## 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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42%
No, 1 ≠ 0.999...
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48%
Other
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:It should be quite obvious that for $$f(x)=0$$, $$f(x)$$ represents the number $$0$$. It also represents a function - more specifically the "way" (a doing) to get to the number (a being) that it already represents. Depending on the number in the set with which there is this binary relation - as in your copy/paste from Wikipedia - the function might represent a different number, but it represents a number regardless.

I really don't see any point in continuing this, you're not going to get it, nor ask questions if you don't agree, which you won't.

Maybe you don't see the point in continuing this discussion but I do.

Consider a word such as "bow". The word "bow" has multiple meanings. Here are two of them:

1) a long wooden stick with horse hair that is used to play certain string instruments such as the violin

2) a weapon to shoot projectiles with (e.g. a bow and arrow)

Here's a question:

Does this mean that a long wooden stick that is used to play certain string instruments such as the violin is the same thing as a weapon used to shoot projectiles with?

Of course not. They are two different things.

Similarly, $$f(x) = 0$$ might have more than one meaning. For example:

1) a function that maps every natural number to $$0$$

2) $$0$$

Does that mean that a function that maps every natural number to $$0$$ is the same thing as $$0$$?

Of course not. They are two different things.

In one sense, functions are not numbers. In another sense, functions are numbers. In this other sense, the word "function" means no more than "number". This is not what the word "function" normally means and what it originally meant. It's a subsequent development. And this kind of thing (taking existing words that have an established meaning and then giving them a different meaning) can lead to equivocation. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

wtf wrote:
iambiguous wrote:But the problem here for mathematically challenged dummies like me, is that I can't connect the dots between the arguments and the world of actual material/phenomenal interactions.

That's because there is no connection!

There is of course this age-old debate:

https://www.sciencefocus.com/science/wa ... iscovered/

And, if you google mathematics and nature, there are many scholarly inputs --- attempts to at least connect the dots:

So, aside from it being a "philosophical mystery", I'm not entirely sure what you mean here by "no connection". But, as you note, out in the either/or world there seem to be countless connections.

Though I suspect that here we may be getting closer to the points that folks like Wittgenstein raised. The relationship between words and worlds. Language itself.

Besides, just on a day to day practical level, it's important to accept that if John owns two horses and buys two more, he now has four horses and not five.

But 1 horse vs. .999... horse? That's the part I can't wrap my head around. Though I am perfectly willing to acknowledge that this revolves far more around my ignorance in regard to mathematics on this level.

No, where dasein comes in for me is when someone will insist that no one should be permitted to buy and sell horses. Period. That owning animals is immoral.

Indeed, I know someone who insists that this includes cats and dogs.
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles

Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382 iambiguous
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:I understand very well that addition is a function that inputs two numbers and outputs a third number.

However, I don't see how such a definition (together with other relevant definitions) isn't enough for us to logically derive the meaning of the expression that is $$0.9 + 0.09 + 0.009 + \cdots$$.

For purposes of discussion, suppose I grant you that point. Then show me any such logical derivation.

Previously you've said that no definition is needed because the meaning is known to everyone or obvious or intuitive or whatever, I apologize if I'm not quoting you perfectly but this is the feeling I get.

But now you are saying is that yes, you AGREE that infinite sums aren't defined UNTIL we LOGICALLY DERIVE their meaning.

So, I do ask you to simply outline such a logical derivation.

What do you think modern real analysis is, other than a logical derivation from first principles of what limits and infinite sums are?

Magnus Anderson wrote:
How does the conclusion (that addition is not defined for infinite sums) follow from the premise (that addition is a function that takes two numbers and returns a single number)? Where's the link?

Addition is defined ONLY for finite sums. One of the axioms for the real numbers says that:

* The sum of any two real numbers is a real number. In symbols, if $$x$$ and $$y$$ are real numbers, then so is $$x + y$$.

So when we see an expression, or string of symbols such as $$x_1 + x_2 + x_3 + \dots$$, we have no rules with which to untangle or parse that string. If it's the sum of 47 elements, it's the sum of the first 46 plus the 47th, and that's an instance of adding only two things. So by induction, any finite number of additions are possible.

But the theory says nothing about infinite sum. No meaning is ascribed to them. We need additional ideas and formalizations, namely the modern theory of real analysis.

Of course if you have a different way of doing it, I'm all ears. I have said that I am openminded. I'm not defending mathematical orthodoxy, only describing it. If you have another way to define infinite sums, let's hear it.

If you have a vending machine that lets you put in two coins, you'll break it if you try to stuff in infinitely many coins.

Magnus Anderson wrote:Hopefully, you are NOT saying that just because addition is a function that takes TWO inputs and not AN INFINITE NUMBER of inputs it follows that addition is not defined for infinite sums.

LOL. But that's exactly what I'm saying. And if you disagree, then tell me what is the definition of an infinite sum, based on the axiom that the sum of any TWO real numbers is a real number?

Magnus Anderson wrote:Using that kind of logic, one can also conclude that addition is not defined for any sum that involves any number of terms other than $$2$$ which you agree is not the case.

A simple inductive argument serves to generalize addition to any finite number of summands, as I outlined a couple of paragraphs ago.

It's very much like parsing a computer language. If we see x + y + x + w we know we can parenthesize (x + y + z) + w to reduce the problem to a sum of two things, which we have a procedure for.

But if we have an infinite sum, the procedure never terminates. We do not have a computable function that allows us to determine the sum of infinitely many terms.

Do you see that? A machine could compute 1/2 + 1/4 + 1/8 by continually breaking it down into smaller sums, a process that is guaranteed to terminate. But if the input is an infinite sum, the evaluation never terminates and no such sum can be reached. We do not have a procedure or algorithm to calculate an infinite sum that is guaranteed to terminate after finitely many steps.

That's why we need a new idea. The idea of limits. The theory of limits goes beyond the theory of computation, you can look at it that way. Indeed, the difference between arithmetic and analysis is exactly that. Arithmetic is everything you can do with finite operations. Analysis deals with infinitary operations.

In fact, finite addition can be defined in Peano arithmetic. The theory of limits requires the axiom of infinity.

You are failing to appreciate this conceptual leap. Finite sums do not require the axiom of infinity. Infinite sums do.

Magnus Anderson wrote:The point was to make my position clear and visible so that other people can address it and not address a strawman.

Then by all means restate your thesis; rather than provide argument or evidence.
Last edited by wtf on Mon May 25, 2020 1:57 am, edited 2 times in total.
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:To wtf:

Here's a hypothesis as to what you're trying to do.

What you're trying to do here is you're trying to force the expression to be something other than it is.

You are asking "What terminating decimal is equivalent to this non-terminating decimal?" The answer to this question is "None". But you can't accept this because you find it unsatisfying. You want it to be something else because you have a need for something else.

Consider a different example.

You want to know the result of $$5 \div 2$$ but you want it to be an integer. The result is not an integer but you nonetheless want it to be an integer. You know that $$5 \div 2$$ is $$2 + 5 \div 10$$ but since you want the result to be an integer you need to do something about it.

That's not an example of anything. The integers are a ring. In a ring you can add, subtract, and multiply, but not necessarily divide.

If you enlarge the ring to a field, then you can always divide (except by 0). This is perfectly well understood. Your post was a little off the mark IMO.

Now if one is interested in integer division, we know that 5/2 is 2R1; that is, quotient 2 and remainder 1. That's the domain of number theory.

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

iambiguous wrote:So, aside from it being a "philosophical mystery", I'm not entirely sure what you mean here by "no connection". But, as you note, out in the either/or world there seem to be countless connections.

Oh yes of course. Better if I'd phrased it "not necessarily any known connection." I only meant that people trying to understand .999... should not try to connect it with physical reality, since in general math isn't. I didn't mean to discount the entire philosophy of the subject. That was the philosophical mystery part.

iambiguous wrote:Though I suspect that here we may be getting closer to the points that folks like Wittgenstein raised. The relationship between words and worlds. Language itself.

I don't know much about his work, and what little I've read I didn't understand. I've heard that he was wrong about a couple of mathematical things.

iambiguous wrote:Besides, just on a day to day practical level, it's important to accept that if John owns two horses and buys two more, he now has four horses and not five.

I am a Platonist during the week and a Formalist on weekends. I did not mean to deny the "unreasonable effectiveness," nor the fact that mathematics came out of the markets of the ancient world and seems like part of nature, or at least part of our own nature.

iambiguous wrote:But 1 horse vs. .999... horse? That's the part I can't wrap my head around. Though I am perfectly willing to acknowledge that this revolves far more around my ignorance in regard to mathematics on this level.

Oh I see. In connection with this thread I've been reviewing the history of infinite sums. Newton was proficient at using infinite sums to solve problems in physics and math. Infinite sums and infinitary processes came into math and science around that time. But you can't apply the real numbers to physical measurements, that's well known. So nobody is saying that .999... horses is one horse. That's not true. What's true is that the real number .999... and the real number 1 are the same real number.

You've touched on the question of whether the integer 1 is the same thing as the real number 1. At best they seem to be two aspects of the same thing that are used for different purposes. Yet their nature is very different, as you note. The discrete and the continuous. Back to Democritus.
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Silhouette wrote:
Similarly, $$f(x) = 0$$ might have more than one meaning. For example:

1) a function that maps every natural number to $$0$$

2) $$0$$

Does that mean that a function that maps every natural number to $$0$$ is the same thing as $$0$$?

Of course not. They are two different things.

They are exactly the same thing. The function that is identically zero is often taken as the zero element in a ring of functions. Would you call that a number or not call it a number? In math there is no universal definition of number!! So you really can't say.

Zero is just the zero element in a group or ring. Whether it's the zero function in the ring of continuous functions; or the zero of the integers; it's just zero. In Platonic heaven there is only one zero, and it has many guises. Zero is not a thing. Zero is characterized by what it does. And what it does is act as the additive identity for any system that has addition and an additive identity! It's the thing that if you add it to another thing, the output is the other thing.

Nobody knows what zero "is." We only know what zero does. If a thing acts like zero, it's zero. Both the zero of the integers and the function that's identically zero in the ring of continuous functions, act like zero. So they're zero.

This by the way is a vague hint of mathematical structuralism: the idea that mathematical objects aren't characterized by what they are; but rather by what they do, and how they stand in relation to other objects.

https://plato.stanford.edu/entries/stru ... thematics/
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### Re: Is 1 = 0.999... ? Really?

wtf wrote:Previously you've said that no definition is needed because the meaning is known to everyone or obvious or intuitive or whatever, I apologize if I'm not quoting you perfectly but this is the feeling I get.

That's correct.

But now you are saying is that yes, you AGREE that infinite sums aren't defined UNTIL we LOGICALLY DERIVE their meaning.

That's not exactly what I said.

What I said is more along to these lines:

The meaning of a sentence is logically derived from the meaning of the individual terms that constitute it.

For example, the meaning of the statement "All bachelors are married" is logically derived from the meaning of the individual terms that consistute it and these are "all", "bachelor", "are" and "married". In other words, you derive its meaning, you do not freely invent a new one e.g. you do not just freely declare that what the statement refers to is a cat.

In the same exact way, the meaning of the expression that is $$0.9 + 0.09 + 0.009 + \cdots$$ is logically derived from the terms that constitute it and these are $$+$$, $$\dotso$$ and numbers of the form $$9 \times 10^{-n}$$ where $$n$$ is a natural number. In other words, you derive its meaning, you do not freely invent a new one e.g. you do not freely declare that what it represents is the limit of the sum that is $$0.9 + 0.09 + 0.009 + \cdots$$.

Addition is defined ONLY for finite sums. One of the axioms for the real numbers says that:

* The sum of any two real numbers is a real number. In symbols, if $$x$$ and $$y$$ are real numbers, then so is $$x + y$$.

You do understand that this axiom says NOTHING other than that the sum of TWO real numbers is a real number? Hell, it does not even tell us the actual result of every (or any) two real numbers, or at least, how to calculate the result of any two real numbers. It merely tells us that the result of every two real numbers belongs to the category of numbers that is "real numbers".

If it's the sum of 47 elements, it's the sum of the first 46 plus the 47th, and that's an instance of adding only two things. So by induction, any finite number of additions are possible.

I agree. The same goes for an infinite number of additions.

So when we see an expression, or string of symbols such as $$x_1 + x_2 + x_3 + \cdots$$, we have no rules with which to untangle or parse that string.

I disagree. Note that you didn't explain WHY. You merely asserted that we have no rules with which to untagle or parse that string.

If you have a vending machine that lets you put in two coins, you'll break it if you try to stuff in infinitely many coins.

But you'd break it by trying to stuff in ANY number of coins other than two. The vending machine lets you put no more than two coins at the same time. So if you want insert more than two coins at the same time, there is no way to do it. But you can insert ANY number of coins (even an infinite number of them) provided you don't try to insert them simultaneously.

It's very much like parsing a computer language. If we see x + y + x + w we know we can parenthesize (x + y + z) + w to reduce the problem to a sum of two things, which we have a procedure for.

I agree.

But if we have an infinite sum, the procedure never terminates. We do not have a computable function that allows us to determine the sum of infinitely many terms.

That's true.

However, I am afraid this is irrelevant. We shall see why shortly.

Do you see that? A machine could compute 1/2 + 1/4 + 1/8 by continually breaking it down into smaller sums, a process that is guaranteed to terminate. But if the input is an infinite sum, the evaluation never terminates and no such sum can be reached. We do not have a procedure or algorithm to calculate an infinite sum that is guaranteed to terminate after finitely many steps.

That's correct. A machine cannot calculate the result of an infinite sum by iterating through an infinite array of terms that constitute it.

But that is a completely irrelevant insight.

There are OTHER ways to figure out the result of an infinite sum. For example, an infinite sum of zeroes is zero. We can know this because zero times infinity is zero (or more generally, because zero times any quantity is zero.) It doesn't matter how many $$0$$'s you add, the result will always be $$0$$. No infinite loop is required in order to calculate the result.

In the same exact way, one can know that $$0.9 + 0.09 + 0.009 + \cdots$$ is greater than $$0.9$$ as well as less than $$1$$. No infinite loop required.

Then by all means restate your thesis; rather than provide argument or evidence.

One step at a time. People must first understand what is it that I'm saying (i.e. what is it that I have to prove.)

wtf wrote:They are exactly the same thing.

I can agree that there is a sense in which it is correct to say that functions are numbers (e.g. if you redefine the word "function" to literally mean "number") but to say this is the original sense, or even worse, to say there is no other sense in which functions are not numbers is something I do not agree with. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

wtf wrote:Nobody knows what zero "is." We only know what zero does.

I am not sure about others but I know very well what it means to have "zero money" i.e. what "zero money" is. On the other hand, I have absolutely no idea what it means to say that "zero money" is doing something.

It's the thing that if you add it to another thing, the output is the other thing.

That's understandable.

However, to claim that the concept of zero is dependent on the concept of addition (or any other operation), in the sense that one cannot know what zero is without knowing what addition (or some other operation) is, is taking it too far. That would be muddying the waters . . .

I can show you a static picture such as this one and tell you "There are zero dragons in it" without knowing anything about mathematical operations.

The concept of function and the concept of number are two INDEPENDENT concepts. You can't reduce one to another, and certainly, you can't say they are synonyms.

The same applies to beings and becomings. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

iambiguous wrote:But 1 horse vs. .999... horse? That's the part I can't wrap my head around. Though I am perfectly willing to acknowledge that this revolves far more around my ignorance in regard to mathematics on this level.

What if the concept "$$1$$" is constructed to signify the quantity of horses that you're seeing, and the concept "$$0.\dot9$$" is also constructed to signify that same quantity of horses that you're seeing?

Either way, the sensory data you're receiving indicates what you know to call "$$1$$ horse", so if some other way of representing that same quantity of horses is "$$0.\dot9$$ horse", what's there to wrap your head around?

Just because the representations look different, and maybe you associate anything that looks like 0.9 with less than 1, does the horse look any different? The two representations of the number of horses are constructed to signify the same quantity, so it shouldn't.

In mathematics, unlike other subjects for which non-mathematicians have much better intuitions, the necessary amount of precision required allows for exactly correct answers.
Fortunately, you have the right attitude, approaching a subject that you regard yourself to be ignorant in with honesty and humility, admitting it's simply something that you struggle to get your head around rather than asserting with unshakable conviction that an understanding known to be false is true just because you don't understand why, which puts you in a position where you could potentially learn if you're interested. Don't you agree that such an approach is the only way to approach philosophy and conduct a productive discussion?

The alternative is apparently to be told why mathematical knowledge is what it is, and to conclude as a non-mathematician that mathematics is wrong regardless.

It's not the case that mathematical knowledge is what it is "because mathematics says so", the whole point is that anyone can construct it from the bare bones of logic and arrive at the same conclusions if they're able and don't suffer from biases and other hindrances to objective thinking. That's why attitude and approach are so fundamentally important, even before mathematical discussion can even get off the ground. Evidently it's a mistake to even begin to engage with anyone on a mathematical level when they lack the most basic ability to think objectively and approach argument with honesty and humility (and not simply "claim" you are, but actually doing so), regardless of their mathematical background or aptitude. Silhouette
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### Re: Is 1 = 0.999... ? Really?

Silhouette!!!!

Stop doing this to people!!!

Fuck, you guys!!! Honestly?

I already gave a proof that the series never equals the limit!!!

I’ll write it for the 10 billionth time!

If the number 1 is divided in half, it equals:

0.5+0.5

If that’s divided it half; it equals:

0.25+0.25+0.25+025

If this sequence continues, then every fucking number equals zero, all rationals, all irrationals and all imaginary numbers!!

That’s a disproof of all you assholes in this entire thread! How many times do I have to state this?!?!

Infinite series don’t converge. I reverse engineered the entire fucking problem to prove this to everyone on earth.
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:Fortunately, you have the right attitude, approaching a subject that you regard yourself to be ignorant in with honesty and humility, admitting it's simply something that you struggle to get your head around rather than asserting with unshakable conviction that an understanding known to be false is true just because you don't understand why, which puts you in a position where you could potentially learn if you're interested. Don't you agree that such an approach is the only way to approach philosophy and conduct a productive discussion?

The starting position is irrelevant. In the case of iambiguous, the starting position is that of ignorance i.e. he has no opinion on the matter. You can't expect people to be EITHER right OR neutral. Most people are opinionated. The most you can hope is that they are willing to EXAMINE their reasoning for potential flaws. (And if they are not, all you have to do is adjust your expectations and act accordingly. Nothing to be frustrated with.) Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:If the number 1 is divided in half, it equals:

0.5+0.5

If that’s divided it half; it equals:

0.25+0.25+0.25+025

If this sequence continues, then every fucking number equals zero, all rationals, all irrationals and all imaginary numbers!!

I agree that $$1 = 0.5 + 0.5$$ and that $$1 = 0.25 + 0.25 + 0.25 + 0.25$$. (Your wording is a bit off. Number $$1$$ divided in half is $$0.5$$, it is not $$0.5 + 0.5$$.) But how does it logically follow that every "fucking" number equals zero? Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:I already gave a proof that the series never equals the limit!!! Silhouette
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:
iambiguous wrote:But 1 horse vs. .999... horse? That's the part I can't wrap my head around. Though I am perfectly willing to acknowledge that this revolves far more around my ignorance in regard to mathematics on this level.

What if the concept "$$1$$" is constructed to signify the quantity of horses that you're seeing, and the concept "$$0.\dot9$$" is also constructed to signify that same quantity of horses that you're seeing?

Either way, the sensory data you're receiving indicates what you know to call "$$1$$ horse", so if some other way of representing that same quantity of horses is "$$0.\dot9$$ horse", what's there to wrap your head around?

In high school, I got an "A" in algebra and an "A" in geometry. But by the time I got to trigonometry and calculus, my brain had failed me. I simply could not think through to the right answers. So maybe your point -- as a technical "concept" -- is right on the money. And it's all about me being a math dummy. I certainly won't deny that possibility.

But here we are 92 pages into this and even with all these sophisticated arguments bursting at the seams with all those sophisticated mathematical symbols, agreement about the one correct answer seems as elusive -- illusive? --as ever.

On the other hand, if a thousand people are standing in front of a horse and are asked how many horses they see, how many are going to say "0.999...horse" instead of "1 horse"?

Or write "0.999... horse" instead of "1 horse" if they are shown a picture of a horse and are answering a test question.
He was like a man who wanted to change all; and could not; so burned with his impotence; and had only me, an infinitely small microcosm to convert or detest. John Fowles

Start here: viewtopic.php?f=1&t=176529
Then here: viewtopic.php?f=15&t=185296
And here: viewtopic.php?f=1&t=194382 iambiguous
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:
Ecmandu wrote:I already gave a proof that the series never equals the limit!!!

Silhouette, Let’s back up here. I’m stating that every rational, irrational, transcendental and imaginary number equals zero if an infinite series equals ANY other number.

I cant remember, e=like 2.7... or some shit as a transcendental number?

That means the e equals zero as well.

Using the number 1 is just an example.
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### Re: Is 1 = 0.999... ? Really?

Magnus! I could be more articulate! Thanks for pointing it out!

1 Split into two parts
1 split into 4 parts
1 split into 8 parts

etc... an algorithm!

I reverse engineered limits that converge to prove that limits don’t converge to a different number.

I’ll use the number 1 as an example for all rational, irrational, transcendental and imaginary numbers!

1 divided by 2 = 0.5+0.5

That divided by 2 = 0.25+0.25+0.25+0.25

That divided by 2 = 0.125+0.125+0.125+0.125+0.125+0.125+0.125+0.125

Obviously, this is an infinite series algorithm!

What happens in this reverse engineered sequence at the limit (the convergence)?

It means that 1=0!!!!!!

Actually, what it means is that if infinite sequences at the limit converge to different numbers, every single possible number equals exactly zero!

That’s a proof that infinite series don’t converge at the limit (equal to) a single number!
————————————

Now let’s get to the disproof of higher orders of infinity:

This is extremely simple: I’ll use variables that I call “my cheat”, placeholders are used constantly in mathematics! If I can’t use my cheat, you can’t use math!!!

1.) rational number (a placeholder / variable)
2.) irrational number (a placeholder / variable)
3.) transcendental number (a placeholder / variable)
4.) imaginary number (a placeholder / variable)
5.) different rational number (a placeholder / variable)
6.) different irrational number (a placeholder / variable)
7.) different transcendental number (a placeholder / variable)
8.) different imaginary number (a placeholder / variable)

This is an algorithm as well. It lists every possible number as variables, representative symbols.

If you’re a stickler for details, make another list of “unlistable” Chaitin numbers!

Those are my proofs through contradiction that infinite series do not converge at limit to single numbers or other infinite series!

The other proof is that all orders of infinity equal each other in size and magnitude: there are no “higher orders of infinity”.!

That’s it for now!
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### Re: Is 1 = 0.999... ? Really?

iambiguous wrote:In high school, I got an "A" in algebra and an "A" in geometry. But by the time I got to trigonometry and calculus, my brain had failed me. I simply could not think through to the right answers. So maybe your point -- as a technical "concept" -- is right on the money. And it's all about me being a math dummy. I certainly won't deny that possibility.

That, to me, suggests that you're probably not as much of a math dummy as you've made out, but rather your trigonometry and calculus teachers probably failed you.

I can tell you that at the heart of trigonometry is circles.
More usefully as a starting point, a unit circle - i.e. a circle with radius "$$1$$" (or $$0.\dot9$$ if you prefer )
This is because you can now draw right triangles starting from the centre of this circle, out to the edge of the circle, then straight down vertically to the point that you can horizontally return straight to the origin.
Now you have a triangle with its longest side (that good old "hypotenuse") the same as the circle's radius (1), and suddenly those "sine" and "cosine" graphs have meaning - the basis of trigonometry.
The sine graph maps how long the triangle's vertical side is for each and any angle made between the horizontal side and the hypotenuse, and the cosine graph maps how long the horizontal side is (for each and any angle made between the horizontal side and the hypotenuse).

Everything trigonometric develops from there - you can scale all sides up or down for different hypotenuse lengths, "tangent" is just the "sine value divided by the cosine value", all the way through other terms you encounter - and you even get an insight into the world of complex numbers when the horizontal side represents the real component and the vertical side represents the imaginary component. All of this becomes accessible from just the few lines I just listed, if you just work your way up from there.

As for calculus, that all just starts at differentiation and integration and you work your way up from there in just the same way. The rest is just familiarity - explore as many different examples of the same concepts that you have the patience for, and you'll develop an intuition that'll help you progress further.
Differentiation is just how the gradient of some curve changes as you go along it, and integration is just the area between the curve and the x axis that you plot it on (for either the whole curve or just a part of it).
Gradient is the y value divided by the x value, and you approach it by drawing a straight line between two points in the curve, working out the gradient of that (starting simple), and seeing what "limit" is approached as you make that straight line smaller and smaller.
The area under a curve is approached by drawing thin rectangles from the x axis to the curve, e.g. with the top left of the rectangles touching the curve, and the top right jutting in or out a little to the side. Keep the widths of these thin rectangles the same, and reduce that width and see what "limit" is approached as you make it smaller and smaller, and add the areas of all the rectangles together. As the rectangles get thinner the true area is approached better and better.

Like trigonometry can lead you to complex numbers, these simple processes for calculus lead you to hyperreal numbers (through limits).

In fact, understanding limits through calculus can correct your intuitions on the title of this thread.
For differentiation, if the straight line gradient "gets to the limit of zero", does it even have a gradient anymore?
For integration, if the widths of the rectangles "get to the limit of zero", does it have an area anymore?

Well, obviously, both methods still approach specific answers, the curves still have gradients at any one point and areas underneath them for any required range. So the answer to both of the above questions is yes! The limit gives the answer.
This thread is no different. The limit of $$0.\dot9$$ is no other number than $$1$$. The consistency of the math that got you to these questions in the first place is maintained.

As such, given "math in the first place" such that we can arrive at this topic's question at all, following it through we get a clear, exact and definite, correct and indisputable answer. Like I said - math allows this by virtue of being strictly precise to the core, such that we can arrive at such a specifically defined question at all.
The conceptual difficulty of a zero length gradient, or the area of a rectangle with zero width is just that - a conceptual difficulty. Mathematics resolves this difficulty by yielding exactly correct answers that make absolutely perfect and consistent sense once you get to them, regardless of any oddness you find in logically getting to an answer that turns out to be true.

iambiguous wrote:But here we are 92 pages into this and even with all these sophisticated arguments bursting at the seams with all those sophisticated mathematical symbols, agreement about the one correct answer seems as elusive -- illusive? --as ever.

On the other hand, if a thousand people are standing in front of a horse and are asked how many horses they see, how many are going to say "0.999...horse" instead of "1 horse"?

Or write "0.999... horse" instead of "1 horse" if they are shown a picture of a horse and are answering a test question.

I mean, I just resolved the entire thread yet again in another single post. And it's no reflection on me, I'm just passing on mathematical knowledge that already exists.

The correct answer was there before this thread even began - all that's elusive is the humility and honesty in approach by the non-mathematicians who think they casually stumbled upon some insight into mathematics that mathematicians got wrong all this time, and relentlessly insist that their inexperienced thoughts have clout (in spite of all valid explanation by actual mathematicians to the contrary), pretending they're open to learning/understanding the actual truth, and are here only for rational debate despite all evidence to the contrary.

It's as much of a poison here as it is in politics and anywhere on the internet, where a bunch of amateurs want a shortcut to tasting what real creative innovation and large scale usefulness is like by pretending to themselves and others that they can be treated like an authority who knows better than countless experts on the subject, because they think they have "a special something" that others don't - or at least they want to think of themselves as such. That's why this thread is not much better than an exercise in identifying psychological biases and logical fallacies - the same as almost every thread on this board in fact. With the notable absence of obvious academic philosophical education here, I only really come here myself to test out layman reactions to certain ideas I have, to test if there's anything obvious that I've missed, or if I actually know a solution to something that others are discussing I can offer them help if they want it and I enjoy trying to teach those who are willing to be taught. I just despise those who do not want to be taught, because they bring everyone else down with them simply out of their own shortcomings and weaknesses - and there are plenty of these types, who collectively make everything worse for everyone.

It's all so human, of course - purpose seems to be a deep need. This desire can inspire one to really study and become a genuine expert in a subject, or it can frustrate the weak who are scared that they might never be able achieve genuine expertise in a subject before they've really even tried, so their wishful thinking and narcissistic eagerness to indulge fantasies of their own greatness get the better of them and they act them out like their very identities depend on it - because they do depend on it. It's a deeply entrenched sickness that I'm hardly going to cure, so I'll simply call it when I see it, and honestly recommend the humility they need, that they will pretend they have, but will probably never truly attain as it's the polar opposite of what's making them sick in the first place. It's everywhere on this forum and ones like it, and if there were any actual philosophical experts here they probably long departed because of all these pretenders who tend to hang around to feed their habit instead of going elsewhere to gain actual expertise.

Ecmandu wrote:Silhouette, Let’s back up here. I’m stating that every rational, irrational, transcendental and imaginary number equals zero if an infinite series equals ANY other number.

I cant remember, e=like 2.7... or some shit as a transcendental number?

That means the e equals zero as well.

Using the number 1 is just an example.

Yep, that's $$e$$, you know the one. Amazing number, pops up everywhere and for very good reason.

If every number of any type actually equals zero, then by "transitivity" every number equals every other number. So there is only one quantity that is "no quantity", with infinite (false?) representations.

If that were the case, then the math would not exist to get you to the start of your reasoning in the first place, that essentially uses math to conclude that math is undone.

So, "given math, there is no math". A contradiction, no?
Either we then conclude that math is therefore bunk, useless, meaningless, doesn't work, is false.
Or we conclude that there's an issue in the process you used to arrive at this conclusion.

You're obviously quite taken with the drama of the first of these potential conclusions. Now I'm suggesting you complete the thought process by investigating the second of these potential conclusions.
Is there an issue with how you arrived at the first potential conclusion?
As in the post of mine that I linked to you, perhaps the plurality of apparent solutions to these equations of yours is a symptom of a flawed methodology. Perhaps doing "more work" could lead you to a different, singular and unique solution that maintains the consistency of the math that you're using to arrive at your conclusion, and which led you to begin your argument at all? There's nothing wrong with noting that an answer is "undefined". Maybe that's the answer, maybe you just need to try a new methodology to give the previous one some better context?

Either way, I don't think your line of thinking has come to an immutable end that everything is zero, which basically just means math is broken - just because of some back-of-a-napkin calculation. Silhouette
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### Re: Is 1 = 0.999... ? Really?

Silhouette,

This is YOUR!!! bias!!!

Math must be BROKEN if infinite series DON’T converge at “limits”.

I gave a VERY simple proof that math is broken if infinite series do converge at limits!!!

Math is not broken if 9/10 + 9/100 + 9/1000 ... etc does not equal the number 1!!!

Not at all!! It just means that you’re wrong on this issue!! That’s all it means!!
Ecmandu
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:1 Split into two parts
1 split into 4 parts
1 split into 8 parts

etc... an algorithm!

That's better. (Except for the part where you call it an algorithm. That's confusing.)

I reverse engineered limits that converge to prove that limits don’t converge to a different number.

You lost me at "reverse engineered" and "limits that converge".

1 divided by 2 = 0.5+0.5

That divided by 2 = 0.25+0.25+0.25+0.25

That divided by 2 = 0.125+0.125+0.125+0.125+0.125+0.125+0.125+0.125

That's all good. (Except for your insistence on saying "divided by 2" when you actually mean "split into 2 equal parts".)

Obviously, this is an infinite series algorithm!

I don't see an infinite series let alone an algorithm.

What happens in this reverse engineered sequence at the limit (the convergence)?

What sequence?

It means that 1=0!!!!!!

And how does that follow?

And that pretty much explains why you don't get much response from me. You are too difficult to understand. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:1 Split into two parts
1 split into 4 parts
1 split into 8 parts

etc... an algorithm!

That's better. (Except for the part where you call it an algorithm. That's confusing.)

I reverse engineered limits that converge to prove that limits don’t converge to a different number.

You lost me at "reverse engineered" and "limits that converge".

1 divided by 2 = 0.5+0.5

That divided by 2 = 0.25+0.25+0.25+0.25

That divided by 2 = 0.125+0.125+0.125+0.125+0.125+0.125+0.125+0.125

That's all good. (Except for your insistence on saying "divided by 2" when you actually mean "split into 2 equal parts".)

Obviously, this is an infinite series algorithm!

I don't see an infinite series let alone an algorithm.

What happens in this reverse engineered sequence at the limit (the convergence)?

What sequence?

It means that 1=0!!!!!!

And how does that follow?

And that pretty much explains why you don't get much response from me. You are too difficult to understand.

Umm... Magnus, I hate to put it to you this bluntly:

For one:

Algorithms are sequences.

Infinite sequences are algorithms as well. That you cannot find an infinite sequence (algorithm) from what I posted is embarrassing to you!!

Let me post the algorithm to you more!! Maybe you’ll understand !!!

1
1/2*2
1/4*4
1/8*8
1/16*16
1/32*32
1/64*64
1/128*128

Etc...

They all equal each other!

At convergence; one side will still be 1 and the other side converges to zero!
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Silhouette,

This is YOUR!!! bias!!!

Math must be BROKEN if infinite series DON’T converge at “limits”.

I gave a VERY simple proof that math is broken if infinite series do converge at limits!!!

Math is not broken if 9/10 + 9/100 + 9/1000 ... etc does not equal the number 1!!!

Not at all!! It just means that you’re wrong on this issue!! That’s all it means!!

So just checking one last time - you're fine using math to work out that math doesn't work?

And you really think math is broken?

Ecmandu wrote:Umm... Magnus, I hate to put it to you this bluntly:

Only just now huh?  Silhouette
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:
Ecmandu wrote:Silhouette,

This is YOUR!!! bias!!!

Math must be BROKEN if infinite series DON’T converge at “limits”.

I gave a VERY simple proof that math is broken if infinite series do converge at limits!!!

Math is not broken if 9/10 + 9/100 + 9/1000 ... etc does not equal the number 1!!!

Not at all!! It just means that you’re wrong on this issue!! That’s all it means!!

So just checking one last time - you're fine using math to work out that math doesn't work?

And you really think math is broken?

Ecmandu wrote:Umm... Magnus, I hate to put it to you this bluntly:

Only just now huh? You’re the person asserting that math is broken if infinite series don’t converge at limits!

I gave you a simple, simple, simple disproof that series don’t converge at limits!

That disproof in no way shape or form disproves math!

Not math!
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Umm... Magnus, I hate to put it to you this bluntly:

Whose credibility? (I believe the answer is yours.)

And why should I care? (I believe that I shouldn't.)

Algorithms are sequences.

Infinite sequences are algorithms as well. That you cannot find an infinite sequence (algorithm) from what I posted is embarrassing to you!!

You are missing the point. There are sequences that aren't algorithms. $$\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dotso$$ is an example of a sequence that isn't an algorithm.

Algorithms are (some algorithms, that is) specific type of sequences, namely, sequences of instructions on how to perform certain task. Most algorithms, however, are closer to being trees than sequences.

Let me post the algorithm to you more!! Maybe you’ll understand !!!

1
1/2*2
1/4*4
1/8*8
1/16*16
1/32*32
1/64*64
1/128*128

Etc...

They all equal each other!

At convergence; one side will still be 1 and the other side converges to zero!

That's not an algorithm. That's a sequence of mathematical expressions (written poorly.)

1 = 1
1 = 2 x 1/2
1 = 4 x 1/4
1 = 8 x 1/8
1 = 16 x 1/16
1 = 128 x 1/128
etc

What do you mean when you say that the other side (the right side, I guess) converges to zero? The right side is at all times equal to 1. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:
Ecmandu wrote:Umm... Magnus, I hate to put it to you this bluntly:

Whose credibility? (I believe the answer is yours.)

And why should I care? (I believe that I shouldn't.)

Algorithms are sequences.

Infinite sequences are algorithms as well. That you cannot find an infinite sequence (algorithm) from what I posted is embarrassing to you!!

You are missing the point. There are sequences that aren't algorithms. $$\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dotso$$ is an example of a sequence that isn't an algorithm.

Algorithms are (some algorithms, that is) specific type of sequences, namely, sequences of instructions on how to perform certain task. Most algorithms, however, are closer to being trees than sequences.

Let me post the algorithm to you more!! Maybe you’ll understand !!!

1
1/2*2
1/4*4
1/8*8
1/16*16
1/32*32
1/64*64
1/128*128

Etc...

They all equal each other!

At convergence; one side will still be 1 and the other side converges to zero!

That's not an algorithm. That's a sequence of mathematical expressions (written poorly.)

1 = 1
1 = 2 x 1/2
1 = 4 x 1/4
1 = 8 x 1/8
1 = 16 x 1/16
1 = 128 x 1/128
etc

What do you mean when you say that the other side (the right side, I guess) converges to zero? The right side is at all times equal to 1.

Every sequence is an algorithm:

1,2,3,4,5 ... is the plus 1 algorithm

And no, they are not always equal to 1 at convergence of limits. One stays the same and the other gets smaller and smaller and smaller.
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### Re: Is 1 = 0.999... ? Really?

Ecmandu wrote:Every sequence is an algorithm

1,2,3,4,5 ... is the plus 1 algorithm

"Not really" to both.

The following is an algorithm:

1) Set $$a$$ to $$0$$
2) Increase $$a$$ by $$1$$
3) Go to step 2

That's a sequence of instructions (an algorithm), whereas $$1, 2, 3, \dotso$$ is a sequence of numbers (not an algorithm.)

And no, they are not always equal to 1 at convergence of limits. One stays the same and the other gets smaller and smaller and smaller.

I have no idea what you're talking about. Unless you make an effort to clarify your position, this is the end of the discussion between the two of us. You have no right to complain at this point. Magnus Anderson
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