## 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...
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No, 1 ≠ 0.999...
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Mowk wrote:The question; Is 1 = 0.999...?

Another way of asking the same question might sound something like; is an infinite number of smaller and smaller "parts" equal to 1 finite "whole" ?

If I had 1 volume of space and I divided it, equally in half, an infinite number of times, the result would be an infinite number of parts. I guess if I could cut a volume of space an infinite number of times, it would still have come from the 1 original volume. Seems like the argument could be made from there, that infinity = 1.

Infinity ($$\infty$$) times infinitesimal ($$\frac{1}{\infty}$$) equals $$1$$. No doubt about that. But $$0.\dot9$$ is a different beast -- one that never adds up to $$1$$.

That’s false Magnus. Infinity times infinitesimal still equals infinity.
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### Re: Is 1 = 0.999... ? Really?

Sure is fascinating to think about.
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:That’s false Magnus. Infinity times infinitesimal still equals infinity.

You need to level up.

It's not either true or false per se but either true or false in relation to the starting premises. If you start with the premise that $$\infty + \infty = \infty$$, then yes, $$\infty \times \frac{1}{\infty} = \infty \div \infty = \infty$$. But if you start with the premise that $$\infty + \infty = 2 \times \infty$$ a different conclusion follows.

The important thing is that, whatever premises you pick, the mentioned Wikipedia proof is invalid. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Mowk wrote:You have added two fractions together from the same decimal position. 0.9 recurring still states there are zero 1s within it's notation.

I want very badly to dismiss this argument, but I am forced to admit it's stronger than I hoped. The best response I have, which is not much response at all, is that it is question-begging: if $$.\dot9 = 1$$, then it isn't true that there are zero 1s when there are 9s in every decimal place.

Magnus Anderson wrote:Somehow, you forgot the fact that $$0.\dot9$$ is an infinite sum, which makes it an infinite number of, non-zero terms.

I think we need to be careful with our language. "An infinite number" is not the same as "the number 'infinity'". The former treats infiniteness as a property of multiple numbers, the latter treats infinity as a number.

This might seem pedantic, but I think it matters. If there's more than one infinity, if it can be a property of a number but not a number itself, then the equations using $$\infty$$ as a number are at best ambiguous, so our answers will be uninsightful.

Magnus Anderson wrote:If you say that $$9.\dot9$$ and $$0.\dot9$$ have the same number of $$9$$'s following the dot, then it follows that
$$9.\dot9\ - 0.\dot9 = 9$$. That's doing arithmetic with infinites.

It isn't. Both $$9.\dot9$$ and $$0.\dot9$$ are finite numbers.

Magnus Anderson wrote:It's pretty clear that infinites and finites overlap. An infinite number of zeroes is a zero. One raised to infinity is one. And so on. If you can calculate the result of $$\infty \times 1$$ what's the reason you can't calculate the result of $$\infty - \infty$$?

I think those answers seem intuitive, but I think they're playing fast and loose with notation that has a formal definition that makes those equations meaningless in a rigorous sense. Using the symbols that way is a sort of expressive math, an uncertain poetry built of the symbols of rigorous expression.

Magnus Anderson wrote:That's just you not willing to play the game.

I'm willing to talk about what $$\infty - \infty$$ might mean, I just don't think it's well-defined ("meaningless" might be too strong). $$\infty$$ can refer to multiple quantities that are provably distinct. I'm not sure what the subtraction symbol means here, given that I don't think $$\infty$$ is a number. My most natural reading is set subtraction.

So let's play, but I don't want to play calvinball.

Magnus Anderson wrote:How about you place $$i$$ in one of the categories that existed before people invented the category of complex numbers? How useful is that?

But we agree that "complex numbers" is not a category, right? Are you defining a new category with $$L$$?
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Carleas
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### Re: Is 1 = 0.999... ? Really?

Mowk wrote:
You have added two fractions together from the same decimal position. 0.9 recurring still states there are zero 1s within it's notation.

I want very badly to dismiss this argument, but I am forced to admit it's stronger than I hoped. The best response I have, which is not much response at all, is that it is question-begging: if .˙9=1, then it isn't true that there are zero 1s when there are 9s in every decimal place.

It is the diminishing value of each place that is filled. I don't know how to notate it but it's the .1 in an infinite number of decimal places where it comes up short to clear the decimal point and increment the digit on the "whole" side of the decimal. In it's form, .9 recurring; it is presented as a fraction and I don't think infinity can change that. It may simply be the largest fraction possible. And with little more then a place holding decimal point between them they could be equal. A fraction turns into a whole through infinite recursion. Fraction or integer?
Last edited by Mowk on Sat Mar 28, 2020 7:39 am, edited 2 times in total.
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### Re: Is 1 = 0.999... ? Really?

My point was that the division works as expected: the long division algorithm produces 0.499999...

And what happens when you multiply .49 x 2, would the result not be the same in an infinite number of decimal places? Try the operations reciprocal.

1 x 2 = 2 and 2 / 2 =1. If .9 recurring is equal to 1 then the math should be that simple.

When you can invoke Murphy's law in math? You're having fun.
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### Re: Is 1 = 0.999... ? Really?

Carleas wrote:I think we need to be careful with our language. "An infinite number" is not the same as "the number 'infinity'". The former treats infiniteness as a property of multiple numbers, the latter treats infinity as a number.

And what exactly is the effective difference?

$$\infty - \infty$$ is the same as $$(1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)$$ which is in the same camp as $$(0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots)$$. If you're going to say that the former is meaningless, what allows you to say that the latter is not? Or vice versa, if you're going to say the latter is equal to $$0$$, what allows you to say that the former is meaningless?

Meaningless: that which has no meaning. When speaking of symbols (such as words) it refers to symbols that have no meaning assigned to them. By whom? Well, by someone, usually the one using those symbols. In most cases, it's a way for people to avoid bothering to understand what the other side is trying to communicate, as in "Look, I can't bother to understand what you're trying to say, it's too difficult and/or time-consuming for me, so I'm just gonna conclude that you're not saying anything, that your statements have no meaning whatsoever, that they mean nothing, that they are meanignless".

You can do arithmetic with $$0.\dot9$$ but you can't do arithmetic with $$1 + 1 + 1 + \cdots$$? Really? You can do arithmetic with convergent series but you can't do arithmetic with divergent series? W-why? Is it perhaps because you're REFUSING to do so? As in, you don't wanna do it because it does not support your present conclusions?

This might seem pedantic, but I think it matters. If there's more than one infinity, if it can be a property of a number but not a number itself, then the equations using $$\infty$$ as a number are at best ambiguous, so our answers will be uninsightful.

Look closer at what you're saying.

You're saying that $$(0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots) = 0$$ makes PERFECT SENSE but that $$(1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots) = 0$$ makes NO SENSE.

It isn't. Both $$9.\dot9$$ and $$0.\dot9$$ are finite numbers.

Are they infinite sums of non-zero terms or finite sums of non-zero terms? They can't be both. Their RESULT can be a finite number, sure, just like how the result of $$1 \times 1 \times 1 \times \cdots$$ is a finite number, but they are nonetheless expressions involving an infinite number of terms. The entire point is that WE'RE DOING ARITHMETIC WITH AN INFINITE NUMBER OF NON-ZERO TERMS. And if we can do it in some cases (such as with $$0.9 + 0.09 + 0.009 + \cdots$$) why can't we do in other cases (such as with $$1 + 1 + 1 + \cdots$$)?

I'm willing to talk about what $$\infty - \infty$$ might mean, I just don't think it's well-defined ("meaningless" might be too strong).

You think that $$(1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)$$ is not well-defined and at the same think that $$(0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots)$$ is well-defined?

But we agree that "complex numbers" is not a category, right?

How are complex numbers not a category? Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Mowk wrote:It is the diminishing value of each place that is filled. I don't know how to notate it but it's the .1 in an infinite number of decimal places where it comes up short to clear the decimal point and increment the digit on the "whole" side of the decimal. In it's form, .9 recurring; it is presented as a fraction and I don't think infinity can change that. It may simply be the largest fraction possible. And with little more then a place holding decimal point between them they could be equal. A fraction turns into a whole through infinite recursion. Fraction or integer?

It is an edge case, and so when it's put this way it seems very ad hoc. But this is kind of an ad hoc way of putting things.

I have a related question: is $$0.\dot9$$ rational or irrational? If it's rational, what's the ratio?

Magnus Anderson wrote:And what exactly is the effective difference?

I can think of a few. The biggest is that "an infinite number" is different from "the infinite number '$$\infty$$'": the former implies multiple infinities.
Another is that numbers have properties and operations, and I don't think any standard operations have an obvious meaning on infinite numbers, and I'm not sure what properties apply or what we could use to infer those properties.

Magnus Anderson wrote:$$\infty - \infty$$ is the same as $$(1+1+1+⋯)-(1+1+1+⋯)$$

Why? Why not the same as $$(2+2+2+...) - (3+3+3+...)$$? Especially considering that you probably don't think $$(1+1+1+⋯) = (2+2+2+...)$$, why should we pick one or the other as a stand-in for $$\infty$$?

So I guess my answer to your question is that I don't think $$\infty - \infty$$ means the same thing as $$(1+1+1+⋯)-(1+1+1+⋯)$$.

Magnus Anderson wrote:In most cases, it's a way for people to avoid bothering to understand what the other side is trying to communicate, as in "Look, I can't bother to understand what you're trying to say, it's too difficult and/or time-consuming for me, so I'm just gonna conclude that you're not saying anything, that your statements have no meaning whatsoever, that they mean nothing, that they are meanignless".

Though I later acknowledge that "meaningless" is perhaps too strong, here I mean it as saying that just because you can string some symbols together doesn't mean that they express a coherent concept. A "square circle" is meaningless in the sense that, even though the words that compose the phrase are perfectly meaningful, the phrase doesn't point to a coherent concept.

Magnus Anderson wrote:You can do arithmetic with convergent series but you can't do arithmetic with divergent series?

2 points:
1) $$.9$$ can be expressed as a convergent series, but it isn't a convergent series.
2) A divergent series is undefined in the limit. So we can do arithmetic with the limit of a convergent series and not with the limit of a divergent series, because we can do arithmetic with things that are defined and not with things that are undefined.

Magnus Anderson wrote:Are they infinite sums of non-zero terms or finite sums of non-zero terms?

They are the finite limits of infinite sums. There is no tension at all there. And again, they can be expressed infinite sums, but they are numbers. I'm about 95% sure that every number can be expressed as multiple infinite sums.

Magnus Anderson wrote:How are complex numbers not a category?

Sorry, typo, should have said "'complex numbers' is a category". I was replying to your rejection of identifying what kind of number $$L$$ is, and you offered the argument that $$i$$ does not fit into any pre-complex-numbers categories. But we agree that complex numbers are a category of numbers. So $$L$$ too should fit into some category. So you should be able to say that it's in some category, or it's some new number like $$i$$ was and it needs a brand new category of numbers, and you have some rigorous definitions and properties of numbers like $$L$$ (or even some vague, hand-wavey, there-should-be-something-$$L$$-shaped-in-roughly-this-vicinity type answer).
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### Re: Is 1 = 0.999... ? Really?

Carleas wrote:Why? Why not the same as $$(2+2+2+...) - (3+3+3+...)$$? Especially considering that you probably don't think $$(1+1+1+...) = (2+2+2+...)$$, why should we pick one or the other as a stand-in for $$\infty$$?

You can pick any.

So I guess my answer to your question is that I don't think $$\infty - \infty$$ means the same thing as $$(1+1+1+...)-(1+1+1+...)$$.

So you don't think that $$1 + 1 + 1 + \cdots = \infty$$?

Obviously, you refuse to understand the meaning that your interlocutors assign to the symbol $$\infty$$ so maybe I should just stop using it, instead substituting it with an infinite series.

What does $$(1 + 1 + 1 + \cdots) - (1 + 1 + 1 + \cdots)$$ equal to? Is it $$0$$ or is it $$1 + 1 + 1 + \cdots$$?

If you say it's $$0$$ then the second part of this post is irrelevant to you and so there is no need for you to respond to it.

Though I later acknowledge that "meaningless" is perhaps too strong, here I mean it as saying that just because you can string some symbols together doesn't mean that they express a coherent concept. A "square circle" is meaningless in the sense that, even though the words that compose the phrase are perfectly meaningful, the phrase doesn't point to a coherent concept.

"Square circle" is a logical contradiction. Are you saying that $$\infty - \infty = \infty$$ is a logical contradiction? I don't see why. Remember that I am not the one claiming that $$\infty - \infty = \infty$$. On this forum, that would be Phyllo. Outside of the forum, that would be mathematicians.

$$0.\dot9$$ can be expressed as a convergent series, but it isn't a convergent series.

That's like saying $$2 + 2$$ can be expressed as $$4$$ but it's not $$4$$. In a way, it's true. They are two different expressions. They aren't the same expression. But also, they are expressions representing the same thing. They are two different expressions representing the same idea. In that sense, $$0.\dot9$$ and $$0.9 + 0.09 + 0.009 + \cdots$$ are two different expressions representing the same idea.

2) A divergent series is undefined in the limit. So we can do arithmetic with the limit of a convergent series and not with the limit of a divergent series, because we can do arithmetic with things that are defined and not with things that are undefined.

The Wikipedia proof I constantly quote does not do arithmetic with the limit of convergent series. It does arithmetic with convergent series.

$$x = 0.999\dotso$$

Then they multiply both sides by $$10$$.

This gives them:
$$10x = 10 \times 0.999\dotso$$

Then they substitute $$0.999\dotso$$ with $$0.9 + 0.09 + 0.009 + \dotso$$ telling us that the number of terms in the sum is $$\infty$$.

$$10x = 10 \times (0.9 + 0.09 + 0.009 + \dotso)$$

How do they calculate the result of the right side of the expression?
Certainly not by asking "What's the limit of $$0.9 + 0.09 + 0.009 + \dotso$$?"

What they do is they multiply every term by $$10$$, like so:
$$10x = 10 \times 0.9 + 10 \times 0.09 + 10 \times 0.009 + \dotso$$

$$10x = 9 + 0.9 + 0.09 + 0.009 + \dotso$$

Note that the number of terms in the resulting sum is the same as before: $$\infty$$.

Then we "split off" the integer part:
$$10x = 9 + 0.999\dotso$$

The number of terms in the resulting $$0.999\dots$$ is $$\infty - 1$$ since the resulting sum is produced by removing one term from the previous sum. $$\infty - 1$$, they tell us, equals to $$\infty$$, which makes the number of terms the same as before. However, saying that $$\infty - 1 = \infty$$ means that $$\infty - \infty \neq 0$$ making $$\infty$$ a non-specific number.

Thus, we can substitute the resulting $$0.999\dotso$$ with $$x$$, like so:
$$10x = 9 + x$$

Then we subtract $$x$$ from both sides:
$$10x - x = 9 + x - x$$

And this is where the problem lies. We don't get what they tell us we get.

We don't get $$9x = 9$$ because $$x - x$$ is not $$0$$ since $$x$$ represents a non-specific number.

AN INFINITE NUMBER OF NON-ZERO TERMS MINUS AN INFINITE NUMBER OF NON-ZERO TERMS IS NOT ZERO NON-ZERO TERMS IF WE ALREADY ACCEPTED THAT INFINITY MINUS INFINITY IS NOT ZERO.

THAT'S THE ENTIRE POINT. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:You can pick any.

... So $$1+1+1+\cdots = \infty = 2+2+2+\cdots = 2\times\infty$$?

But $$\infty -1 \neq \infty$$?

Magnus Anderson wrote:Obviously, you refuse to understand the meaning that your interlocutors assign to the symbol $$\infty$$[...]

It is charitable to assume that I'm capable of understanding, but uncharitable to assume my interlocutors have made their meaning clear.

Magnus Anderson wrote:What does $$(1+1+1+\cdots)-(1+1+1+\cdots)$$ equal to?

I don't know. I agree it seems intuitive, but I think that's deceptive. By associativity, $$1+1+1+\cdots = (1+1)+(1+1)+\cdots = 2+2+2+\cdots$$
$$= 1+(1+1)+(1+1+1)+\cdots= 1+2+3+4+5+\cdots$$ And I have it on good authority that that last expression is equal to $$-\frac{1}{12}$$, but I don't know why.

And $$(1+1+1+\cdots)-(1+1+1+\cdots)$$ could mean taking each term in the one on the right and subtracting it from the one on the left. But normal algebraic manipulation also permits a pattern of $$1+1-1+1+1-1+1+1-1+\cdots$$.

I also take your point that this has implications for $$9.\dot9-9$$, but it is noteworthy that this doesn't have similar problems. There's no way to change the outcome by mixing different parts of the series, because the parts don't overlap: $$9 + .9 + .09 -.9 + .009 + .0009 - .09 + \cdots$$ That difference seems significant, what do you make of it? I have a suspicion that this is related to the divergent/convergent distinction (e.g. that the formal way of saying that is true for convergent series and not for divergent series; that is a weakly held belief, like 65% confidence).

Magnus Anderson wrote:"Square circle" is a logical contradiction. Are you saying that $$\infty-\infty=\infty$$ is a logical contradiction?

I'm saying that it's meaningless in the sense of not pointing to a coherent concept. Another example that doesn't use a logical contradiction would be "the number of dreams it takes to power a Saturn V rocket". All the words are meaningful, but they don't point to a coherent concept.

Magnus Anderson wrote:In that sense, $$0.\dot9$$ and $$0.9+0.09+0.009+\cdots$$ are two different expressions representing the same idea.

Right, but the idea and the expression are distinct. That you can use an infinite series to express the idea does not make the idea infinite. Or, if we want to insist that it does, then it's true of any number that can be expressed as an infinite series, which I'm pretty sure is any number. And so in that case there is no distinction being made.

Magnus Anderson wrote:The Wikipedia proof I constantly quote does not do arithmetic with the limit of convergent series. It does arithmetic with convergent series.

This is not my understanding of that proof.

To give an analogy for the argument I understand you to be making: the number 2 can be expressed as the product of two complex numbers $$-2i\times i$$. So you point out that $$0.\dot9$$ can be expressed as an infinite series, and so you treat any math using $$0.\dot9$$ as "do[ing] arithmetic with covergent series", but I wouldn't call the expression $$2+2$$, "do[ing] arithmetic with [complex numbers]".

Magnus Anderson wrote:Then we subtract $$x$$ from both sides:
$$10x−x=9+x−x$$

And this is where the problem lies. We don't get what they tell us we get.

We don't get $$9x=9$$ because $$x−x$$ is not $$0$$ since $$x$$ represents a non-specific number.

Am I to understand that you don't think any convergent series equals an integer? Otherwise this is completely breaking of like all of algebra.
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Carleas
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### Re: Is 1 = 0.999... ? Really?

It is no mere hyperbole to assert that the seemingly moat simple things are the most complicated?

There must be a transformation at the end of the line, such as was described by meno, in his paradox , such as later treats partial differentiations in a reassamblad Fourier transformations; where the transformation encompasses matter/energy mutations?
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### Re: Is 1 = 0.999... ? Really?

The question is what's the procedure by which we find the result of subtracting one infinite sum from another. There must be such a procedure, right? Otherwise, how can we say that $$0.\dot9 - 0.\dot9$$ is $$0$$?

Given $$(a_1 + a_2 + a_3 + \cdots) - (a_1 + a_2 + a_3 + \cdots)$$, how do we calculate the result? And you seem to be arguing my case (: Do we pair the terms like this $$(a_1 - a_1) + (a_2 - a_2) + (a_3 - a_3) + \cdots$$ resulting in $$0$$ or do we pair them like this $$(a_1 + a_2 - a_1) + (a_3 + a_4 - a_2) + (a_5 + a_6 - a_3) + \cdots$$ which may be non-zero say if every $$a_n$$ is equal to $$1$$. And to make it worse, the number of ways we can pair the terms seems to be infinite . . . so which way is the right way?

How can you say that $$0.\dot9 - 0.\dot9\ = 0$$ without having an answer to that question? Consider that $$(0.9 + 0.09 + 0.009 + \cdots) - (0.9 + 0.09 + 0.009 + \cdots) = 0$$ only and only if we pair the terms like so: $$(0.9 - 0.9) + (0.09 - 0.09) + (0.009 - 0.009) + \cdots$$. But as you say, we can also pair the terms like so $$(0.9 + 0.09 - 0.9) + (0.009 + 0.0009 - 0.09) + (0.00009 + 0.000009 - 0.009) + \cdots$$ in which case the result isn't zero but approaching zero. And this is a serious problem. It would mean we're trying to prove that $$0.9 + 0.09 + 0.009 + \cdots = 1$$ by taking for granted that $$0.09 - 0.0801 - 0.008901 - \cdots = 0$$. That would make it . . . not much of a proof. And certainly, noone ever mentioned this anywhere in the proof. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

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

Meno_ wrote:There must be a transformation at the end of the line

But there is no end of the line.

Magnus Anderson wrote:The question is what's the procedure by which we find the result of subtracting one infinite sum from another. There must be such a procedure, right? Otherwise, how can we say that $$0.\dot9−0.\dot9$$ is $$0$$

Because $$0.\dot9$$ isn't an infinite sum, any more than $$4$$ or $$\pi$$ is an infinite sum. $$0.\dot9$$ is a real number, and 0 is the additive identity, so $$0.\dot9-0.\dot9=0$$

Magnus Anderson wrote:That would make it . . . not much of a proof.

Yes. Also not much of a disproof.

Mowk wrote:Do you think the answer is going to change just cause you've asked it an infinite number of times?

But we aren't doing something repeatedly, and nothing is changing. I think this is unintuitive but crucial: $$0.\dot9$$ is not changing, it isn't moving, it isn't approaching; it is static.

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

Opps, was hoping to get back to edit before someone responded.

So the argument was in evaluation of the question 1 = 0.999... ? Really? And I am continuing with decimal notation.

All you have to question is 9/10ths = 1? and the answer is no. Poof, an infinite number of decimal places filled with not equals.

Do you think the answer is going to change just cause you've asked it an infinite number of times?

The quoted part is the result of the rephrasing.

So sorry, I didn't mean to imply any thing other then our minds were changing.
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### Re: Is 1 = 0.999... ? Really?

In other words change the question from a math question to a true false question.

Is 0.9 equal to 1. False
Is 0.99 equal to .1 False
Is 0.999 equal to .01 False

This is a pattern that is represented infinitely by the expression 0.9 recurring = 1, and the answer is... it is infinitely false that 1 = 0.9 recurring. From the first decimal position to the infinite, all at once.

What happens when logic and math disagree?

And I respect Magnus for smelling something fishy in the mathematical proofs. This idea of the infinite and how it is applied, is mathematically slippery.
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### Re: Is 1 = 0.999... ? Really?

"But there is no end of the line."

The end of the line rests on the argument that has expanded from meno through the geometers through the conical hyperbole of calculation, which lead to the quantic/uncertainty generated partially reconstructed Fourier series which defined the MC2 transformations.

In fact, it could not have been surmised inductively, anyway.

Meno was right, and Leibnitz through Kant showed metaphysics to have been shut reductively

So how the connect, the touted synthesis?

Maybe Russel's sense data is right on point, after all, as right as the well repeated proposition:
'if God does not exist, he had to be created'

The point about transformation is significant because in a sense, something does come from an appearent Nothingness, and that is the point to the underlying functional math analysis , where partial differentiations are subsumed by the calculus of indiscernibles.

So the 'end of time's is inherent in the declaration that -(.99=1.00) ; because of the same partially derived reconstruction by Fourier transform, validates such a pro-position.

The end is transformed from a logical level, (Russel-Wittgenstein) to one that is inductively not reduceable, is subsumed by the differential levels of quantifiability.

Put it in linear language, upon whose architecture the modality of it has to accorded to: simply: the end(s) justify the means of the reconstruction- that, the original transformation can be signified,

The transformed end can not be defined within but without recourse to a reconstructed generation of progression: from meno through Leibnitz/ Kant, all the way to the missing key of Principia Mathematica:

The 'sense data'.

This hotly contested concept, overcomes the epoche within which liberal arts still vanquishes, unable to mimic its more quantitatively able cousin , the human brain.

The end of any road is the preintegrated calculus of missing pieces minus the ones left unfilled, but such a presumption fails, unless a transformative filler can replace it.

So, the end of the road could not be in a one dimensional map, but does exist in the calculable certainty in the existence of the perfect monad.

Without that, atomism would/ could not have come about signaling the need: to overcome it.

I shared this with St.James on occasion, and I think there was some concurrent albeit partially limited agreement, there.
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### Re: Is 1 = 0.999... ? Really?

The transformed end can not be defined within but without recourse to a reconstructed generation of progression: from meno through Leibnitz/ Kant, > and all the curious minds asking the questions here at ILP, and everywhere < all the way to the missing key of Principia Mathematica: >and beyond <

Come on everyone... in your best Buzz Lightyear impersonation.

Amen.

I have just changed my vote from other to no; pseudo confidently.

Curious if the argument has come up else where in the discussion.
Mowk
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### Re: Is 1 = 0.999... ? Really?

Carleas wrote:Because $$0.\dot9$$ isn't an infinite sum

Really? Are you telling me that $$0.\dot9 \neq 0.9 + 0.09 + 0.009 + \cdots$$?

Note: I am well aware of the fact that $$0.\dot9$$ itself is not an infinite sum of numbers -- not even a sum. It's just a single number. The point is that it's a number that can be represented as an infinite sum of non-zero terms. That's the entire point and the only thing that matters.

any more than $$4$$ or $$\pi$$ is an infinite sum.

$$4$$ is not an infinite sum. It's not a finite sum either. That's because it's not a sum. It's a single number. However, it is equal to a finite sum such as $$2 + 2$$ and an infinite sum such as $$2 + 2 + 0 + 0 + 0 + \cdots$$. This means you can substitute every occurence of $$4$$ with $$2 + 2$$ or $$2 + 2 + 0 + 0 + 0 + \cdots$$.

How do you know that $$10 \times 0.\dot9 = 9.\dot9$$? What's the procedure? You avoid this sorts of question like a plague.

$$0.\dot9$$ is a real number, and 0 is the additive identity, so $$0.\dot9-0.\dot9=0$$

By definition, $$0.\dot9$$ is a bit more than you'd like to admit. Yes, it is a real number but . . . it's also an infinite sum of non-zero terms, specifically, $$9 \times 10^{-1} + 9 \times 10^{-2} + 9 \times 10^{-3} + \cdots$$. Indeed, every real number can be represented as an infinite sum but not every real number can be represented as an infinite sum of non-zero terms. $$1$$, for example, can be represented as $$1 + 0 + 0 + 0 + \cdots$$. It's an infinite sum (i.e. there is an infinite number of terms) but the number of non-zero terms is finite.

Yes. Also not much of a disproof.

So you admit that Wikipedia proof is not much of a proof? Good. At least we agree on something. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

James S Saint
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Sun Jun 26, 2016 12:05 am Post
Carleas wrote
In the standard set of real numbers, excluding the hyperreals, do you agree that .999... = 1?

Even more certainly not.

If allowed to proceed to absolute zero (the limit of the hyperreals), it might be arguable that the number string reaches that absolute zero difference between itself and 1.0. But if confined to the standard reals, the string cannot get even down below a single order infinitesimal difference (the rule being that you either cannot divide an infinitesimal or that if you divide an infinitesimal, you get the same infinitesimal remaining). So in the standard reals, there is always a first order infinitesimal difference between the two numbers.

The deeper truth is that even if including the hyperreals, the very definition of "0.999..." still forbids any remaining difference to ever be totally consumed into the accumulated sum. Thus by definition, there must always be a difference between the two numbers.

The "..." simply means, "you can't get to the limit from here".

.. Sorry to have to be so stubbornly affixed. ---------- ------------ ----- ------------ ---------

{Arguably, the reasoning becomes circular, if no allusion can be made to be a priority.} The circular becomes both: an infinite repetition of what finally is defined as tautological.
The difference between .99999=1 & -(.9999=1) overcomes it
(The tautology) by transforming the function of differentiation. The same way that real numbers can not contain hyperreal numbers, yet hyperreal can contain real numbers.

No presumptions intended here, except philosophical ones, for which I need broadly to review this forum.
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### Re: Is 1 = 0.999... ? Really?

What happens when logic and math disagree?

This is one of those issues that display the clear distinction between a good philosopher and a expert mathematician.

Well James S Saint. From this point on, I would like to include myself within the ranks of a "good" philosopher, even if I suspect you beat me to the punch... with your permission. I am a lousy mathematician that believes it accomplishes a great many results. But that mono polar magnetic thing?

And WHY?

Note that you can change your poll-vote later if you wish.

For one you gotta respect a question formed through belief? In both directions. Will leave it up to you all whether philosophy is more capable than mathematics.
Last edited by Mowk on Mon Mar 30, 2020 7:00 pm, edited 1 time in total.
Mowk
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### Re: Is 1 = 0.999... ? Really?

I want very badly to dismiss this argument, but I am forced to admit it's stronger than I hoped. The best response I have, which is not much response at all, is that it is question-begging: if .˙9=1, then it isn't true that there are zero 1s when there are 9s in every decimal place.

Different from Uciscore, Urwrongx1000 and a bully mentality, your response offered encouragement and insight to question further. I respect that as grace. Thank you.
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### Re: Is 1 = 0.999... ? Really?

So we have two arguments that challenge the truth of the statement.

The convention of decimal notation, the expression itself informs; with a zero to the left of the decimal point, there are zero 1's. 0 x 1 is not equal to one.

is that it is question-begging: if .˙9=1, then it isn't true that there are zero 1s when there are 9s in every decimal place.
This challenge comes from the perspective that .9 recurring equals 1 is a true statement. The decimal notation "0.9 recurring" declares in it's own notation, the falseness of the claim.

and

The logic which recognizes the expression 0.9 recurring = 1, represents an infinite falseness in it's pattern of recurrence as result of offsetting decimals.
0.9 does not equal 1.0, 0.99 does not equal .1, 0.999 does not equal .01 ... an infinite string of offsetting decimals that are not equal. From the first to the infinitive.

Interesting that mathematics can "prove it" >within a convention<, while both the expression itself and logic say it just ain't so. Likely, it has something to do with what is being "accepted" as a mathematical "truth" out of convention.

When an equals sign is used to relate two values to each other the claim is made that it is true there is value equivalency. The two arguments presented establish that it's truth is falsifiable.

ILP's very own offset decimal paradox.
Mowk
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### Re: Is 1 = 0.999... ? Really?

$$0.999...$$ represents a number that is larger than every number of the form $$0.9_{1}9_{2}9_{3}...9_{n}$$ (where $$n \in N$$) but smaller than $$1$$.

Interestingly, whether or not such a number exists is completely irrelevant. That's because if it does not exist, then it's not $$1$$, since $$1$$ exists. Remember that the subject of this topic is whether or not $$0.999\dotso$$ is the same number as $$1$$.

Pretty much EVERY single person on this board defending the widely accepted position has been forced to argue that the two numbers, $$0.999\dotso$$ and $$1$$, are MERELY APPROXIMATELY EQUAL. And noone disputes this. But approximately equal is not the same as strictly equal. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

0.9 recurring; the largest possible fraction that isn't a whole. 1 - 0.9 recurring would be the smallest possible fraction greater than zero, leaving all possible fractions existing between None and 1.
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