutely have to, I will apologize as well (: Sorry I find you funny, Silhouette.

1966

Because we “let” it be, that is what theoretical mathematicians do. And that is the only way we’re going to be able to prove it.

But is that the only choice we have in the reasoning?

Seems like it has been boiled down to a rather simple true or false statement and the mathematicians wrote it, as a true statement. Which rather seems to require that a mathematician can’t imagine it as a false statement because then the mathematician could not be certain it is true, they can only claim it is true and prove it mathematically.

But can some other branch of study question it.

So as we live in a rather linear time format we shall first look at the statement itself. 1 = 0.9 recurring.

Does 1 = 0.9 ? no, that is not true, oh, but there is another 9 next to it.
Does 1 = 0.99 ? no, that is not true as well. oh, but there is yet another 9 next to it.
Does 1 = 0.999 ? no, that is also not true. oh, but there is yet another 9 next to it. … My… this goes on forever, I don’t think it will ever be true.

Because the recurring pattern is untrue it is infinity untrue. There is nothing that changes the pattern given the form of the expression.

Both a mathematician and a philosopher should be able to see this argument presents a challenge to the truth of the statement.

So what happens when logic and math disagree?
The mathematician exclaims “that amateur philosopher can’t see it is mathematically true… what a dolt. I am such a great mathematician compared to him.”

1967

The word “posturing” is commonly used to mean “pretending to be something you are not”. What exactly am I pretending to be? I stated explicitly that I am not a mathematician – not even an amateur one – so it can’t be the case that I am pretending to be a mathematician, correct?

The word is also used to mean “behaviour or speech that is intended to impress or mislead”. I am certainly not trying to impress or mislead anyone. Not sure how you’re going to prove that one. But it might be the case that I am unintentionally impressing and/or misleading people. But that’s no posturing.

I think it’s better for you to stick to words such as “stupid”, “misleading” and “arrogant”. You can say I am stupid because I don’t know the correct answer to the question. You can say I am misleading because I am actively defending a mistaken position. And you can say I am arrogant in the sense that my disagreement with people who are right persists.

This is very touching. My heart almost melted.

The fact is that, all too often, what people think they are doing is different from what they are actually doing. Even if you truly want the best for others, even if your intentions are the best that can be, the consequences of your actions might be terrible.

What people do, not what people say, is what ultimately matters.

And as far as I can tell, you’re more of an evangelist, a preacher, than what you claim to be – a benevolent teacher.

1968

I came out as a true, real and bona fide god in my recent post. Do you want to know what a god thinks of you? Of course you do! So here’s the deal. I think every human is a genius! In the human species, if someone doesn’t understand something, that means that there’s nothing worth understanding about it! The “stupidest people” are the smartest people on earth.

That’s my evaluation. Take it or leave it as you prefer.

1969

(1 + 0 + 0 + 0 + \cdots) must be an algorithm too (:

1970

That’s funny.

But! It requires inference. If you take it literally, it never stops.

These are problems that emerge with sequences (algorithms).

Now let me ask you this?

Do you see a tangible difference between

1

And

1+0+0+0…

???

I do, in a very literal sense, I do.

And it all comes down to function:

Consider a person who is evaluating a number without the property of inference:

1 (done and done… takes a fraction of a second)

1+0+0+0… (they never finish!)

These things matter in math and in real life

1971

You seriously need to rigorously define the word “define” given how much you use it and in how many different contexts.

Until then, I’ll just repeat that it is certainly the case that the word “infinite” is well defined.

If the word “infinity” isn’t defined, which means it has no meaning assigned to it, then you can’t say that (0.\dot9) is equal to (1).

(0.\dot9) is defined using the word “infinity”. If the word “infinity” is meaningless then so is (0.\dot9). And if (0.\dot9) is meaningless then it ISN’T equal to (1).

But you just said that infinity is undefined.

So it’s no longer undefined?

That’s flattering.

Basically, unless we’re professional mathematicians, we should make no effort to answer the question ourselves. Instead, we should just accept what mathematicians tells us it’s true. Any independent thought is punished with “You’re being arrogant, thinking that you, being a self-confessed not-even-an-amateur, know better than professionals”. That’s how you instill irrational fear of independent thought in people. That’s also how you turn them into drones. Drones that might have the right answers, but still drones.

1972

Thinking… recurring?

1973

Oh absolutely. Of course you don’t realise it - that’s the whole problem.

The posturing is you presenting yourself as someone with obviously convincing arguments that are clearly better than the mathematicians’ arguments who are trying to correct you. And I’ve noted there’s been others and not just me attempting to do the same!

If you weren’t posturing you’d come across as uncertain whether your material is actually valid, but you don’t - at all. You’d ask questions and come across even in the slightest bit humble. That would match your admission that you are in fact not qualified on the subject. Yet not even a hint.

And yes, I know the approach I’m taking to teaching this subject sucks, even though everything I’m saying is perfectly valid I just can’t be arsed to say it with an amenable attitude. I shouldn’t need to for adults but obviously I should have pandered to your ego to be more effective, because that’s the only way to get “certain people” to listen.

Absolutely not!

Make all the effort you can to answer questions yourselves! Just don’t present whatever you come up with like it couldn’t possibly be wrong, and act like the people who do know what they’re talking about are laughably wrong if they try to correct you.
Stop with the reductio ad absurdum. Start with the temet nosce.

Define is already defined just fine.
That’s how I’m using it.

Being able to use a term doesn’t mean it’s defined.
If something is “undefined” that doesn’t mean it’s defined - by definition. And yet we still “know what it means” - but only by defining its opposite and then thinking “not that”. That’s not the same as defining the actual word. So you can use things like “infinity” operationally just fine, knowing what it means to do so, and yet infinity remains undefined.
Do you understand or not?

Everything I’ve been saying makes so much sense if only you didn’t put all your efforts into assuming it doesn’t and trying to present it as completely wrong at every step, just so you can carry on with the same flawed arguments.

But I’m trying to tell you I’ve given up on this actually happening, and trying to not engage with you as a result, but you’re so keen to engage with me you’re making it hard!

This sounds very Zeno.

That arrow is never going to reach the target! And yet it does…

1974

…I guess >this< is what happens when logic and math disagree.

Some mathematician can’t accept their arrows fall short, they then insists where the arrow fell was the target they were aiming at.

“Let” us, also assume that all mathematicians are also great archers in addition to their prowess with philosophy.

They are all such great archers they simply can’t imagine missing or even considering it as a possibility. Yeah the arrow reached that target for sure.

I don’t think so… maybe you would consider listening again… I don’t hear the paradox, and certainly wasn’t trying to sing it.

But thanks for the compliment all the same. That’s pretty cool to have one’s voice compared with Plato’s, by such an esteemed archer, mathematician and philosopher as you.

:arrow_right: On second thought… not so much, as you don’t seem to evidence listening so well.

But… that’s right… the greatest of archers, mathematicians and philosophers also have the great listening skills as well. We should add that to the list of assumptions we have “let” be true as well. For what… certainly not what might actually be true.

Your noise makes a sound too. And it sounds like some mathematicians that are singing out of tune.

1975

I will admit the arrow landed. I will agree the arrow landed where it was aimed. But has the combination of the draw force of the bow in combination with the weight of the arrow and the infinite distance to the target ever held any hope of the arrow actually getting there? Logic concludes the arrow fell where it was aimed but never actually reached the target it was aimed at.

In the listening that doesn’t sound like Zeno to me. Lets try re-tuning our instruments.

Have you explored the website Ukebuddy.com ukebuddy.com/ukulele-tuner. Explore… enjoy… but I haven’t found an assumptive scale anywhere on the site.

Pick from any of the conventions available, the result is we may play in tune from there. And honestly we can not even depend on that.

Western and Eastern musical tradition doesn’t agree on a convention in that.

But damned if western musical convention hasn’t landed a few hits in the process. And eastern musical tradition produces beauty in it’s results as well.

Does it escaped you… that which is both metaphor and analogy?

I am not looking for friends either and it has nothing to do with a popularity contest or democratic process.

Accepting the conventions of how pieces can be moved within the rules of chess… it looks from here like you are in check. You may/must now move only your king to a different position within the application of the rules of chess…

Your turn.

1976

If I think I have better arguments, I will present them as such. That’s not posturing. I am not really sure you understand what the word “posturing” means.

Why not simply admit you are merely frustrated by the fact that the world isn’t bending to your will? Even if you are right, what makes you think the world must bend to your will? Entitlement?

So you’re complaining that I’m confident because you think I’m wrong. And you’re calling that posturing. So confidence + being wrong = posturing. Since when? Posturing means pretending to be something you are not in order to attract attention, get respect and/or impress. Simply being confident and wrong does not cut it. There must be an adequate motive.

We are not discussing the ENTIRETY of mathematics but only one EXTREMELY SMALL PART of it. And this small part of it has very little dependence on the rest of the mathematics. That’s why you don’t need to be a mathematician in order to answer this question.

But most importantly, who you are is completely irrelevant. What matters are your arguments – even if they are mistaken in every imaginable way. Last I heard, this forum is free for everyone to express their opinions and to learn at their own pace.

Now let’s go back to discussing something that is at least in some way related to the subject.

Agree.

Indeed, if you can use a term it does not mean it’s a meaningful one.

Agree.

If you only know what the term does not mean, then you do not know what the term means. The term is, for all practical purposes, meaningless.

The argument you’re putting forward is that the word “infinity” is one such term: we only know what is NOT infinite.

But you make no effort to prove this claim, you are mereley asserting it.

How can we make a sensible claim that the set of all natural numbers is infinite if we do not know what the word “infinite” means? How can we if all we know is what the word does NOT mean?

It makes no sense and it reeks of pseudo-philosophy but you’re welcome to prove me wrong. I don’t see you trying to do that.

1977

I am not sure what you mean by “tangible difference” but there certainly is a difference between the two expressions. They are two different expressions. But they are two different expressions that represent one and the same number. So if you’re arguing that (1 + 0 + 0 + 0 + \cdots \neq 1) then I disagree.

You are one of the few making the case that (1 + 0 + 0 + 0 + \cdots) never stops and that it is therefore not equal to (1). I am not really sure how to respond to that.

I can, for example, say that (1 + 0 + 0 + 0 + \cdots) is equal to (1) because it attains (1) and maintains it indefinitely. I don’t have to speak of its value at the end of the calculation.

On the other hand, I also believe that it is perfectly logical to speak of an infinite sequence that has an end. Because of that, I can also say that (1 + 0 + 0 + 0 + \cdots) is equal to (1) at the end of the summation.

The sum can be represented algorithmically like so:

  1. Let (x) be (0).
  2. Let (i) be (1).
  3. If (i) is equal to or less than (\infty) go to step 4. Otherwise, go to step 6.
  4. If (i) is equal to (1), add (1) to (x).
  5. Go to step 3.
  6. The output of this algorithm is equal to (x).

This algorithm will never halt because there is no (x \in N) such that (1 + x >= \infty). The problem is with the algorithm. The algorithm itself restricts us from reaching infinity. It is not that infinity itself is unreachable, it is that the algorithm is asking us to reach infinity by starting at the first position and then taking a finite number of steps in the positive direction. You can never reach infinity this way. It’s akin to Zeno asking us to reach (1) by walking half the remaining distance at each step. This isn’t possible, not because (1) is unreachable, but quite simply because you can’t reach it the way he wants to reach it.

1978

You didn’t really take in all that I stated. I stated IN THE ABSENCE OF AN INFERENTIAL PROOF, there is a difference between 1 and the algorithm!

You didn’t quote, and ignored the salient points in my post.

1979

Could be the case. I wouldn’t be surprised if I failed to understand your point.

By the way, there is no need to quote me if you’re responding immediately after my post. It makes it difficult to read.

1980

I’m neurotic about that Magnus, silhouette already knows this… that’s why I almost always quote myself instead of editing. This is one of my quirks, perhaps not an admirable one by far.