Is 1 = 0.999... ? Really?

[wtf]

=
"10 multiplied by itself infinitely" is another expression for \(10^{\infty}\).

\(10^{\infty}\) is undefined, it's not a well-formed formula in any mathematical system I know. Have you looked at the definition of an infinite product I linked? At best you could define \(10^{\infty}\) as \(\displaystyle \lim_{n \to \infty} \Pi_{k =0}^n 10^k\), but that infinite product diverges. That is, it's the limit of 1, 10, 100, 1000, 10000, ...

If you want to call it \(\infty\) in the extended real numbers you could do that. But it doesn't mean anything significant. Certainly not the way you're trying to use it.

[Magnus Anderson]

They define infinite product as the limit of partial products. That's a different concept.

[obsrvr524]

You merely repeated what I had already rebutted. Repetition doesn't count as a rebuttal - it indicates that there is no more argument to be presented and the end of that encounter.

I don't think I repeated myself. I stated something I never stated before -- what I mean by "implicit digit". I thought you did not understand the term.

I understood the first time. Did you understand that what I said was that I did not see the implication of the 0 - I didn't say that I merely did not see the 0.

Again, you are repeating yourself. I already explained why that is not acceptable

if the "..." is allowed to go below zero [the decimal point] then your number "999..." is undefined

You seem to think that if a meaning is contradictory that it cannot possibly be associated with a word.

"is contradictory"? If the meaning of a word is undefined (or worse self-contradictory) then the word is ambiguous and cannot be used to prove anything. To prove is to remove doubt, question, or alternative.

It's like saying the meaning of "square-circle" is not "a shape that is both a circle and a shape" because a shape cannot be both a circle and a shape.

Except that "a number" was presented - not an oxymoron. If you want to claim that the "999..." isn't really a number that is a different argument.

Again you are repeating what I have already explained away - your number cannot be of a higher cardinality than the natural numbers because that is all we are dealing with and the ellipsis is only about the natural "countable" number set.

Yes, that's what you think. I disagree with that.

Then why haven't you simply said "I disagree that it must be within the natural number set"?
We could go from there.

and that means that your first "9" of your "999..." figure is AT index \(infA\) (or \(infA - 1\) if you prefer), else the entire number doesn't make sense.

That does not follow. That's merely you stitching things together so that they make sense to you.

If you want to claim that the "999..." is an oxymoron then that constitutes a definitional premise disagreement and is in opposition to the implied definition of my proposal - because in my proposal it is NOT meant as an oxymoron.

You are not allowed to change definitions intended by the OP although asking for them or clarifying them is certainly permitted. Like I said - finding agreement is the issue - not trying to win an argument.

[Magnus Anderson]

I understood the first time. Did you understand that what I said was that I did not see the implication of the 0 - I didn't say that I merely did not see the 0.

You said there is no implicit \(0\) that comes before the leftmost digit in \(99\dot9\).

And you reasoning was:

your "..." restricts you from exceeding your upper "9" to any supposed preceding "0"

In \(99\dot9\), the three dots ("...") say no more than "repeat \(9\) endlessly in the direction of the least significant digit". They say NOTHING about the digits that comes before the leftmost \(9\).

And if the index of the first \(9\) is \(infA\), they are also not saying anything about the digit associated with \(10^0\) (the \(9\)s do not extend enough to reach that point.)

Except that "a number" was presented - not an oxymoron. If you want to claim that the "999..." isn't really a number that is a different argument.

Right, so if it's a number, it's not an oxymoron. But if it's a shape, then it might be an oxymoron (:

My task is merely to prove that \(999\dotso\) is not \(\sum_{i=0}^{i->{\infty}} 9\times10^0\).

Then why haven't you simply said "I disagree that it must be within the natural number set"?
We could go from there.

I think I said that long time ago. (But not during the Resolution Debate we started.)

If you want to claim that the "999..." is an oxymoron then that constitutes a definitional premise disagreement and is in opposition to the implied definition of my proposal - because in my proposal it is NOT meant as an oxymoron.

We're trying to DEDUCE the meaning of \(999\dotso\) using existing mathematical definitions, not merely invent a new one.

If your claim is merely "This is how I define \(999\dotso\)" then there is nothing to debate.

[obsrvr524]

I understood the first time. Did you understand that what I said was that I did not see the implication of the 0 - I didn't say that I merely did not see the 0.

You said there is no implicit \(0\) that comes before the leftmost digit in \(99\dot9\).

And you reasoning was:

your "..." restricts you from exceeding your upper "9" to any supposed preceding "0"

In \(99\dot9\), the three dots ("...") say no more than "repeat \(9\) endlessly in the direction of the least significant digit". They say NOTHING about the digits that comes before the leftmost \(9\).

Ok then you are arguing with the definition intended in the OP.

Since I am the one who presented that proposal - my intended definition was that "999..." is an infinite quantity in the set of natural numbers - with no implicit digits to the left of the leftmost 9 nor to the right of any decimal point."

End of game.

Unless you have an argument against what you now understand the original post to mean.

[wtf]

They define infinite product as the limit of partial products. That's a different concept.

That's how it's defined.

If you have a different definition: please supply it forthwith.

We're trying to DEDUCE the meaning of \(999\dotso\) using existing mathematical definitions, not merely invent a new one.

You haven't done so. And now you say you are using existing mathematical definitions, but you just rejected the ONLY existing mathematical definition of an infinite product, which is the limit of partial products. And which DOES happen to make sense in this case, except that it trivially diverges to infinity.

So please show me your definition of \(10^\infty\) that does NOT involve the limit of partial products; but DOES rest on existing mathematical definitions.

[Magnus Anderson]

Ok then you are arguing with the definition intended in the OP.

Since I am the one who presented that proposal - my intended definition was that "999..." is an infinite quantity in the set of natural numbers - with no implicit digits to the left of the leftmost 9 nor to the right of any decimal point."

End of game.

I don't exactly remember on which page but at one point you made a claim that the meaning of \(99\dot9\) according to standard mathematical definitions (not according to your own definitions) is \(\sum_{i=0}^{i->{\infty}} 9\times10^i\). It is this claim that I suggested be the subject of ILP's first Resolution Debate ever (remember that it was me who suggested we debate this subject.) But now, you're trying to bail out on the ground that your proposal merely spoke of your own personal definition ): I mean, anyone can define words any way they want, and unless there's a reason to think they are lying about the way they are defining them, there is no reason to disagree with their definitions. It's a category of beliefs least suitable for a debate (since the point of debates is to resolve disagreements.)

[Magnus Anderson]

And now you say you are using existing mathematical definitions, but you just rejected the ONLY existing mathematical definition of an infinite product, which is the limit of partial products.

I rejected a definition that mathematicians developed because they didn't know how to properly deduce the meaning of certain category of expressions from existing definitions.

An analogy would be a man who due to his inability to calculate the result of "2 + 2" says "Fuck it, let's just define it to be 10!"

When I say that an infinite product is this or that, I am not merely saying "This is a personal definition of mine". I am actually saying "This is what the term means according to the standard mathematical definitions".

[obsrvr524]

Ok then you are arguing with the definition intended in the OP.

Since I am the one who presented that proposal - my intended definition was that "999..." is an infinite quantity in the set of natural numbers - with no implicit digits to the left of the leftmost 9 nor to the right of any decimal point."

End of game.

I don't exactly remember on which page but at one point you made a claim that the meaning of \(99\dot9\) according to standard mathematical definitions (not according to your own definitions) is \(\sum_{i=0}^{i->{\infty}} 9\times10^i\).

If you think you are going by "standard definitions" then we disagree on that too.

[obsrvr524]

They define infinite product as the limit of partial products. That's a different concept.

That's how it's defined.

If you have a different definition: please supply it forthwith.

That could only apply to convergent partials.
The limit of divergent partial products would remain undefined.

[wtf]

That could only apply to convergent partials.
The limit of divergent partial products would remain undefined.

I agree with you. But I want to be charitable and allow extended reals so that we can call the limit of 1, 10, 100, 1000 ... \(\infty\). But @Magnus will not even agree to that. So I'm at a loss.

[wtf]

When I say that an infinite product is this or that, I am not merely saying "This is a personal definition of mine". I am actually saying "This is what the term means according to the standard mathematical definitions".

Great. Then tell me what \(10^\infty\) means "according to the standard mathematical definitions," and please make sure to reference those definitions. Because frankly I'm highly familiar with "standard mathematical definitions," and you have not presented one yet.

[Certainly real]

Yes and no. Too many issues have been cast into this and you are getting them confused.

As I stated earlier - any real thing can be divided however you want. But what is represented by "1/3" is NOT the exact same as what is represented by "0.333..." (the same for "1" and "0.999...").

Exact ratios often cannot be represented in digital form - that is why we have to use a non-digital symbol "..." to indicate our inability to express the ratio while still giving a potentially useful number.

Yes, but the question is can a circle be divided into 3 wholly equal parts? I know I can divide a circle into 3 non-equal parts. I am not certain if I can divide it into three wholly equal parts. I think it's doable, and given pi, this is supposed to be doable. As far as I can see, if you accept pi, then consistency would have it that you accept 1/3 to = 0.333...

[Certainly real]

Of course pi is finite. It's a real number between 3 and 4 on the number line.

You are right that the decimal representation of pi has infinitely many digits. But that doesn't make pi infinite. In fact pi only contains a finite amount of information.

Note that I completely expressed pi using only 15 characters. In other words if I wanted to completely transmit all the digits of pi to a friend on Mars, I would not have to send infinitely many digits; I'd only have to send the 15 character formula and they could use that to calculate as many digits as they want.

I don't see how we can say that pi only contains a finite amount of information. I understand we can represent this infinite amount of information in a finite manner. But this is not the same as pi itself containing a finite amount of information. If pi consists of an infinity of digits, then that is an infinite amount of information because each number is a bit of information.

Where we can divide a circle into 3 wholly equal parts using pi, then pi is infinite. 1/3 is a finite part of something. Pi is more like a tool to do something with than a finite part of something.

Google says nothing of the kind. Pi is a finite number, it's a real number between 3 and 4.

I meant that it says that it consists of an infinity of digits.

Well 1/3 is exactly expressed with only three characters. Again you are confusing a number with its representation.

See my above point on the representation of pi versus the value of pi.

If you prefer, and I explained this several times in my previous post and urge you to slow down and comprehend this point: I do not need to talk about infinitely many 3's at all, and from now on I won't do that anymore.

What I am saying is that I can get AS CLOSE AS I WANT to 1/3 by taking a large enough FINITE number of 3's, such as 0.333333333 or 0.3333333333333333. I never have to talk about infinitely many 3's and from now on I won't do so.

I mean no offence by the following, just constructive feedback: I think you are too focused on representation and not focused enough on semantics.

I agree that you can get as close as you want. We were never in disagreement on this. And that was my point to you when you first said 1/3 = 0.333...

If you remember, I said that 1/3 cannot equal .333... because an infinity of 3s are impossible and an infinity of 3s are needed to fulfil the semantic of 1/3 as opposed to just get close to it. But then you made that point about circles and I reconsidered my position.

To emphasise, you say you do not need to talk about infinitely many 3s, but then semantically speaking you have not fulfilled the semantic of 1/3 have you? When the 3s are finite, you have only fulfilled close to 1/3. The only way you can say 1/3 = .333... is if you are saying that an infinity of 3s follow.

[obsrvr524]

Yes and no. Too many issues have been cast into this and you are getting them confused.

As I stated earlier - any real thing can be divided however you want. But what is represented by "1/3" is NOT the exact same as what is represented by "0.333..." (the same for "1" and "0.999...").

Exact ratios often cannot be represented in digital form - that is why we have to use a non-digital symbol "..." to indicate our inability to express the ratio while still giving a potentially useful number.

Yes, but the question is can a circle be divided into 3 wholly equal parts?

Certainly - 120° each slice = exactly 1/3 of a circle.

I know I can divide a circle into 3 non-equal parts. I am not certain if I can divide it into three wholly equal parts. I think it's doable, and given pi, this is supposed to be doable. As far as I can see, if you accept pi, then consistency would have it that you accept 1/3 to = 0.333...

Still - you are confusing the digits involved with the value they represent. A very small number might be represented by an infinity of digits - that does NOT make the number infinite. And it does not stop anyone from dividing something into 3 equal parts.

The decimal system of listing numbers has issues - problems. Some values (such as 1/3) cannot be entirely and precisely represented with decimal numbers. The issue is only in the language being used (base_10 digits) - not with the real values.

Pi is a value that cannot be exactly represented by decimal digits ever. The value of Pi doesn't care whether you try to represent it in decimal form. The value of it doesn't change to match your estimations. The value is a little over 3.14 and always will be even if you list an infinity more digits in the effort to be 100% exact. Pi will always just be a little over 3.14.

Similar with 1/3. It doesn't matter how many digits your want to throw together to try to represent it. It will always remain merely 1/3 or a little over 0.33. Add a billion more 3's at the end and you do not change what 1/3 is at all. You only change the accuracy of your string of decimals.

The whole thing is a language issue - the language of decimal numbers. It has nothing at all to do with reality or real values.

[phyllo]

The whole thing is a language issue - the language of decimal numbers. It has nothing at all to do with reality or real values.

The language keeps saying that 1/3=0.333... and 1=0.999... through series convergence and the many other techniques that have been demonstrated in this thread.

The results are completely consistent.

[Certainly real]

Still - you are confusing the digits involved with the value they represent. A very small number might be represented by an infinity of digits - that does NOT make the number infinite. And it does not stop anyone from dividing something into 3 equal parts.

The decimal system of listing numbers has issues - problems. Some values (such as 1/3) cannot be entirely and precisely represented with decimal numbers. The issue is only in the language being used (base_10 digits) - not with the real values.

Take the semantic of '21' and the semantic of 'divide' and the semantic of '3' and the semantic of 'equals'. What does 21 divided by 3 equal? The answer is not 21/3 because that is the question that is being asked. It is not the answer. The answer is the semantic of 7. 7 is the answer in its actual purest form.

Now, what is 1 divide by 3 equal to? The answer is either absurd or .333... (where ... stands for an infinity of 3s)

You cannot say the decimal system of listing numbers has issues. It is the semantic of 'one' and the semantic of 'three' and the semantic of 'divided by' such that when you ask what is one divide by 3? the answer is either you cannot divide 1 by 3 or it is zero point followed by an infinity of 3s.

Similar with 1/3. It doesn't matter how many digits your want to throw together to try to represent it. It will always remain merely 1/3 or a little over 0.33. Add a billion more 3's at the end and you do not change what 1/3 is at all. You only change the accuracy of your string of decimals.

The whole thing is a language issue - the language of decimal numbers. It has nothing at all to do with reality or real values.

Yes but semantics dictate that either 1/3 is absurd, or 1/3 is 0.333...
It is semantically inconsistent to say that 1 divide by 3 is 0.333.. (follow by a vary large but finite number of 3s).

[phyllo]

These forms are interchangeable because they are equivalent. They refer to the same value. 21/3=7=7/1=700/100=49/7

[Certainly real]

These forms are interchangeable because they are equivalent. They refer to the same value. 21/3=7=7/1=700/100=49/7

I see them as having the same identical answer. The semantic of 7. I don't see them as being semantically identical. As in 21/3 is not semantically identical to 7/1. Their answers are semantically identical. This does not make them semantically identical.

[phyllo]

That's part of the point and usefulness. You can move from one form to another. Gaining understanding and insight in the process.

Like

the cat = le chat

Let's you move from the society of English speaking people to the society of French speaking people. And back again.

If you focus purely on the differences in the words, then you limit yourself.

[obsrvr524]

Certainly real,

It might help us (and you) if you would describe what certain words mean to you - define them in your own words - such as "semantics", "infinite", "...", and "infinity".

I don't have any doubt that this whole thread is nothing but arguments over words and symbols.

[obsrvr524]

The whole thing is a language issue - the language of decimal numbers. It has nothing at all to do with reality or real values.

The language keeps saying that 1/3=0.333... and 1=0.999... through series convergence and the many other techniques that have been demonstrated in this thread.

Would you agree that if the string of 3s in 0.333... actually got to infinity then 0.333... = 1/3 ?

[phyllo]

The whole thing is a language issue - the language of decimal numbers. It has nothing at all to do with reality or real values.

The language keeps saying that 1/3=0.333... and 1=0.999... through series convergence and the many other techniques that have been demonstrated in this thread.

Would you agree that if the string of 3s in 0.333... actually got to infinity then 0.333... = 1/3 ?

I already covered this. There is no time or process involved.

There is no "got to infinity" or "getting to infinity". It's at infinity. It's always equal.

[obsrvr524]

Would you agree that if the string of 3s in 0.333... actually got to infinity then 0.333... = 1/3 ?

It's at infinity. It's always equal.

That is all I meant. So if 0.333... represents the 3s being at infinity then you agree that 0.333... = 1/3.

[Certainly real]

That's part of the point and usefulness. You can move from one form to another. Gaining understanding and insight in the process.

Like

the cat = le chat

Let's you move from the society of English speaking people to the society of French speaking people. And back again.

If you focus purely on the differences in the words, then you limit yourself.

It might help us (and you) if you would describe what certain words mean to you - define them in your own words - such as "semantics", "infinite", "...", and "infinity".

I don't have any doubt that this whole thread is nothing but arguments over words and symbols.

Semantic = meaning
Infinite = that which has no beginning and no end
Infinity = that which no greater quantity can be conceived of

If I ask the question what is the answer to a third of 21 exactly, the answer I would get is 7. Not 21/3. Perhaps someone might say you will find the answer when you divide 70 by 10, but they are still saying you will find the answer after you do this division.

To convey what I'm trying to convey more clearly, if I ask the question what is a third of a 1 meter ruler, I would not get the answer, well that's just a third of a one meter ruler. I would either get the answer you cannot have a third of a meter, or it's .333...m or 33.333...cm