## 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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40%
No, 1 ≠ 0.999...
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Other
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### Re: Is 1 = 0.999... ? Really?

You seem to think that 0.99 is smaller than 0.9

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

Isn't a fraction of one, the numerator 1 over a number as a denominator it is being divided by?

The larger the number that fills the role as denominator the smaller the fraction of one you are referring to. So yes the fraction 1/9 is much larger than the fraction 1/99.

1 is over here .9 1/2 .5 .25 nearer to here .1 .99 and zero is over here.

I am fairly confident that .9 is larger than .99
Last edited by Mowk on Fri Apr 10, 2020 3:35 am, edited 1 time in total.
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### Re: Is 1 = 0.999... ? Really?

That's not how the decimal system works.
$$0.9=\frac{9}{10}$$
which can also be written as
$$0.9=\frac{90}{100}$$
Compare to
$$0.99=\frac{99}{100}$$
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### Re: Is 1 = 0.999... ? Really?

You are thinking in terms of percentages.

The range between 0 and 1 in a percentage system has divided all possible values between 0 and 1 into only 100 places. 99 percent of 1 is very close to one.

But 1 of 99 pieces of a whole is not 99% of the whole it represents, just 1 percent of 1 whole number. 1 divided by 99 represents just 1 piece of of the whole 1, divided into 99 parts.

1 part of 10 parts, is 10 percent of 1 whole. 10 parts of 100 parts is still 10 percent of 1 whole.....
Last edited by Mowk on Fri Apr 10, 2020 6:40 am, edited 2 times in total.
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### Re: Is 1 = 0.999... ? Really?

I am going to attempt a mapping of several numeric system as far as my understanding goes. Rather than to tell you, your are wrong I invite you (in the collective sense that you doesn't mean me) to tell me where I am wrong, because, frankly I am trying to make some sense of it all and I will readily admit I can be wrong. Like that hasn't happened .9 recurring times before. One range in the map will include the first ideas of math I can imagine took place. The counting numbers.
Another will include fractions as it seems to be the next evolution in how the range is divided.
Decimal notation; I would imagine came after fractions, as result of a practiced use of fractions.
Integers opened up a whole notion that a numbering system can work both in the positive and negative vectors and required the idea that 0 had to exist between them.
Just for grins I will also include binary numbers.
For shits and grins hexadecimals, just because Sil mentioned them. But I am not aware of anyone that applies them besides computer programmers. The emergence of computers, and the algorithms used in programming. Which algorithm is being used when. An algorithm is sort of like shorthand notation for complex processes.

6 lines that express each of these numbering system and how I understand the mapping between them. You may challenge my understanding of each of these numeric systems. Please.

One request... please focus on just these numbering systems or I will surely become overwhelmed. I will not be illustrating them in chronological order. But rather in order of complexity.

Stay tuned... same Bat time... same Bat channel. Please support our sponsor.

In a social contract of all for one and one for all, disagreement causes stress. Lacking a social contract, a me, myself, and mine camp is less stressed, that is until it becomes "only" about "my" survival, then that camp goes ballistic, and then it is just flat out, all about the individuals survival. Understandable.
Mowk
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### Re: Is 1 = 0.999... ? Really?

Mowk wrote:On one side of the equals is the representation of a single whole unit with nothing to it's left and nothing to it's right, on the other is a decimal expression of 9/10th divided by 9/10 infinitely, filling all possible digits with a non-zero number. Infinitely less than 1.

All the math you want to throw at the question as proof is irrelevant, smoke and mirrors... a distraction.

That's pretty much it. The problem is that noone dares to say the emperor is naked for fear that they will be seen as stupid.

Silhouette wrote:I've already addressed this issue, but let me tie it specifically to what you're saying to spare you from claiming that others such as phyllo are not paying attention when they are.

I do agree with Mowk that, on average, and within this thread at least, Phyllo is not paying attention. Ever since the creation of this topic, his mindset has been "I don't find this question interesting so I'm going to answer it any way I like no matter how stupid it is because . . . why not?" thereby automatically disqualifying himself as a serious interlocutor. It's only slightly better than what Biguous is doing here and elsewhere.

When I speak of the difference between representations and quantities I am speaking of the difference between concrete and abstract, the signifier and the signified.
Something like "1" is the "concrete signifier" that represents an "abstract signified" known as quantity.
Putting a value in the units column is one way of representating the same quantity as putting a zero in the units column and 9s in infinitely many orders of magnitude less than a unit (everything after the decimal point).
Two different representations: same quantity.

Another example to perhaps clarify what I'm getting at here:
In hexadecimal, the digits 0-F extend the digits 0-9 in decimal. "10" in decimal is "A" in hexadecimal.
Now, if we were to "borrow" this A from hexadecimal, which means "10" in decimal, and put it into a decimal number like so: "0.A", there is a zero in the units column, and yet the quantity represented could also be represented as a decimal "1".
Two different representations: same quantity.

I think that we all understand that two different symbols can mean one and the same thing. There is absolutely no need to point this out over and over again.

The question is whether the two symbols $$0.\dot9$$ and $$1$$ do in fact represent the same quantity. Of course they can, in theory, but do they, in reality?

And yes, $$0.A$$ in base-10 with 16 digits and $$1.0$$ in standard base-10 represent one and the same quantity but that's only because we're dealing with two different numeral systems.

The point is that within base-10 system two different representations necessarily represent two different numbers.

Can you give us an example of two different base-10 representations that represent one and the same number?

I don't think you can. All you gave us so far is:

1) $$2\pi rad$$ and $$0\pi rad$$ represent the same angle
2) $$0.A$$ in decimal with 16 digits and $$1.0$$ in decimal with 10 digits represent the same number

Both statements are true but neither proves that it is possible for there to be two different decimal representations that represent the same number.

Getting caught up in the appearance of each representation being different is merely superficial. By itself it doesn't say anything about the quantity "in itself".
Do you understand this?

Sure. If two symbols look different it does not necessarily follow they represent two different things. Very deep. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Mowk wrote:You are thinking in terms of percentages.

The range between 0 and 1 in a percentage system has divided all possible values between 0 and 1 into only 100 places. 99 percent of 1 is very close to one.

But 1 of 99 pieces of a whole is not 99% of the whole it represents, just 1 percent of 1 whole number. 1 divided by 99 represents just 1 piece of of the whole 1, divided into 99 parts.

1 part of 10 parts, is 10 percent of 1 whole. 10 parts of 100 parts is still 10 percent of 1 whole.....
In the decimal system, the digits to the right of the decimal point represent fractions having a denominator which is a multiple of 10. The first digit is 1/10 fraction, the second is 1/(10x10), the third is 1/(10x10x10), etc.

So here is an example :
$$0.54321 = \frac{5}{10} +\frac{4}{100} +\frac{3}{1000} +\frac{2}{10000} +\frac{1}{100000}$$
which can also be written as
$$0.54321=\frac{54321}{100000}$$
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### Re: Is 1 = 0.999... ? Really?

I do agree with Mowk that, on average, and within this thread at least, Phyllo is not paying attention. Ever since the creation of this topic, his mindset has been "I don't find this question interesting so I'm going to answer it any way I like no matter how stupid it is because . . . why not?" thereby automatically disqualifying himself as a serious interlocutor. It's only slightly better than what Biguous is doing here and elsewhere.
Correction. I don't pay much attention to some posts and some posters.

For example, you are unable to make up your mind about whether infinity is a number or not. You wobble back and forth as it suits you. You don't have a consistent point of view and so you don't have a consistent argument.

You don't understand convergence and divergence. Yet you insist on using series representations. The result is more inconsistent arguments.

Then there are the posts where you accuse people who disagree with you, of being blind followers of convention. I have no interest in those kinds of personal and manipulative posts.
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:I do agree with Mowk that, on average, and within this thread at least, Phyllo is not paying attention. Ever since the creation of this topic, his mindset has been "I don't find this question interesting so I'm going to answer it any way I like no matter how stupid it is because . . . why not?" thereby automatically disqualifying himself as a serious interlocutor. It's only slightly better than what Biguous is doing here and elsewhere.

His answers seem perfectly relevant to me.
He's a lot more curt than I am, but generally straight to the essence of the misunderstanding.
Though I guess that's the whole problem - mathematicians can see the core of the misunderstanding very easily, but the whole reason the non-mathematician is a non-mathematician is because they can't.

The gap is clearly very tricky to bridge.
How do you get someone who thinks they're right to see they're not right when they're wrong?
The whole reason they think they're right when they're wrong is because they can't see why they're not right... - if they could see why, then they'd not present wrongess as rightness...

Magnus Anderson wrote:Can you give us an example of two different base-10 representations that represent one and the same number?

Yes. $$1$$ and $$0.\dot9$$

You get the concept using symbols across number bases, and in radians - so clearly you understand how the appearance of the numbers differing doesn't necessarily matter - they can still represent the exact same quantity.

Thanks for confirming.

$$0.\dot3$$ and $$\frac1{3}$$ are likewise just different representations of the same quantity. Both represent the division of exactly 1 by exactly 3 to get a third exactly.
There's no problem dividing 9 by 3 in exactly the same way as any rational number with however many decimals - so using the exact same division there's no problem dividing 10 by 3 to get exactly $$3.\dot3$$
Why is the result of division okay sometimes but not others? It's exactly the same concept and any methods or processes to successfully achieve it do so in exactly the same way to the same success.
As such the decimal representation of $$3.\dot3$$ ONLY works precisely because the 3 is recurring, never ending, infinite, with an undefined number of repetitions. It's not a problem that the remainder is never 0 to terminate a finite (rational) decimal. As with all irrational numbers, the remainder never being 0 is the only way in which they can be exactly correct - you just have to respect the involvement of the undefined nature of infinity and what it means. That's where this mysterious ambiguous "remainder" is, which you're looking for - it's in your lack of respect and understanding of infinity/the undefined. It sure as hell isn't anywhere in the realm of the defined!

Mowk, I fear your teachers have failed to instill some basic building blocks in your mathematical understanding, at least when it comes to division. If you like, I will gladly go through the process with you - as I'm sure phyllo will too.

You seem willing to learn, which is very encouraging. Silhouette
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:For example, you are unable to make up your mind about whether infinity is a number or not.

It's a number. Thought it's obvious.

The way I see it, you are simply not interested in the subject, and yet, here you are participating. People do that when they are bored and it's not a good thing. The lack of patience necessary to thoroughly process what people are saying is a tell. And I can understand that. I don't expect people to share the same interests as I do. Different strokes for different folks and all that but if you're not interested in the subject, and you're still participating, then that's a problem. Joining a thread only to tell others they are not meeting your expectations does nothing but communicate to others that you're frustrated.

You wobble back and forth as it suits you. You don't have a consistent point of view and so you don't have a consistent argument.

I suppose that's your general impression. Well, I don't agree with it.

You don't understand convergence and divergence. Yet you insist on using series representations. The result is more inconsistent arguments.

That's because convergence and divergence are completely irrelevant.

Then there are the posts where you accuse people who disagree with you, of being blind followers of convention. I have no interest in those kinds of personal and manipulative posts.

I can understand that. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

Magnus Anderson wrote:That's because convergence and divergence are completely irrelevant.

So if you say an essential thing to understanding the topic is irrelevant, you can imagine it's not an issue?

I've been getting this impression from you for a long time now. You throw around the accusation of irrelevance as soon as a problem comes up. It's either intentional out of bad faith or unintentional from incompetence - I think a mixture: except I don't think you are explicitly trying to argue in bad faith, I think it's just the usual psychological protections that people usually fall foul to that are getting in your way of changing your mind. If you weren't merely trying to bolster your own (lack of) argument, and instead you tried to understand see the sense in the arguments of others - take a leaf out of Mowk's book for this one - then we could get somewhere, but I think the inability also plays a part against you here too.

You like to say these psychological analyses of you are also "irrelevant", but they really aren't - for the same reasons as above ^ Silhouette
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:Yes. $$1$$ and $$0.\dot9$$.

The problem is, if this is truly enough, why did you feel the need to talk about radians and unusual numeral systems (such as the base-10 system with 16 digits)? I believe it's because . . . it's actually not enough. Hopefully, you will agree the quoted is a redundant statement. It's a statement that has to be proven, so you can't use it as a premise.

You get the concept using symbols across number bases, and in radians - so clearly you understand how the appearance of the numbers differing doesn't necessarily matter - they can still represent the exact same quantity.

Thanks for confirming.

Thanks for noting.

$$0.\dot3$$ and $$\frac1{3}$$ are likewise just different representations of the same quantity. Both represent the division of exactly 1 by exactly 3 to get a third exactly.

Alright, that's your position, and I will confirm (I believe once again, it's not the first time) that I understand it.

There's no problem dividing 9 by 3 in exactly the same way as any rational number with however many decimals - so using the exact same division there's no problem dividing 10 by 3 to get exactly $$3.\dot3$$

The problematic part is your claim that dividing $$10$$ by $$3$$ is the same exact process as dividing $$9$$ by $$3$$. That's where we disagree. I don't think it's the same exact process.

Why is the result of division okay sometimes but not others? It's exactly the same concept and any methods or processes to successfully achieve it do so in exactly the same way to the same success.

That's a good question. On the other hand, I do not really think it's important. Certainly, it's not necessary to answer it. But I'll do my best anyways.

But . . . I already explained what's wrong with the long division proof, didn't I? So I'll be repeating myself in all likelihood. Be warned.

When you divide $$9$$ by $$3$$, you get to a point when every subsequent digit is $$0$$. When you divide $$1$$ by $$3$$, no such point is ever reached, simply because there is no such point. And this is a signficant difference even though it may not appear so at first.

In plain terms, in the case of $$1 \div 3$$, the long division process is never finished. (And this is quite simply because $$\frac{1}{3}$$ has no decimal representation.)

The question I need an answer to is: why do you disagree with this? This will help me proceed with more accuracy. Less redundancy, fewer assumptions about what you think and so on.

As it is, I can only guess . . . shoot blindly.

As such the decimal representation of $$3.\dot3$$ ONLY works precisely because the 3 is recurring, never ending, infinite, with an undefined number of repetitions. It's not a problem that the remainder is never 0 to terminate a finite (rational) decimal.

The never-ending process of long division, such as the one in the case of $$1 \div 3$$, tells us that there is no number of digits used in that particular numeral system (such as $$3$$s) that can satisfy the condition. This is a subtle but very important point.

The long division tells us that the number of $$3$$s necessary to satisfy the condition is greater than any number we can think of. The problem is . . . there is no such number.

Suppose that such a number exists. Being greater than every number, it means it's greater than itself. But a number cannot be greater than itself. Hence, there is no such number.

That tells us that there is no number of $$3$$s that can satisfy the condition. It does NOT tell us that the number of $$3$$s necessary to satisfy the condition is infinite since infinity has a different meaning: that of a number greater than every integer (not every number in general.)

As with all irrational numbers, the remainder never being 0 is the only way in which they can be exactly correct - you just have to respect the involvement of the undefined nature of infinity and what it means. That's where this mysterious ambiguous "remainder" is, which you're looking for - it's in your lack of respect and understanding of infinity/the undefined. It sure as hell isn't anywhere in the realm of the defined!

You keep using the word "undefined" in your own particular way while refusing to define it when other people tell you they have trouble processing it.

Note that if a word is meaningless there is generally no place for it in mathematics (such words can only be useful in the art of deception and perhaps cryptography.)

If you mean that the word "infinity" is undefined in the sense that its meaning has not been verbally described (at least not properly) then I disagree. The meaning of the word "infinite" is perfectly captured by the sentence "greater than every integer". Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

It is really hard to illustrate base-10 decimal system mapping as it is a logarithmic scale not a linear one.

The distance between each number isn't the same distance apart on a line. As an example the distance between 0 and 9 are separated by 9 units, the distance between 9 and 99 is 90 units, the distance between 99 and 999 is 900 units, the distance between 999 and 9999 is 9000 units and so on. To the right of the decimal point it is logarithmic as well. The distance along the line between 1 and 0 is divided into 10 segments, the line is divided into 100 segments then a thousand segments, then 10,000 segments, then 100,000 segments and on and on and on. Well actually the line has all these divisions all at once because the distance between 1 and 0 can be divided into an infinite number of parts. So if we plot the 9/10ths, the next digit is 9 represents 9/100ths, the next 9/1000ths, and the next 9/10,000ths and so on. As you plot the points along the line the segment distance between iterations of 10 gets smaller and smaller the closer to 0 you get. But as a result of having 9/over what ever number of powers of 10 you are working with, you get closer and closer to zero but never get there.

This is why I believe that .9 recurring is actually a better description for zero then it is for one but it doesn't work well for zero either. While it is remarkably close to zero it will never get to zero because of the infinitely recurring 9. It is really difficult to plot the point along a line that is divided infinitely.

Look up decimal notations if you don't believe me.
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### Re: Is 1 = 0.999... ? Really?

So Phyllo,

Yes the numbers you have represented are what I thought too, but how are you mapping them on a line between 1 and 0?

1.0 9/10 9/100 9/1000 9/10,000 9/100,000 and zero is over here.

By this time you are very close to zero already. So how does it work that 1.0 = 0.9 recurring? When by the time you've gotten to the 9/100,000th digit in the number you're already close to zero not 1?

Hey Sil,
Yeah I had really shitty math teachers growing up. Really. Plot .9 on a line representing 1.0 and 0, then plot .999999 on the same line.
I did OK in algebra did really well in geometry. Calculus I barely passed. Didn't take Trig.

Never took a computer class in my life, but I managed to realize plenty. Enough to be the "go to" guy in a tech college with over 1500 employees. My understanding of graphics applications is intense. I've saved companies I've consulted for hundreds of thousands of dollars. So yeah, I really appreciate your condensing attitude. But I don't let it go to my head.
Last edited by Mowk on Fri Apr 10, 2020 11:05 pm, edited 1 time in total.
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### Re: Is 1 = 0.999... ? Really?

So you still don't think that 9/10, 99/100, 999/1000, 9999/10000 is getting closer to 1?
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### Re: Is 1 = 0.999... ? Really?

Silhouette wrote:So if you say an essential thing to understanding the topic is irrelevant, you can imagine it's not an issue?

I don't think it's relevant and I'm not the only one. JSS didn't think it's relevant either. Phyllo probably knows this.

Now, I may be wrong, but if I am, I'll have to realize the relevance of these concepts, and if others want to show me their relevance, they must make an adequate effort.

It's either intentional out of bad faith or unintentional from incompetence - I think a mixture: except I don't think you are explicitly trying to argue in bad faith, I think it's just the usual psychological protections that people usually fall foul to that are getting in your way of changing your mind. If you weren't merely trying to bolster your own (lack of) argument, and instead you tried to understand see the sense in the arguments of others - take a leaf out of Mowk's book for this one - then we could get somewhere, but I think the inability also plays a part against you here too.

This is very much in bad taste. Complaining about others not behaving the way you want them to behave (even if it's better for them to behave in such a way) is a negative self-portrait (:

Suppose you're right and I'm wrong. Either you want to help me or you don't. If you do, you have to make an effort to do so, and most importantly, you must not be surprised if you fail. You make a decision to try to help, you own what follows. If you don't -- and NOBODY and I really mean NOBODY -- is holding a gun to your head, then you simply don't and you mind your own business.

Otherwise, redundancy ensues in the form of unnecessarily many unnecessarily long posts that say nothing of value.

. . . unless, of course, you want to psychoanalyze others, which can be of value, but has nothing to do with the subject.

You like to say these psychological analyses of you are also "irrelevant", but they really aren't - for the same reasons as above ^

Well, if you want to stick to the subject, they are irrelevant.

I don't spend my time psychoanalyzing Ecmandu, Carleas, wtf, Uccisore, Gib and others who are in disagreement with me, not because I am stupid and don't realize this is a useful thing to do if you want to explore the subject, but quite simply because I am on-topic and don't see much value in psychoanalyzing random people on the Internet. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:So you still don't think that 9/10, 99/100, 999/1000, 9999/10000 is getting closer to 1?

To my understanding you are repeating the 9's and not representing them as single digits. 9 10ths the next digit 9 is in the 100th column, the next digit 9 is in the 1000th column.

In whole numbers the number 5,324 has just 1, 5 in the thousands column, just 1, 3 in the hundreds column, a 2, in the tens column and a 4, in the ones column.

To the right of the decimal point, .9999 has just 1, 9 in the 10ths column, just 1, 9 in the 100ths column, and just 1, 9 in the thousandths column.

Nope I don't think what you have written is getting anywhere.

9 divided by 10 = .9
99 divided by 100 = .9
999 divided by 1000 = .9

All of your representations are going no where, stuck to the fist digit. You aren't plotting different numbers. It isn't getting any closer to one or zero.

I could make a snide remark here but I'll just keep that to myself. That'll learn me.

9-10th + 9-100ths + 9-1000ths + ..... now plot the numbers on a line between 1 and 0.
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### Re: Is 1 = 0.999... ? Really?

Nope I don't think what you have written is getting anywhere.

9 divided by 10 = .9
99 divided by 100 = .9
999 divided by 1000 = .9

All of your representations are going no where, stuck to the fist digit. You aren't plotting different numbers. It isn't getting any closer to one or zero.
No.
9/10=.9
99/100=.99
999/1000=.999
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### Re: Is 1 = 0.999... ? Really?

It's a number. Thought it's obvious.
Okay, then you should have no problems relating it to other numbers.

For example :
$$?= \frac{\infty}{2}$$
Is infinity an odd or even number?
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:Is infinity an odd or even number?

From Wikipedia (emphasis is mine):
https://en.wikipedia.org/wiki/Parity_(mathematics)

In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd. An integer is even if it is divisible by two and odd if it is not even.

A formal definition of an even number is that it is an integer of the form n = 2k, where k is an integer; it can then be shown that an odd number is an integer of the form n = 2k + 1 (or alternately, 2k - 1). It is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201.

The standard notion of parity, as indicated by Wikipedia, applies only to integers -- and infinity is not an integer. Of course, you can extend the concept to other kinds of numbers, but in such a case, you'd have to define what parity means. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

The standard notion of parity, as indicated by Wikipedia, applies only to integers -- and infinity is not an integer.
Both interesting and odd, since you are using it to count digits and elements in a series and since you defined it as (1+1+1+...)

So what's the value of
$$\frac{\infty}{2}$$
or of the general
$$\frac{\infty}{n}$$
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### Re: Is 1 = 0.999... ? Really?

did you map the points to a line divided into ten and into one hundred and one thousand with reference to 1 and 0?
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### Re: Is 1 = 0.999... ? Really?

phyllo wrote:Both interesting and odd, since you are using it to count digits and elements in a series and since you defined it as (1+1+1+...)

Yes, you can use it to count digits and elements in a series and though necessary it's not a sufficient condition to classify something as an integer.

From Wikipedia, once again:

The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, ...), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, ...).

No mention of infinity. Hyperreals aren't considered integers either.

What would it mean for an infinite quantity to be even or odd?

Why is that even important?

So what's the value of $$\frac{\infty}{2}$$

"The value of an expression" generally means "An equivalent expression that is of one's interest". "The value of 4 + 4 is 8" means that "8" is equivalent to "4 + 4" and that "8" is the kind of expression that one needs at that point in time (which is usually a single number.)

So I suppose you're looking for an equivalent expression . . . but what kind of equivalent expression? There are many expressions that are equivalent to $$\frac{\infty}{2}$$ e.g. $$\frac{\infty}{4} + \frac{\infty}{4}$$. Magnus Anderson
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### Re: Is 1 = 0.999... ? Really?

What would it mean for an infinite quantity to be even or odd?

Why is that even important?
More inconsistent properties.

"The value of an expression" generally means "An equivalent expression that is of one's interest". "The value of 4 + 4 is 8" means that "8" is equivalent to "4 + 4" and that "8" is the kind of expression that one needs at that point in time (which is usually a single number.)

So I suppose you're looking for an equivalent expression . . . but what kind of equivalent expression? There are many expressions that are equivalent to ∞2 e.g. ∞4+∞4
.
Given a number 'n' we have ways to calculate n/2 which is also a number. If infinity was a number we ought to be able to calculate infinity/2 and it ought to be a number.

You only gave expressions with pending operations. You gave no resulting number.
phyllo
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### Re: Is 1 = 0.999... ? Really?

Mowk wrote:did you map the points to a line divided into ten and into one hundred and one thousand with reference to 1 and 0?
I don't know why I would do any sort of mapping when we don't even agree to the value of 99/100. If you think that it's .9 and I think think that it's .99 then we have a major problem that needs to be resolved before moving on.
phyllo
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