I agree with abgemacht. multiply and dividing infinity doesn't hold water because you could easily make any number equal any other number by multiplying both by infinite.

This is the same reason why 1/inf is not 0.0000...1
You're implying that infinity is a multiple of 10, or 10000... which it isnt, or at least hasn't been proven.

However 0.0000...1 is in fact 0 because you're not doing any arithmetic with infinity, just an infinite number of repeating numbers, the same as working with 1/6.

I have decided to no longer go in depth on this stuff however, I would encourage you to look at my theory that there is a standard infinity in every situation from the last page. If it works it would mean that you were multiplying both sides by the exact infinity, and then, if you believe that there are multiple infinities, one side would be bigger. Example: the amount of rational numbers between 0 and 1 multiplied by 2 would be the number of rational numbers between 0 and 2 which would be bigger. Lastly, 1/i=0 multiply both sides by I, 1=0 ...

For clarity, what is your theory, exactly?

Tvrocks, I agree that there are multiple infinities. It's actually quite common in the mathematics. The number 99999... is infinite as is the number 88888..., by common sense, one would be larger than the other but it isn't that simple. Say you have a billion 9s and a billion 8s, the number with the 9s is larger but 1 billions 9s is not infinity, nor is 1 billion 8s. You couod easily place another 8 in the front, making it larger. Since there are an infinite number of digits in each, neither is larger since there is no last digit. They are just unique values of infinity.

"the amount of rational numbers between 0 and 1 multiplied by 2 would be the number of rational numbers between 0 and 2 which would be bigger."

The set of rational numbers from 0 to 1 and the set of rational numbers from 0 to 2 are both countably infinite and thus the same size.

Abe, it is one page back. Basically it says that as there are multiple infinities in order to make it so that i is not >=< i it says that there is a standard infinity that may depend on the situation, and that you can then use the standard infinity to express other things such as 2i. This can also apply to you yoyo.

"Also, here's a theoretical situation I posted to a friend: Hey Dane, I have a question: so there are the number of real numbers between 0 and 1 red seashells on a beach, ok? There are also the number of real numbers between 0 and one people who want to pair up a blue seashell with each of those red seashells, ok? The number of blue seashells is equal to the number of real numbers between 0 and 2, ok? So each of the people take one of the blue seashells, and pair it up with a red seashells until all of them are paired up, ok? However, they only used the number of real numbers between 0 and 1 blue seashells in order to pair them up, so what happened to the number of blue seashells between 1 and 2? The people would only need to use the number of real numbers between 0 and 1 in order to pair up all the red seashells with a blue seashell, so they wouldnmt need to use any of the number of blue seashealls between 1 and 2, correct? Do you have any problems with this conclusion?". Basically, there would be the same amount of rational numbers between the 0 and 1 of the second one as there were total of the first one, and there would remain a number equal to the number of rational numbers between 1 and 2.

So, first, I think these thought experiments are great, but it's important that you're willing to think critically and honestly enough to realize when you are wrong. Please consider the following:

Let's start with the set of natural (counting) numbers {0,1,2,3,...}. We shall use this as the basis to determine the "size" of other sets of infinite numbers. We do this by creating a mapping of each element in the set of natural numbers to the elements in the set that we are comparing.

For instance, we can create a mapping of the set of "numbers" in the alphabet such that:
0->a
1->b
...
26->z

As we can clearly see, the alphabet set is not infinite because it can be mapped using a finite number of natural numbers.

Now let's look at the set of integers {...,-3,-2,-1,0,1,2,3,...}. It would be reasonable to assume that this set is twice as big as the set of natural numbers. Just looking at it, it appears twice as large. But, let's see if we can create a mapping as we did above.

0-> 0
1-> 1
2-> -1
3-> 2
4-> -2
5-> 3
6-> -3
7-> 4
8-> -4

What we quickly see is that there is actually a very simple mapping from the natural numbers to the integers. If there is a natural number for every integer than these two sets must actually be of the same size.

If the above is clear, then it is quite simple to get the same result in your seashell example and find that, in fact, the number of red and blue seashells is the same.

If the amount of blue seashells were only the number of rational numbers between 0 and 1 would they still be the same?

Yes, the number of rational numbers from 0 to 1 is the same as from 0 to 2. They are both countably infinite. In fact, all the rational numbers are countably infinite.

To answer your question, I'll answer a similar question. There is the # of values between 0 and 1 hotel rooms in a very large hotel. The number of values between 0 and 2 people show up to book a room. What does the hotel manager do?

He could place the number of values between 0 and 1 people in the hotel and then tell everyone in the rooms to shift over one place to fit one extra person. This can happen because the rooms never get filled, it is infinite. There is no end to the number. To prove this, lets say he tells the number of values between 0 and 1 to move into their rooms, and then stops the last person from moving it yet. Is there one room left? No because there is no such thing as the number that comes before infinity. So after the second to last person "fills" the room, there will be an infinte more rooms to fill.

So the smartest move would be to assign the group of people from 0 to 1 a number and then tell them to move into the room number that is double their assigned number. The first one moves into 2 the second into 4... and so on. Then he tells the second half to move into the spaces between, so 1, 3, 5...

In the seashell thing, you could pair them up while still having half of the blue shells left over. (Blue shells 0-1 with red shells 0-1. 1-2 is untouched and remains.)

Lol I'm late

The problem with the hotel thing id that you would notmactuslly be having thrm esch have their own room, half of them would be waiting for a room at any given time.

"In the seashell thing, you could pair them up while still having half of the blue shells left over. (Blue shells 0-1 with red shells 0-1. 1-2 is untouched and remains.)"

No, the point is you can't. Make a mapping that proves what you're saying. It doesn't exist. Both (0,1) and (0,2) map to the natural numbers and so the transitively map to each other.

Half would not be waiting for a room. Every other roomis empty in my example.

Here are the two mappings:

For 0-1:

0->0
1->1
2->1/2
3-> 1/3
4-> 1/4
5-> 1/5
...

For 0-2:

0->0
1->1
2->2
3->1/2
4->3/2
5->1/3
6->4/3
7->1/4
8->5/4
9->1/5
10->6/5

As you can see, both 0-1 and 0-2 map to the natural numbers, so they must also map to themselves.

they must also map to *eachother.

Uh, @tvrocks are you seriously trying to math by... Ignoring actual math? This is very silly. All of your OP conjectures are solidly provable in a particular direction.

1. There are multiple infinities, go look up Cantor's Diagonal Proof.

2. No. You can't divide a real or rational number by an infinite number, they aren't on the same number plane. But go look up Transfinite Cardinals if you want to check out math using infinite numbers.

3. People proved this fairly trivially using algebra in this thread. There are multiple ways to prove it too.

You may not like it, that doesn't make the several proofs magically wrong. Sorry.

@basvan the difference between 0.9(infinite repeat) and 1 doesn't merely tend toward zero, it *is* zero.

@abgemacht actually your first map is wrong it will never map the 2/5

the right thing is just

For 0-1:
0-0
1-1/1
2-1/2
3-1/3
*4-2/3*
5-1/4
*6-3/4*
....

For 0-2:

0-0
1-2/1
2-2/2
3-2/3
*4-4/3*
5-2/4
*6-6/4*
...
you just double the former map, and then you'll need twice the time to arrive at any given rational between 0 and 1


@tvrocks, sorry if I went to sleep in the meantime, the book I mentioned to you is not an easy book, and no, it probably has not the specific answer to the specific question. Actually, you might not like it so much at first, because this is reallly formal mathematics, and it's not easy to get used to it. Anyway, if you ever get those free chapters, I'd say that from chapter 1 (the intro) you'd read only the first 3 questions and the first 2 examples, the others you won't nor need understand. Then yo really start at chapter 2, and as it says, it starts at the beginning cos it happens that not only the real numbers are usually misunderstood, but natural numbers are too. From there, you'd get to a point relevant to our current discussion like in 3 months or so, but then you''ll have a better understanding of maths than many math undergraduates. Actually, if I reccomend this to you, it's rather an experiment, because this book is supposed to be directed to guys like 24 or 25 years old that have already studied formal maths. But I'm convinced that if you are determined enough, you can actually follow it and if you get used to the formaI maths, you can actually really enjoy it. If I reccomend this, is also because it's the best introduction to analysis I've ever met (and our discussion is about analysis)... All of this, of course, if you're seriously interested on all of this.
https://terrytao.wordpress.com/books/analysis-i/

If you wanna get to the point a bit faster (but maybe not as clear), however, I'd follow the reccomendation of basvanopheusden: that you read about the construction of real numbers, and the link he gave seems nice enough
http://tinyurl.com/qzroluv

If it helps I can tell you that good math graduates are some of the best paid people, at least, that's the case in my country.

Also, I'll repeat that the arguments you're giving are very similar to the greek Zenon (or Zeno) arguments. I don't know which version of the Aquilles and toroise paradox you met. But think about this. Suppose you'r a t your bed right now and that you wann go out of your room and the door is at 2 meters distance form your bed.

Well, before you reach you door, you need to cross half the distance, (1 meter). But then agai, you need to cross half the distance of the remaining meter (1/2 meter). But then you still need to cross the half of the remaining 1/2 meter which is 1/4, and then you need to cross 1/8, then 1/16, then
1/32,
1/64,
1/128,
...
1/1024
1/2048
...
1/134217728
...
1/6739986666787659948666753771754907668409286105635143120275902562304

and so, there's always a half of the half of the half of the half of the half of the half of the..... of the half that you still need to cross before yo reach your door.

You see? YOU'LL NEVER REACH YOUR DOOR

@tvrock by the way, the "theory" you are talking about actually formally exists, as I pointed many comments above, it's called "nonstandar analysis" (it's just a *bad* name, don't take the "nonstandard" word too seriously)

https://en.wikipedia.org/wiki/Non-standard_analysis

Of course the formal defs are... well much more formal than your hardly understandable defs...

Stop going on and on about infinity

@ principians

Thanks for the correction. I guess I shouldn't make maps at 3am.

@abge: i was not sure what you wre doing with a map, though after looking at it it is incorrect. A correct map would be something like this
For 0-1:
0-0
1-1/1
2-1/2
3-1/3
*4-2/3*
5-1/4
*6-3/4*
....

For 0-2:

0-0
1-1/1
2-1/2
3-1/3
*4-2/3*
5-1/4
*6-3/4*

@principians: it makes sense if you graph the part of 0-2 that is between 0 and 1, and you will see that both would have the same results, eithout ever having to use the parts between 1 and 2. The main problem with yours is that you assume that the 0-1 part of second is not equal to the base 0-1. I am currently in the middle of reading basvan's article, though do find it interesting. I also think thatbyou misunderstand me. I agree with the idea that if you are only trying to cross half the distance at a time, that you would never actually reach the door. The concept of .5+.25+.125+.0625... Going on forever is in the same situation as .999..., both will go on forever and will become infinitesmally close to 1, hoeever, neither kf them will actually ever reach it. (Or if they did they would keep going.) i actually agree with that situation, thank you for mentioning it.

"by the way, the "theory" you are talking about actually formally exists, as I pointed many comments above, it's called "nonstandar analysis" (it's just a *bad* name, don't take the "nonstandard" word too seriously)" i am actually very happy to hear this, i came up with the "theory" just using careful thought and came to that conclusion, this was without actually doing any in depth research on the theory. (Kind of like the concept of 1/i being the way to write infinitesmal.) I'm glad that it was a real thing though (and, of course, am not surprised that the explanation in the theory is better then me just trying to explain it quickly using mainly theories/ laws. It was actually the first time that i had shared the idea, sos did not expect it to be very clear.

@thorfi: 1. i agree, and have mentioned this concept multiple times. The only thing i was saying is that it seems that there is a standard infinity in every situation that would only represent one of the many as, otherwise, i could be greater than, less than, or equal to another i.

2. Go look up infinitesmals.

3. The algebra they used was flawed. Setting 1/3=.333 then multiplying by 3 means nothing unless the 2 numbers are actually equal. For example: 1=2 . Multiply both sides by 4 and you figure out that 4=8 . This is not true, correct? It is the same situation. If you divide 1 by 3 you will see that it wikl be .333... With a remainder while .999... Divided by 3 woukd be .333... Without a remainder. The presence of the remainder means that 1/3 is actually bigger than .333..., one has a remainder while the other does not. If we subtracted one of them from each other, the remainder would remain. The "proof" x=.999... 10x-x=9 thing is also flawed as when you multiply something by 10, you move the decimal point to the right and there will be one less sig dig behind it. Ex: 9.9(10)=99 .37(10)=3.7. In both of those cases if you subtracted one of them by the other it would yield a result with a remainder because there is one less sig dig behind the decimal point. It os the same situation. In the first one it would have an i number of 9s behind it and in the second one it would have an i-1 number of 9s behind it. Then if you set it up as an equation i>i-1 snd subtract by i on both sides you will see that it ends up as 0>-1 which it true. The second one does have an infinite number of 9s, however, that infinity is smaller than the infinite number of 9s in the original equation. This would be the only conclusion that would make sense if 1/3 was not equal to .333... (Which i proved earlier) as .999... Cannot both be equal to and be less than 1.

I have a question for you though, as .999 is less than 1, and .99(with a billion 9s) is also less than 1, would it be accurate to say that you guys believe that because it goes on forever that, with enough 9s, it would eventually become 1?

@thorfi: this is for you as an answer to #2. The English mathematician John Wallis introduced the expression 1/∞ in his 1655 book Treatise on the Conic Sections. The symbol, which denotes the reciprocal, or inverse, of ∞, is the symbolic representation of the mathematical concept of an infinitesimal.

https://en.m.wikipedia.org/wiki/1/%E2%88%9E#History_of_the_infinitesimal

Reasoning about 'infinity' can be counter-intuitive, which is why it took so long to be formally introduced (late 19th C, if I recall correctly), and even longer before it was generally accepted. If you're unfamiliar with the concept, then better be cautious with using it, otherwise you'll get into trouble sooner or later (this happened to the greatest Greek philosophers and mathematicians who tried). One important thing to have clear, is that 'infinity' itself is not a number (as opposed to 0, 1, π, e, i, etc), nor is 1/∞.

Then the 1=?=0.999... "problem". There are many ways to prove they are just two different ways of notating exactly the same number, and because it's mathematics, all proofs are principially equivalent.
Here's another one, perhaps easier to fathom:

6+2=5+3; proof: (6+2)-(5+3)=0; thus there is no difference.
Or to put it differently, if the difference is 0, then two things are exactly the same thing. By definition.

(tvrocks, I hope you don't disagree with me so far)

Mutatis mutandis: 1=0.999...
Proof: 1-0.999... = 0. QED

@tvrocks
Sorry, didn't understood your comments regarding my second map, but I can saw you're missing the point of the second map by looking at yours, which never will cove the rationals between 1 and 2.

My map is right, but I have to say my claim "you'll need twice the time to arrive at any given rational between 0 and 1" is wrong.

Once again, I'm afraid I won't convince you... well, another instance of why we NEED rigoruous definitions.

I see that you are still thinking that the
.999...
=
.9 +
.09 +
.009 +

gets built one term at a time. No, the same as you don't try to cross one half at a time, you don't try to add one .00...9 at a time: you cross the halfs of your room at once, and you add the terms of the 0.999 at once.

I'v been reviewing the material of the book I've been referring to, and it's even better than I made it appear in my previous comment.

Here an extract of the 3rd page of chapter 2:

"We will aso forget that we know the decimal system, which of course is an extremely convenient way to manipulate numbers, but it is not something which is fundamental to what numbers are. (For instance, one could use an octal or binary system instead of the decimal system, or even the Roman numeral system, and still get exactly the same set of numbers.) Besides if one tries to fully explain what the decimal number system is, it isn't as natural as you might think. Why is 00423 the same number as 423, but 32400 isn't the same number as 324? How come 123.4444... is a real number but ...444.321 isn't? ... Why is 0.999... the same as 1? What is the smallest positive real number?"

So you'd read trhough chapters 2, 3, 4, 5 (this is a big bad new, cos ch. 5 is not among the free chapters) and then proceed to appendix 13 whose execise 13.2.2 asks you to prove that the only "decimal representations" of 1 are 1.000... and 0.999... It's still something that could require various days (if not weeks) of focused work, but if you are into anything related maths, it will be worth the time :)

Finally, the *nonstandard* analysis is rather equivalent to the *standard* one (the names are because usual math tradition is based on the latter, had Leibnitz lived in England and Newton in Germany, the names would be the other way around), yes, there are some subtleties, but basically the only important difference between 2 is that one is based on infinitesimals, and the other is based on limits (and yes, in the nonstandard one, 1 might not be equal to .999... but that would depend on how you define .999... anyway)

@nescio: your 1-.999... "Proof"is flawed as it assumes that .999... Is equal to 1. It is actually just infinitesmally close, which would be written as 1/infinity. 1-1/infinity=.999... . They exhibit different patterns, such as one being eivisbke ny 3 whike the other yields a remainder, and one being divisibke by 5 while the other does not. I also assume that you would be advocating for 1.999... Being 2 and 2.9999... Being 3. On at least the example of 2.999 you can see using the rule for checking whether a number is divisible by 3 (add all the digits, 3rd grade concept) that 3 will be even,y divisible by 3 while 2.999... Will not. 2 will be evenly divisible by 2 while 1.999 will not. It is like this with LITERALLY ALL terminatiipng decimals and their "non-terminating" forms. When you use the things that are infinitesimally close to them and divide, add, subtract, multiply, lr do anything else with them, you will get a different result. This idea proves that they are not the same.

@propincipians: i think of .999... As an infinite number of 9s rather than an infinitely repeating series. (Although all numbers could theoretically be infinite.y repeating series. The very concept of limits is that it will always be approaching it. If it ever actually reaches it then that thing was not the actual limit.