## Is 1 = 0.999... ? Really?

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## Is it true that 1 = 0.999...? And Exactly Why or Why Not?

Yes, 1 = 0.999...
13
42%
No, 1 ≠ 0.999...
15
48%
Other
3
10%

Total votes : 31

### Re: Is 1 = 0.999... ? Really?

Infinity cannot be defined as the smallest number greater than any integer because it is not a number as such
And adding one onto the largest integer makes that number the largest integer and it carries on ad infinitum

Where two infinities are identical the answer is 0 when they are subtracted
And so for example the infinite set of primes - the infinite set of primes = 0
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surreptitious75
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### Re: Is 1 = 0.999... ? Really?

surreptitious75 wrote:
$$\infty$$ + I = $$\infty$$
$$\infty$$ - I = $$\infty$$

$$\infty$$ + $$\infty$$ = $$\infty$$
$$\infty$$ - $$\infty$$ = 0
Actually $$\infty$$ - $$\infty$$ = $$\infty$$
$$\infty$$ + ANY NUMBER = $$\infty$$
$$\infty$$ - ANY NUMBER = $$\infty$$

All the results can be derived from the starting equation $$\infty$$ + I = $$\infty$$
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### Re: Is 1 = 0.999... ? Really?

when $$\infty$$ = $$\infty$$ then $$\infty$$ - $$\infty$$ = 0
when $$\infty$$ < $$\infty$$ then $$\infty$$ - $$\infty$$ = < 0
when $$\infty$$ > $$\infty$$ then $$\infty$$ - $$\infty$$ = > 0
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surreptitious75
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### Re: Is 1 = 0.999... ? Really?

surreptitious75 wrote:
when $$\infty$$ = $$\infty$$ then $$\infty$$ - $$\infty$$ = 0
when $$\infty$$ < $$\infty$$ then $$\infty$$ - $$\infty$$ = < 0
when $$\infty$$ > $$\infty$$ then $$\infty$$ - $$\infty$$ = > 0
If you accept that $$\infty$$+1= $$\infty$$ then none of those three equations makes sense. That's because they all depend on some sort of definite value/quantity for "each" infinity.
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### Re: Is 1 = 0.999... ? Really?

The size of an infinity is relative to how many members it has
For example the infinite set of integers is smaller than the infinite set of irrationals
So even though they are both infinite one is demonstrably larger than the other one
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surreptitious75
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### Re: Is 1 = 0.999... ? Really?

surreptitious75 wrote:The size of an infinity is relative to how many members it has
For example the infinite set of integers is smaller than the infinite set of irrationals
So even though they are both infinite one is demonstrably larger than the other one
Those equations deal with infinity as a numeric concept. They are not about sets or the number of elements in a set.

What do you get if you add 1 to infinity? Is the result infinity or is it something else?

Your initial equations started with $$\infty + 1 = \infty$$

If you accept that as true then the other results follow directly. If you subsequently deny those results, then you are not being consistent.
Last edited by phyllo on Thu Jan 23, 2020 4:50 pm, edited 2 times in total.
phyllo
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### Re: Is 1 = 0.999... ? Really?

Had some irl stuff take priority over this joke of a thread. A lot of persisting nonsense to catch up on, unfortunately.
Interesting to note that phyllo, already established as a competent mathematician here, has re-joined to argue the same points as me. It's almost as if it's only mathematicians who understand the maths behind this topic.

Magnus Anderson wrote:
Silhouette wrote:Btw your notation made me laugh

That's probably because you're deeply insecure and have a strong need to see flaws in people around you in order to feel good about yourself. And you're looking for any kind of flaws, so as long they are flaws -- big or small, significant or insignificant, etc.

Normal people don't do that.

Compare and contrast this response to the following 2 we've seen literally just before your accusation:

phoneutria wrote:I stand corrected.

A simply explained correction of her math, and yet(!) - honesty offered and repsonsibility taken.

Ecmandu wrote:I meant to type 9 instead of 8 Silhouette, glad you caught it.

An insignificant typo that I pointed out, and yet - gratitude.

Magnus, the reason your notation made me laugh is because you're pretending to be something you're not and your misuse of terms unfamiliar to you gives it away as obviously (to adults) as a child pretending to be an adult. It's funny because it's adorable in the same way - though I don't mean to discourage you from trying to learn. Making mistakes and making a fool of yourself will help you learn faster. This doesn't make me "deeply insecure" It's your flaw and you owned it many posts ago before I was applying pressure on you to stop acting like you're a mathematician when you already know you're not. Ever since I did this, your insecurities have been triggered strongly enough to provoke responses such as this - of psychological projection onto others, who you understand deep down to be most threatening to your perception of yourself as capable, which I've already said you probably are in other areas than mathematics. Everyone has their strengths and weaknesses - including me. I have no issue telling you and everyone else all about my weaknesses - I've admitted several times that there are far better mathematicians out there than me, you're just not one of them and you know it. In contrast to you, one of my strengths is to stay relatively silent on issues where I'm no expert, and not pretend they're my strengths.

Through this display you further evidence your deficit of mathematical thinking by failing to think Bayesian - as in "Bayes' theorem".
In Bayesian probability, one considers their observations in terms of their context in order to conclude more accurately.
Your projection of your insecurity onto me is based on two factors: not only that I've pointed out a relatively insignificant error of yours, but also the frequency with which I point out relatively insignificant errors for anyone. Bayesian thinking places the former within the context of the latter and although I've just now cited two very recent examples of me pointing out relatively insignificant errors on top of yours, this thread and forum as a whole is littered with small mistakes: spelling, grammar etc. which I do not point out. Far more often I do not call these into question. However in your case, your continued pretense of mathematical capability is hugely affecting this thread to its detriment and thus cannot be counted as insignificant, and even if it was, it is not frequent for me to point out such things given the far higher frequency with which I notice them. And either way, your willingness to take a specific instance and transform it into a general rule - and an ad hominem one too - betrays several fallacies and cognitive biases in the space of just one sentence.

So in response to your conclusion that my behaviour is not normal, I can confidently say that your condensed series of mistakes is unfortunately very normal.

Enough though.

Magnus Anderson wrote:Are you saying that my logic is invalid?

Are you saying that it's not true that we can know that an infinite product of $$0$$ is $$0$$ if we know that $$0$$ raised to any number (whether finite or infinite) is equal to $$0$$?

How about an infinite product such as $$1 \times 1 \times 1 \times \cdots$$? How do you know the result of this product is $$1$$? Is it because we know that $$1$$ times any quantity (whether finite or infinite) is $$1$$? Or is it because we know that $$1$$ raised to any quantity (whether finite or infinite) is equal to $$1$$?

0 raised to any finite number greater than 0 will be 0, because as I said, zero lots of anything, whether zero multiplied by itself several more times or not, is zero.
I specify "greater than 0", because 0 to powers less than 0 results in division by 0, which is undefined, just the same as 0 to the power of 0.
So the truth of 0 raised to any finite number being equal to 0 is already in dispute.
As for 0 raised to an infinite number, already the above shows that "0 raised to an undefined quantity being equal to 0" is not as simplistic as your understanding might suggest.

Take, for example, your other mention of 1 to the power of an infinite number. This doesn't even have the same problem that 0 has: of the power being 0 or less than 0 for finite numbers. And yet:
$$\lim_{x\to0}({\frac{sin(x)}x})^{x^{-4}}=0$$

The base, $$\frac{sin(x)}x$$, tends towards 1 as x approaches 0.
The index, $$x^{-4}$$, tends towards infinity as x approaches 0.
The whole expression of this "1 to the power of infinity", as x approaches 0... approaches 0.

A mathematician will be aware of examples where infinities seem to cause unintuitive answers as a result of infinity being undefined.

Your intuitions about infinities are that they have to follow the patterns that finites look like they're tending towards.
This is why it's so significant and laughable that you're not a mathematician and yet you're pretending to be one.

Magnus Anderson wrote:
We know from $$\prod_{n=1}^\infty{\frac1{10_n}}$$ that indeed any partial product has a different value - so that's no help

It's of no help if what you're doing is looking for a number that does not exist e.g. a finite number that is equal to the result of the infinite product $$\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \cdots$$. But if you're merely trying to figure out whether such a number exists, then it's quite a bit of help. It tells you that such a number does not exist.

It's only of help to define what you're looking for such that it's actually possible to find it and verify it.

It's of no help to suggest that something "ought" to exist because your adolescent mathematical intuitions seem like they might point that way - and to criticise the idea of trying to define such a thing to replace this "ought" with an "is".

It's not a coincidence that trying to define a quantity between $$0$$ and $$\prod_{x=1}^\infty\frac1{10_x}$$ is undefinable. No matter how much "special pleading" you apply to your naivety on the topic, "looking for a number that does not exist" is all that is possible to do here.

Magnus Anderson wrote:The limit of an infinite product is not the same thing as its result. They are two different concepts.

Thank you for explaining to me what I've already explained to you.
Again.

Magnus Anderson wrote:It tells us that the result of the infinite product is smaller than every real number of the form $$\frac{1}{10^n}$$ where $$n \in N$$. Most importantly, it tells us that no matter how large $$n$$ is, the result is always greater than $$0$$.

Your argument is basically that there are no numbers greater than $$0$$ but smaller than every number of the form $$\frac{1}{10^n}$$ where $$n \in N$$.

That's one of our points of disagreement.

Actually, if you'd read my post, you'd realise it's one of our points of agreement - at least for finite natural numbers:

Silhouette wrote:the argument remains that whilst $$\forall{n}$$ where $$n\in\Bbb{N}$$, $$\frac1{10^n}-0\gt0$$, $$\prod_{x=1}^\infty\frac1{10_x}\not\gt0$$

In short, of course for all finite natural numbers, 10 to their negative power is greater than $$0$$.
And of course for the infinite product, the undefined element leads to an undefined answer that tends towards a defined limit, which just so happens to back up the lack of possibility to define a quantity between zero and the difference between $$1$$ and $$0.\dot9$$, resulting in there being no defined distinction between the two representations.

Magnus Anderson wrote:It does not merely "look like". It will never ever get to zero for the simple reason that there is no number $$n$$ greater than $$0$$ that you can raise $$\frac{1}{10}$$ to and get $$0$$.

You can say that $$0.\dot01$$ is approximately equal to $$0$$, and that is true and noone disputes that, but that misses the point of this thread. We're asking whether the two numbers are exactly equal not merely approximately equal.

You can say that $$\frac{1}{0}$$ can be substituted with $$\infty$$ for practical reasons (given that $$0 \approx \frac{1}{\infty}$$) but you cannot say that $$\frac{1}{0} = \infty$$ given that there is no number that you can multiply by $$0$$ and get anything other than $$0$$.

So from your point of view, the only conclusion that should make sense is that $$0.\dot01$$ is a contradiction in terms, and thus, not equal to any quantity. By accepting such a conclusion, you'd have to agree that $$0.\dot9 \neq 1$$. So at least one point of our disagreement (really, the main point of disagreement) would be resolved.

Still, one point of our disagreement would remain, and that would be your insistence that $$0.\dot01$$ is a contradiction in terms based on the premise that there is no quantity that is greater than $$0$$ but less than every number of the form $$\frac{1}{10^n}, n \in N$$.

There's nothing approximate about the impossibility for any defined quantity to between $$1$$ and $$0.\dot9$$.

$$\frac{1}{0}$$ can be substituted with "undefined". $$\infty$$ suggests the direction it's headed to an undefined degree. It's practical for mathematicians who understand these implications to swap these things out for display purposes, because they all know what is meant by it, which isn't that $$\infty$$ is a defined quantity that can be definitely operated upon by dividing 1 by it.
As such, as I keep having to clarify for you, I'm not saying that they're equal, given any reason - "numbers you can multiply by $$0$$ and get $$0$$" or not.

$$0.\dot01$$ is a contradiction in terms because it pretends that you can bound the boundless. It's not a contradiction in terms because there's no quantity greater than $$0$$ and the infinite product of tenths - that's just a symptom of it being a contradiction in terms. It's just another piece of evidence, alongside the limit of the infinite product of tenths being 0, even though infinite series never terminate to a defined answer. Combine the evidence of the limit with there being no definable quantity between 0 and the infinite product of tenths, and with the various proofs that you're trying to dismiss as merely "Wikipedia proofs", and the contradiction in terms is simply verified from all sides and validly falsified from none. Thus there's no difference between $$1$$ and $$0.\dot9$$. Done. Easy. Simple.

You keep telling me your straw men are my point of view.
I keep telling you to stop this.
You never listen.
If you have a question about my point of view, ask it. Don't tell me the answer.

Magnus Anderson wrote:You have yet to explain where's the contradiction.

Statement 1: "At some point in time at some point in space, there exists an infinite line of apples."

Statement 2: "At some other point in time, no apples exist anywhere in space."

How do the two statements contradict each other?

Certainly, the word "infinite" does not mean "not being able to be something else at some other point in time".

Another red herring:
"infinite" $$\to$$ "an example involving properties" $$\to$$ "infinite doesn't mean not conforming with afore-mentioned properties in some expected way".
Certainly indeed, irrelevant conclusion indeed.

Physical constraints aside - I know you're not a fan of them - even a conceptual infinite line of apples being added or removed from the universe brings with it problems of "the undefined" by virtue of the line being infinite. It is necessarily going to be undefined whether or not even a conceptual infinite "anything" has been/can be constructed/destroyed: it's a contradiction to suggest the termination of an infinite process, whether adding or removing an infinite "anything".

Magnus Anderson wrote:Silhouette keeps insisting that $$0.\dot01$$ is equal to $$0$$.

The rest of this post has already been dealt with: you're using your intuitions about finites. Still. Continue your mathematical learning, then get back to me once you've understood and accepted the undefined nature of infinity.
I'm saying that $$0.\dot01$$ is a contradiction and doesn't exist. This is not saying it's equal to anything.
There's no definable difference between what's "intended" by this contradiction, and 0. The gap you insist "ought" to exist from the suggestions of finites, cannot. Done. Easy. Simple.

Magnus Anderson wrote:So which one is it? Is $$N = \{1, 2, 3, \dotso\}$$ the same size as $$2N = \{2, 4, 6, \dotso\}$$ or is it actually smaller?

The answer is that $$N = \{1, 2, 3, \dotso\}$$ does not specify the size of the set. The size of the set is something that is specified separately (usually merely assumed, without any kind of explicit specification.)

I can't believe that I'm having to explain that f(x) = 4x is not f(x) = 2x. The former won't have elements 4x-2 even though the latter will, so suggesting surjection on those grounds is just another red herring.

Yes, infinity has nothing to do with size - I'm going to pretend you're finally on board and have convinced yourself that you came to this conclusion all by yourself as though it countered the same point I've been making this whole time.

MagsJ wrote:..the exact point at which infinity becomes self-defining, so yes.. anything infinite is not bounded within a defined measurable set.

"Anything infinite is not bounded within a defined measurable set" indeed.
Doesn't that contradict what you said immediately before about "infinity becomes self-defining"?
How can infinity be accepted as not boundable by a defined set, and yet also be self-defining?

MagsJ wrote:
Silhouette wrote:The term infinite defies definition by the definition of "definition" and of "finite". Either something is infinite or it is not - of course. Infinite is only "definable" insofar as we can easily define finite... and then saying "not that". This says what you don't have and not what you do have. The analogy I used is that this "defines" what's in a hole by defining the boundaries of the hole and what's outside of it (i.e. it doesn't define what's inside the hole at all).

A good definition.. it’s not what you’ve got, it’s what you ain’t got. I like it.

Not really what I said - definition needs "what you've got" as well as "what you ain't got". But I'm glad you liked it.
"Infinity" only has "what you ain't got" so doesn't qualify as being defined as what it is (only what it isn't i.e. finite, which is fully definable).

MagsJ wrote:..but then wouldn’t that simply mean that something is either infinite or not? which I ‘think’ Silhouette (I don’t want to put words in his mouth) is also saying.

I wouldn't object to finitude and infinitude being binary opposites - though I have extensively covered combinations of the two and how they must be strictly separated in order to get at the exact sense in which different infinite series can "appear" to have different sizes: it's the number of finite constraints around the infinity of a series that determine any difference in size, and not the infinity itself that has difference in size. This is why different "infinite series" all being collectively and homogenously referred to as "infinite series" is misleading, when they only differ by their finite constraints and they are a mix of these finite constraints with infinity - "infinite series" are not "only infinite". There is an infinite element that is side by side with finite elements, and each are only one and not the other (no overlap or middle ground).

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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:I realized that silhouette’s argument hinged on 1/10*1/10... (repeating) hinged upon 0.0...1 being equal to zero.

As I just explained to Magnus, "0.0...1" is a contradiction so can't even exist, never mind be equal to anything.

The intended implication of the contradiction is indistinguishable from zero though - yes.
As you say, you never ever ever get to any terminating "1". It never comes into existence, leaving you only with the "0.0... = 0"

Ecmandu wrote:So anyways,

No response yet from Silhouette, so I’ll just post it.

Silhouette and I agree that 1/10*1/10 (repeating) never expresses the 1 at the tail end!

Ok, so far so good.

That means the only way that you can shift the decimal is not from right to left, but from left to right!

That means that:

0.999... + 0.111... must equal 1.111... if 0.999... equals 1

There’s a problem with this!

0.999... + 0.111...

Equals: 1.1...0

And we know that 0.0...1 is the number that makes 0.999... equal one.

That means that there is a discrepancy of 0.0...2 which makes not the smallest possible number (equal to zero) that can possibly be made!

Thus, Silhouette’s argument thus far, has been falsified.

I made edits.

Concerning this "0.999... + 0.111... = 1.1...0":
in the same way that you never ever ever reach the "terminating" 1 in "0.0...1" leaving you with only "0.0...",
you never ever ever reach the "terminating" 0 in "1.1...0" leaving you with only "1.1...".
I'm sure you have no objection with $$\frac{9}9+\frac{1}9=\frac{10}9$$, which is the fractional representation of the decimal sum you're demonstrating.
Perhaps you'd like to quantify the difference between $$\frac{9}9$$ and $$0.\dot9$$, $$\frac{1}9$$ and $$0.\dot1$$ and $$\frac{10}9$$ and $$1.\dot1$$ individually?

I get that you're trying to accumulate these suggestions of "terminating" digits in non-terminating decimals such that they seem to become significant, but the problem is in the inherent contradiction of trying to do so. There are no terminating digits to accrue into something significant.

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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:
Ecmandu wrote:I realized that silhouette’s argument hinged on 1/10*1/10... (repeating) hinged upon 0.0...1 being equal to zero.

As I just explained to Magnus, "0.0...1" is a contradiction so can't even exist, never mind be equal to anything.

The intended implication of the contradiction is indistinguishable from zero though - yes.
As you say, you never ever ever get to any terminating "1". It never comes into existence, leaving you only with the "0.0... = 0"

Ecmandu wrote:So anyways,

No response yet from Silhouette, so I’ll just post it.

Silhouette and I agree that 1/10*1/10 (repeating) never expresses the 1 at the tail end!

Ok, so far so good.

That means the only way that you can shift the decimal is not from right to left, but from left to right!

That means that:

0.999... + 0.111... must equal 1.111... if 0.999... equals 1

There’s a problem with this!

0.999... + 0.111...

Equals: 1.1...0

And we know that 0.0...1 is the number that makes 0.999... equal one.

That means that there is a discrepancy of 0.0...2 which makes not the smallest possible number (equal to zero) that can possibly be made!

Thus, Silhouette’s argument thus far, has been falsified.

I made edits.

Concerning this "0.999... + 0.111... = 1.1...0":
in the same way that you never ever ever reach the "terminating" 1 in "0.0...1" leaving you with only "0.0...",
you never ever ever reach the "terminating" 0 in "1.1...0" leaving you with only "1.1...".
I'm sure you have no objection with $$\frac{9}9+\frac{1}9=\frac{10}9$$, which is the fractional representation of the decimal sum you're demonstrating.
Perhaps you'd like to quantify the difference between $$\frac{9}9$$ and $$0.\dot9$$, $$\frac{1}9$$ and $$0.\dot1$$ and $$\frac{10}9$$ and $$1.\dot1$$ individually?

I get that you're trying to accumulate these suggestions of "terminating" digits in non-terminating decimals such that they seem to become significant, but the problem is in the inherent contradiction of trying to do so. There are no terminating digits to accrue into something significant.

As I was trying to explain to Magnus, there’s two ways to look at this:

1.) completed infinity (on this you are 100% correct)

2.) process towards infinity (on this I am 100% correct)

So... I posit this to you:

Does endlessness ever become complete?!?!

I think you’ll like that train of thought!
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### Re: Is 1 = 0.999... ? Really?

As I was trying to explain to Magnus, there’s two ways to look at this:

1.) completed infinity (on this you are 100% correct)

2.) process towards infinity (on this I am 100% correct)

So... I posit this to you:

Does endlessness ever become complete?!?!

I think you’ll like that train of thought!
There is no concept of time in mathematics.

Ideas like process and completion don't make any sense.
phyllo
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
As I was trying to explain to Magnus, there’s two ways to look at this:

1.) completed infinity (on this you are 100% correct)

2.) process towards infinity (on this I am 100% correct)

So... I posit this to you:

Does endlessness ever become complete?!?!

I think you’ll like that train of thought!
There is no concept of time in mathematics.

Ideas like process and completion don't make any sense.

“T” (time) is a variable in all branches of mathematics.

I know you’re trying to sound profound —- thing is, without time we couldn’t possibly have or understand this discussion on ANY level!
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:If you agree that $$\sum_{i=1}^{n} \frac{9}{10^i} < 1$$ for every $$n > 0$$, and if you agree that $$\infty > 0$$, then it necessarily follows that $$\sum_{i=1}^{\infty} \frac{9}{10^i} < 1$$. Everything else is irrelevant. The only condition is that $$n$$ is greater than $$0$$. No need to satisfy Gib's definition of the word "quantity".

That's really careless. Dismissing the definition of "quantity"? Really? Well then, I guess we can put in whatever we want for n. How 'bout "cow"? Is "cow" > 0? If so, then I can prove that $$\sum_{i=1}^{cow} \frac{9}{10^i} < 1$$. But I think even you can appreciate that there is a limit to what we can substitute for n.

I'm going to demonstrate how you're confusing intuition for logic. Your argument is:

1) $$\sum_{i=1}^{n} \frac{9}{10^i} < 1$$ for every $$n > 0$$
2) $$\infty$$ > 0
3) Therefore $$\sum_{i=1}^{\infty} \frac{9}{10^i} < 1$$

^ Seems logical, right? But you're missing a crucial step between 2) and 3): $$\infty$$ is a valid value for n. I know you feel intuitively that $$\infty$$ must be a valid value for n because $$\infty$$ is a quantity, so no need to prove it. But when you're arguing with someone who disagrees with you on that, you do need to prove it. You can't just run on intuition. If I argued that "cow" > 0, and therefore the formula applies to cows as well, you would insist I prove that "cow" is a valid value for n, wouldn't you? Same onus falls on you to prove that $$\infty$$ is a valid value for n. But so far, all I've seen from you is re-assertion after re-assertion that $$\infty$$ is a number--no proof--which tells me you believe it on intuition, not logic. You need to prove this just as much as you need to prove that what applies to finite sets also applies to infinite sets (and I wonder if this is just a special case of the same thing). But seeing as how you refuse to prove your point when I ask you too, I'm guessing you'll cower away from this one too.

Magnus Anderson wrote:$$\sum_{i=1}^{n} \frac{9}{10^i} < 1$$ holds true for every $$n > 0$$. It does not only apply to integers. It literally applies to anything greater than zero.

Well, it applies to any real number greater than 0. But what that means is: grab any number on the number line. $$\infty$$ is not on the number line. It's a direction in which the number line extends.
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### Re: Is 1 = 0.999... ? Really?

“T” (time) is a variable in all branches of mathematics.
That's a mathematical representation of time when modeling some physical situation. There is no time within mathematics itself. Nothing within mathematics requires time to complete.

I can have a variable "U"(unicorns) but it does not mean that unicorns are a part of mathematics.
I know you’re trying to sound profound —- thing is, without time we couldn’t possibly have or understand this discussion on ANY level!
Humans exist within time and therefore need time to understand things.
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:As I was trying to explain to Magnus, there’s two ways to look at this:

1.) completed infinity (on this you are 100% correct)

2.) process towards infinity (on this I am 100% correct)

So... I posit this to you:

Does endlessness ever become complete?!?!

I think you’ll like that train of thought!

The explanation I gave earlier in the thread for any "concept of time in mathematics" was that there's a difference between quantity and representation of quantity. The former is "already there" (for infinite quantities, it's undefined where exactly "already there" is), but representations of quantities as well as representations of how to construct/deconstruct quantities have ordered steps that occur before or after other ordered steps - according to that order and therefore implying temporality.

When I say endlessness in reference to quantities, I'm referring to the quantitative boundaries (ends) that specfically define a quantity as distinct from higher or lower quantities. Not having this, as with an undefined/indefinite/infinite, means it has no ends. Mathematical construction of such endlessness would take an endless amount of time to complete, just as its deconstruction to a definite quantity would never happen. The best you can do is imply endlessness by using a symbol/notation that only looks like it's "bounding" boundlessness, which has to be treated very carefully and separately from defined, bounded finites, which have specific ends. Otherwise you can be fooled into thinking there's more than one kind infinity, and/or that each one can have a different size, when in fact it's the finite constraints around infinity only that affect anything to do with size: there's only one way in which endlessness can be endless.

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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
“T” (time) is a variable in all branches of mathematics.
That's a mathematical representation of time when modeling some physical situation. There is no time within mathematics itself. Nothing within mathematics requires time to complete.

I can have a variable "U"(unicorns) but it does not mean that unicorns are a part of mathematics.
I know you’re trying to sound profound —- thing is, without time we couldn’t possibly have or understand this discussion on ANY level!
Humans exist within time and therefore need time to understand things.

Nothing in mathematics requires time to complete ?!?!?!?!?!

Really!?!?!?!? Please explain!
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:
phyllo wrote:
“T” (time) is a variable in all branches of mathematics.
That's a mathematical representation of time when modeling some physical situation. There is no time within mathematics itself. Nothing within mathematics requires time to complete.

I can have a variable "U"(unicorns) but it does not mean that unicorns are a part of mathematics.
I know you’re trying to sound profound —- thing is, without time we couldn’t possibly have or understand this discussion on ANY level!
Humans exist within time and therefore need time to understand things.

Nothing in mathematics requires time to complete ?!?!?!?!?!

Really!?!?!?!? Please explain!
What is there to explain?

If you are presented with the fraction 1/3 , then it takes some time for you to do the division. It might take you 10 minutes, it might take you 5 seconds to realize that 1/3=0.333... (Or you might never realize it.)

It takes you time because time applies to you. Mathematics is not a "being" who has to "do" something. It doesn't "need" to "get to the end". It doesn't need a process to "do" the calculation. Mathematically 1/3=0.333... - that's it.

Time is just not applicable to some stuff.
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### Re: Is 1 = 0.999... ? Really?

Phyllo,

Even if 1/3 is equal to 0.333... that means that 0.333... *3 is equal to 0.999..., which looks ALOT different than 1!

If you say they are equal, you’re making the claim that EVERY counting number is equal to an infinity of infinities (contradiction)

If you say they’re not equal, then you are drawing a line which states “counting numbers are finite” (which is correct)
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### Re: Is 1 = 0.999... ? Really?

1/3=0.333...
3*(1/3)=1
3*(0.333...)=0.999...
Therefore 1=0.999...

If that's not true, then simple division and multiplication don't work.

You can even take out the multiplication: 1/3+1/3+1/3= 0.333... + 0.333... + 0.333... = 0.999... = 1

It's not rocket science.
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:1/3=0.333...
3*(1/3)=1
3*(0.333...)=0.999...
Therefore 1=0.999...

If that's not true, then simple division and multiplication don't work.

You can even take out the multiplication: 1/3+1/3+1/3= 0.333... + 0.333... + 0.333... = 0.999... = 1

It's not rocket science.

Phyllo, you don’t understand what I’m saying when I say the implication is that all finite numbers are infinite (in fact you avoided it)

Let’s say we hypothetically live in a world where fractions don’t exist... it would be unfathomable that 0.999... = 1.

In a decimal world. 0.111... * 9 equaling 1 is impossible.

The problem is not with my logic, the problem is how operators work with 1 minus base, to make them ‘appear’ equal... but then again, they don’t appear equal at all do they ?!?!?!
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### Re: Is 1 = 0.999... ? Really?

If you say they are equal, you’re making the claim that EVERY counting number is equal to an infinity of infinities (contradiction)

Phyllo, you don’t understand what I’m saying when I say the implication is that all finite numbers are infinite (in fact you avoided it)
There seems to be a lot of confusion here about what "infinite" means. "All finite numbers are infinite" doesn't make any sense as a statement.
Let’s say we hypothetically live in a world where fractions don’t exist... it would be unfathomable that 0.999... = 1.

In a decimal world. 0.111... * 9 equaling 1 is impossible.
That's simple enough. All you need to do is to restrict yourself to whole numbers, natural numbers or integers.
There is no integer which represents 1/3 or 1/2 or 8/9. When evaluated, those fractions are equal to :
1/3=0 in integer
1/2=0 or 1/2=1 in integer
8/9=0 or 8/9=1 in integer

The two values given for 1/2 and 8/9 depend on whether truncation or rounding(up/down) is the standard procedure when evaluating the results.

A similar thing happens with 0.999... when using real numbers. It "jumps" up to 1.
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### Re: Is 1 = 0.999... ? Really?

gib wrote:Fixed Cross

Fixed Cross wrote:The first part of the disagreements comes down to whether infinite also means indefinite.
If it does, then it is not a quantity but rather something like a condition, of a set or whatever.

As far as I know, "indefinite" means something like "we don't know when or if it will ever end." Infinite means "we definitely know it won't end."

Yes but in as far as it pertains to quantity, its pertaining makes quantity indefinite.

Infinite is the property of being endless, it's not a quantity. Quantities are the things you find on the number line. Infinity is a property of the number line itself. It is where the number line extends to (or more accurately, the property of its extension being unlimited).

We agree on that.

It's LaTeX.

You can see how we do it by quoting our posts and looking at in the text editor.

LaTeX.png

This board doesn't seem to have all LaTeX features enabled though. I know the mars symbol ♂ can't be posted in LaTeX.

Magnus seems to be the real LaTeX guru. He uses it even to say $$n > 0$$. I'm not that hardcore. I'd rather just type out n > 0.

Thanks gib.

Fixed Cross wrote:So: "keep adding 9s after the decimal point but you stop at some number n of 9s, then you will still be below 1" is wrong. The formula doesn't provide for a "stop at some number", it rather says to keep going indefinitely.

That's a bit ambiguous, turning on what exactly is meant by "indefinitely".

Yes and I have noticed during the years that the main problem when discussion infinity is its indefinite-ness.
This is the hot core all these debates centre around and it not being made explicit perpetuates the confusion.

How are you not going to be confused if you want definite results out of something that is by definition indefinite?

Fixed Cross wrote:But, not to keep going indiscriminately. You have to keep going with a specific task which by definition precludes any step from altering the result of the previous step. Which is what would have to happen for 1 to be reached.

What $$\sum_{i=1}^{n} \frac{9}{10^i} < 1$$ for any n means is: pick any integer from the number line. You are completely unlimited in which number you pick. But it does not mean: pick infinity. And not just because infinity isn't a number on the number line, but because substituting $$\infty$$ for n actually means: don't pick a value for n. Just keep adding forever.

That seems legit.
The strong do what they can, the weak accept what they must.
- Thucydides

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### Re: Is 1 = 0.999... ? Really?

Phyllo,

I’ll press you on this for now.

Phyllo wrote: “all finite numbers are infinite doesn’t make any sense as a statement”

EXACTLY!!!!!!!!

That’s my whole point. It makes no sense!

It makes no sense that 0.999... EQUALS 1!!!!

In this formulation, 1 by definition is an INFINITE number!! By equality!!!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Phyllo,

I’ll press you on this for now.

Phyllo wrote: “all finite numbers are infinite doesn’t make any sense as a statement”

EXACTLY!!!!!!!!

That’s my whole point. It makes no sense!

It makes no sense that 0.999... EQUALS 1!!!!

In this formulation, 1 by definition is an INFINITE number!! By equality!!!
Who the heck knows what you mean by "INFINITE number". I certainly don't.

Infinite digits doesn't mean infinite number.
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:
Ecmandu wrote:Phyllo,

I’ll press you on this for now.

Phyllo wrote: “all finite numbers are infinite doesn’t make any sense as a statement”

EXACTLY!!!!!!!!

That’s my whole point. It makes no sense!

It makes no sense that 0.999... EQUALS 1!!!!

In this formulation, 1 by definition is an INFINITE number!! By equality!!!
Who the heck knows what you mean by "INFINITE number". I certainly don't.

Infinite digits doesn't mean infinite number.

Doesn’t matter how you word it... you’re like a squirrel running from slingshots right now. You’ll dodge for a while, but, more likely than not, one will eventually connect! *To make this rated g, the pebble never hurts the squirrel*

You’re still making the claim that 1 is ....

EQUAL!!!

To “infinite digits”

EQUAL!!!
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:
phyllo wrote:
Ecmandu wrote:Phyllo,

I’ll press you on this for now.

Phyllo wrote: “all finite numbers are infinite doesn’t make any sense as a statement”

EXACTLY!!!!!!!!

That’s my whole point. It makes no sense!

It makes no sense that 0.999... EQUALS 1!!!!

In this formulation, 1 by definition is an INFINITE number!! By equality!!!
Who the heck knows what you mean by "INFINITE number". I certainly don't.

Infinite digits doesn't mean infinite number.

Doesn’t matter how you word it... you’re like a squirrel running from slingshots right now. You’ll dodge for a while, but, more likely than not, one will eventually connect! *To make this rated g, the pebble never hurts the squirrel*

You’re still making the claim that 1 is ....

EQUAL!!!

To “infinite digits”

EQUAL!!!
Sure. It looks counter-intuitive but the math equations show that it must be true.

What's wrong with the argument using 1/3 fractions? Nothing. Unless you want to argue that dividing 1 by 3 doesn't work.
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