The answers to 1^(1/2) and 1/0

[exfret]

Through the laws of the general relativity of a testtubegames forum post to the topic at hand,
"A post made after a pause in the advancement of a thread will be ignored by the next poster, who will choose instead to continue the topic that had been at hand before the pause, even though he/she was aroused by the post he/she is ignoring."

Edit/Adding-on: 2nd law: Pauses tend to coincidentally happen when the next post will be on the next page...

[robly18]

I didn't add anything because I didn't think of anything to contribute :p
We've come to an agreement. 1/0 is infinity, except all kinds of it.

Speaking of which, further proof for 1/0 not being positive or negative:
Positive number * Positive number = Positive number
Negative number * Negative number = Positive number
Positive number * Negative number = Negative number
Number * Number that is neither negative nor positive(0) = Number that is neither negative nor positive(0)

These rules apply for division as well. However, the last one doesn't quite work because we know that 1 / b != b for b not being 1. So we end up with a non negative nor positive number that isn't zero.

[exfret]

Wait, you agree with me?!? Well, I still don't have concrete proof of my definition of ∞ satisfying the property that it should be greater than all other numbers. Do you have any ideas of how I could present this to the general public? I still think it's full of formal flaws that would ruin it if it was to be scrutinized by someone like my teacher. Actually, if I were to choose whether or not to believe that 1/0=∞ when presented with only my proof, I would probably decide making 1/0 undefined as the best choice for mathematicians. Only when looking at how this fits in with the rest of mathematics do I see that this would really make sense, but that doesn't mean it's right. I was just trying to be more convincing to you than I am to myself in my past proofs so that you could "see my side of the story". An analogy would be me trying to pull you from one side of the mountain to another so that you can observe from the peak for a short period of time to see which side is greener. Just because the grasses of 'my' side of the mountain are arranged in a better pattern for rainfall doesn't mean they'll be more green (figuratively, of course). Still, in a way, undefined isn't technically incorrect, while my model could be. You can just avoid defining anything and you'll still never be incorrect. I'm now starting to think that a better way to put my result(s) of 1/0 would be to call it(/them) a number, ( stands for a constant in this case or should I use the letter ?), in which is a number that is any integer followed by an endless stream of zeroes before the decimal point. Basically,
                 _                  _
= n 0 . 0 = n * 1 0 . 0

I am using a letter for shorthand. It's like how i is √(-1), but we don't just write √(-1) all the time because that would obscure meaning. So, basically, 1/0 is definitely defined, it's , we just don't know if = ∞. Anyways, I'm really wondering what uses could sprout out of this. I feel so much like writing lots of new numbers like + 1, and i, which sadly are likely both equal to (well, i definitely equals , but there's still hope, however so small, . How could this expand to help other areas of mathematics or even possibly create it's own field of mathematics?

I wrote a lot in this post. The questions I am really asking are:
-What errors in my proof should I correct?
-How should I prove = ∞ if possible?
-How should I present my findings to the general public?
-What should I do about my math teacher last year teach x^(1/2)=√(x)?

[robly18]

This post is a reminder for future me to respond. Gotta go right now, but when I get the time, note to self: respond!

[testtubegames]

Latecomer to the party here

One thing I've really liked about watching this discussion is how deeply you've been exploring the topics at hand. I'm sure Joe Mathematics and Anne Arithmetic had similar discussions on this topic when the question of 1/0 first came up.

exfret -- one problem I have with your proof is in the starting definitions. You say that is 'any number followed by an endless stream of zeros before the decimal point'. But beware. No number has an endless string of zeros before the decimal point. Any single number that you can define is a definite number of steps away from zero. (And having complex numbers doesn't help us out on this front, either... the imaginary portion of a complex number adheres to this fact, too.)

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Sidebar about limits. I'm a physics dude, so I, too, once preferred to play fast-and-loose with infinity... leaving limits at the wayside, treating infinity as a quasi-number. But as I learned in college, you do that at your peril. The concept of infinity is inextricably linked to limits. Without limits, you can't have the concept of infinity. And if you use limits carefully -- they can be really beautiful. They help you simply and elegantly get answers to otherwise confusing topics. (Like the one we're talking about.) exfret, you seem to be pretty comfortable with math and eager to push the boundaries of your knowledge. Which makes me thing that you, too, would really enjoy learning more about limits. Gotta think if I know any good resources...

[exfret]

Sorry for taking so long to reply, but I just wanted the time to reply with a well-written response.

This post is a reminder for future me to respond. Gotta go right now, but when I get the time, note to self: respond!

Here's another of those reminder posts. Hope this one works!

Latecomer to the party here

Eh, only 9 days 'late'. I'm posting this 7 days afterwards, so I guess that makes both of us late.

One thing I've really liked about watching this discussion is how deeply you've been exploring the topics at hand

Didn't we do that to your poor gravity simulator as well? !!!Ohh, poor, poor gravity simulator, being picked on like ~TES~!!! By the way,

(Sorry, the punctuation… it’s obligatory!)1

.

Ah, so I see you're a double-space guy.

I'm sure Joe Mathematics and Anne Arithmetic had similar discussions on this topic when the question of 1/0 first came up.

I just want to inquire about the meaning of what you're saying here to make sure I understand the full meaning of this statement. You're just saying that this topic has been covered before, right? Well, that's part of my reason for posting it: it obviously goes against the status quo. (This is actually a bit of a good thing because I don't have to do any research to see if anyone else has came up with the same idea ).

exfret -- one problem I have with your proof is in the starting definitions. You say that is 'any number followed by an endless stream of zeros before the decimal point'. But beware. No number has an endless string of zeros before the decimal point. Any single number that you can define is a definite number of steps away from zero.

This is from http://en.wikipedia.org/wiki/Number: A number is a mathematical object used to count, label, and measure.
And this is from https://www.google.com/search?q=number+ ... 1&ie=UTF-8: an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification.

Well, the agreement here seems to be that a number is a thing expressed in a figure that represents a quantity. My number is obviously a figure (t :His is quite obvious, in fact), which makes it a thing, and it represents a quantity. Sure, the quantity may not be obvious or could even be undefined or whatever or multiple quantities, but this doesn't mean it doesn't have a quantity. You're the one making the claim here, not me. Yes, you may think that I am the one making a claim by basically implying its number-ness by creating this proof and saying that it's the result of 1/0, but why would I ever have to say that had to be a number for 1/0 to be equal to it? I mean, come on, we set it as undefined before and don't you think that's pretty number/quantity-less, too? We didn't say "not a number/value," we said "undefined," and unless you explain how why my "number" has to be a number, I don't see why I can't just pick it right out of my imagination. Also, here's something from that wikipedia page mentioned above:

In mathematics, the definition of number has been extended over the years to include such numbers as 0, negative numbers, rational numbers, irrational numbers, and complex numbers.

So, basically, the definition of what a number is has grown, and there's nothing to say it can't grow to include my number as well. Oh, and here's a dilemma for you:
When the concept of zero was first created, there was some disagreement on whether it should be considered a number. Don't you think it's a number? Well, according to your definition, a number is

Any single number that you can define is a definite number of steps away from zero.

Well, it's a single number, it's define-able, and it's a definite number of steps away from zero, right? Well how many steps away from zero is zero? Just think about that. So, doesn't it seem like a circular argument to defend zero's number-ness? You see, we invented numbers, so one more definition won't mess anything up.

(And having complex numbers doesn't help us out on this front, either... the imaginary portion of a complex number adheres to this fact, too.)

Are you talking about the complex number plane? That thing was only invented to help us imagine the complex numbers. It doesn't represent their actual distances from zero. Sure, their absolute values are some distance away from zero, but the actual distance they are from zero on a complex plane is measured inches (or some other measurement of length, but you get my point). The "definite number of steps" i is away from zero is i, regardless of how close you place it to zero on the complex plane or what operations you put on it (like absolute value) to get something that's positive.

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Sidebar about limits. I'm a physics dude, so I, too, once preferred to play fast-and-loose with infinity... leaving limits at the wayside, treating infinity as a quasi-number.

I take your advice. I've seen that infinity isn't really treated as a number like it should be (in my opinion). I mean, it's the number that's greater than all numbers. That's basically it's definition. Besides, you can't approach something that isn't a number, or if you can, then that thing you're approaching should at least be considered a quasi-number. Also, this is part of the reason I chose to just say instead of infinity was because I realized in this discussion that I don't have a proof that 1/0 is infinity, but I do have one that defines 1/0. I think my proof shows that we can't just say it's "undefined" anymore. There's an actual number (or "non-number" or whatever it is) that 1/0 is equal to, and I just defined it, so leave it undefined no more, because doing otherwise would be as ignorant as the ignorance of the people who despised Cantor simply because their "common sense" went against his claims.

But as I learned in college, you do that at your peril.

Also, there's a reason I'm leaving limits at the wayside. I don't want to know what the limit of 1/x is as you approach x, I want to know the actual value. Oh, and here's an example of a case where limits are deceiving: What's the sum of the sequence 2^x at infinity (e.g. 1+2+4+8+16...)? Well, limits might lead you to believe it's infinity, but it's actually -1 (http://www.youtube.com/watch?v=4MUOdUvtf8o, which is actually a very good video for this argument in other ways, too). I used properties of numbers to find out what 1/0 actually is, and trying to say it's not would be like saying that i isn't sqrt(-1).

The concept of infinity is inextricably linked to limits. Without limits, you can't have the concept of infinity.

Actually, you can have infinity without the concept of limits. It's the number larger than any other number. Okay, I would actually go into more depth on this and the following things, except your forum thing, like, logged me out automatically for trying to preview my post (just because I had quoted you, BTW), so I just lost half, maybe a full hour of work. That would have quadrupled if I hadn't been smart enough to save, so please, (tell me how to) get rid of the thing that automatically logs me off. You have no idea how stressful it is to do something like creating a 5-or-so-page long proof, and almost forgetting to save it before logging back in to submit it. On the other hand, this is good practice to get me to save my work.

And if you use limits carefully -- they can be really beautiful. They help you simply and elegantly get answers to otherwise confusing topics.

Well, they're great with approximations, and I used to have a big analogy about holes in walls an all here, but I guess there's just a hole in the wall of the forum now. Too bad. I need to get to sleep, so I don't really have any time to elaborate over the details of it again, but basically, limits are extremely useful in doing things for practical solutions, like in physics, but in pure mathematics, the concept isn't full-proof. Also, not having limits in my proof doesn't make it less credible. In fact, it's almost the other way around.

(Like the one we're talking about.)

Did I explain it in a confusing way?

exfret, you seem to be pretty comfortable with math and eager to push the boundaries of your knowledge. Which makes me thing that you, too, would really enjoy learning more about limits. Gotta think if I know any good resources...

Basically, I said here that I would enjoy reading bout limits and they would probably be relevant here, but there was a reason I didn't use limits (well, I actually also didn't know about limits when I created this proof in the first place, either), and it was because they're limited ( ) in the fact that they don't give you what actually happens at a particular spot, but instead what might happen if a trend continues, and I wanted to know what happened, period.


Here I just said, "Pwease wespond to my conscerns about scare woots, too," except without that cute accent added in that I added in just because I was rushing and didn't have the time to stop my self and say, "Is a cute accent really appropriate in here, or is it just too silly?"


And, finally, I put here some words telling you not to be offended if I had been 'too passionate' in any part of this post.


Also, if you ever reach a confusing, contradictory, incorrect(ly formatted), or just plain bad part of my post, please alert me so I can fix it, for I did not have the time to revise or edit this like I usually do to long posts, because it was getting late, and I did not think it would be a good idea to spend more time on this post (and my post was killed because I was automatically logged out ), so you may see a variety of errors here that need to be fixed. Just alert me of said errors, and I will respond to whatever you said by fixing what you said I said in a way I shouldn't have said by placing the thing I should have said so that no one will have said something about it, like the said part where I said all the saids that you could barely tell what I said, so I said something you didn't understand, making that said part I said a bad part in this said post, meaning you should have said something about all my saids and the fact that even I said it was too confusing so I should take it out like I promised, which is a synonym, for said.


1http://testtubegames.com/blog/shockto-progress/.

[robly18]

Here's another of those reminder posts. Hope this one works!

Mine didn't D:

This is from http://en.wikipedia.org/wiki/Number: A number is a mathematical object used to count, label, and measure.
And this is from https://www.google.com/search?q=number+ ... 1&ie=UTF-8: an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification.

Well, the agreement here seems to be that a number is a thing expressed in a figure that represents a quantity. My number is obviously a figure (t :His is quite obvious, in fact), which makes it a thing, and it represents a quantity. Sure, the quantity may not be obvious or could even be undefined or whatever or multiple quantities, but this doesn't mean it doesn't have a quantity. You're the one making the claim here, not me. Yes, you may think that I am the one making a claim by basically implying its number-ness by creating this proof and saying that it's the result of 1/0, but why would I ever have to say that had to be a number for 1/0 to be equal to it? I mean, come on, we set it as undefined before and don't you think that's pretty number/quantity-less, too? We didn't say "not a number/value," we said "undefined," and unless you explain how why my "number" has to be a number, I don't see why I can't just pick it right out of my imagination.

Hold on, let me try a proof by contradiction here...
Axiom list, for reference: http://quizlet.com/363753/algebra-1-pro ... ash-cards/

First, let's start off with 1/0, and multiply it by zero. So we get:
(1/0)*0
This is equal to 1/0 * 0/1. As the axiom of multiplicative inverses says, a number multiplied by its inverse equals one. So we reach the conclusion that (1/0) * 0 = 1.
Then we move on to the axiom of the multiplicative property of zero, which states that any number multiplied by zero equals zero. So (1/0) * 0 = 0.
Afterwards, we move on to the axiom of substitution principle, which states that if (1/0)*0 = 0 then they can be used interchangeably. So with the formula (1/0) * 0 = 1 we can replace the former by zero, getting 0 = 1
This is obviously false.
Therefore, 1/0 is either not a number, or is one that doesn't obey the axioms set in mathemathics. I'll let you be the judge of which one is most likely.

Well, it's a single number, it's define-able, and it's a definite number of steps away from zero, right? Well how many steps away from zero is zero? Just think about that. So, doesn't it seem like a circular argument to defend zero's number-ness? You see, we invented numbers, so one more definition won't mess anything up.

Zero is the same numbers away from zero as one is away from one, or any number is away from itself. That is the definition of zero.

Are you talking about the complex number plane? That thing was only invented to help us imagine the complex numbers. It doesn't represent their actual distances from zero. Sure, their absolute values are some distance away from zero, but the actual distance they are from zero on a complex plane is measured inches (or some other measurement of length, but you get my point). The "definite number of steps" i is away from zero is i, regardless of how close you place it to zero on the complex plane or what operations you put on it (like absolute value) to get something that's positive.

I'm not exactly sure what to say about this one.

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I take your advice. I've seen that infinity isn't really treated as a number like it should be (in my opinion). I mean, it's the number that's greater than all numbers. That's basically it's definition. Besides, you can't approach something that isn't a number, or if you can, then that thing you're approaching should at least be considered a quasi-number. Also, this is part of the reason I chose to just say instead of infinity was because I realized in this discussion that I don't have a proof that 1/0 is infinity, but I do have one that defines 1/0. I think my proof shows that we can't just say it's "undefined" anymore. There's an actual number (or "non-number" or whatever it is) that 1/0 is equal to, and I just defined it, so leave it undefined no more, because doing otherwise would be as ignorant as the ignorance of the people who despised Cantor simply because their "common sense" went against his claims.

My counter-proof, above, shows it is indeed not a number.

Also, there's a reason I'm leaving limits at the wayside. I don't want to know what the limit of 1/x is as you approach x, I want to know the actual value. Oh, and here's an example of a case where limits are deceiving: What's the sum of the sequence 2^x at infinity (e.g. 1+2+4+8+16...)? Well, limits might lead you to believe it's infinity, but it's actually -1 (http://www.youtube.com/watch?v=4MUOdUvtf8o, which is actually a very good video for this argument in other ways, too). I used properties of numbers to find out what 1/0 actually is, and trying to say it's not would be like saying that i isn't sqrt(-1).

That series actually converges towards infinity, because, for n terms in that sequence you sum, you get the number:
2^(n)-1
So if you sum only the first term, you get 1
Sum the first two, and you get 1+2 = 3
First five and you get 31
What that proof forgets is that, while there is an infinite number of numbers to sum (?), some infinities are greater than other infinities. The number of odd numbers isn't the same as the number of integers, despite the fact that for every integer you have an odd number and vice versa. In the end, the difference between odd numbers and integers is infinity, or the number of even numbers.

Well, they're great with approximations, and I used to have a big analogy about holes in walls an all here, but I guess there's just a hole in the wall of the forum now. Too bad. I need to get to sleep, so I don't really have any time to elaborate over the details of it again, but basically, limits are extremely useful in doing things for practical solutions, like in physics, but in pure mathematics, the concept isn't full-proof. Also, not having limits in my proof doesn't make it less credible. In fact, it's almost the other way around.

You don't necessairly need approximations. There are ways.
For instance, I know that sum(4^n/9^n) starting at zero growing towards infinity equals 9/5, and this isn't an approximation. If you watch vihart you know what I'm talking about. If not... Who am I kidding, you have.

Basically, I said here that I would enjoy reading bout limits and they would probably be relevant here, but there was a reason I didn't use limits (well, I actually also didn't know about limits when I created this proof in the first place, either), and it was because they're limited ( ) in the fact that they don't give you what actually happens at a particular spot, but instead what might happen if a trend continues, and I wanted to know what happened, period.

That's the thing, they give you what happens at a particular spot.
I seriously recommend you look up something on calculus, it's quite interesting. It also helps you fend off those who don't understand the beauty of infinitely repeating nines.

Here I just said, "Pwease wespond to my conscerns about scare woots, too," except without that cute accent added in that I added in just because I was rushing and didn't have the time to stop my self and say, "Is a cute accent really appropriate in here, or is it just too silly?"

Square roots (Took me a while to spell that right, darn your cute accents!) are simple:
sqrt(a) = b
b^2 = a
Of course, due to -(-x) = x, when you square something, you drop the plus sign. Which is why every number has two square roots.

Das basically everything I've an answer to. Your move.

[exfret]

Hold on, let me try a proof by contradiction here...
Axiom list, for reference: http://quizlet.com/363753/algebra-1-pro ... ash-cards/

First, let's start off with 1/0, and multiply it by zero. So we get:
(1/0)*0
This is equal to 1/0 * 0/1. As the axiom of multiplicative inverses says, a number multiplied by its inverse equals one. So we reach the conclusion that (1/0) * 0 = 1.
Then we move on to the axiom of the multiplicative property of zero, which states that any number multiplied by zero equals zero. So (1/0) * 0 = 0.
Afterwards, we move on to the axiom of substitution principle, which states that if (1/0)*0 = 0 then they can be used interchangeably. So with the formula (1/0) * 0 = 1 we can replace the former by zero, getting 0 = 1
This is obviously false.
Therefore, 1/0 is either not a number, or is one that doesn't obey the axioms set in mathemathics. I'll let you be the judge of which one is most likely.

There are several problems with this proof. First of all, as I said, who ever said that or 1/0 both had to be numbers for them to be equal to each other? Also, as you pointed out, the fact that they violate laws of numbers can only mean 2 things: the laws are wrong, or the laws don't apply to those numbers. It can't be that it 'isn't a number.' How would it 'not be a number'? Even then, if it weren't a number, what would that mean, and what significance would that have? Just that it doesn't act like other numbers? Doesn't i not act like other numbers? Just showing that it doesn't act like most numbers doesn't justify it 'not being a number.' [Insert transition word here] The laws obviously aren't wrong, but it is possible that they don't apply to all numbers. The 'property' that n/0= (or undefined, it doesn't matter for my purposes here and now) doesn't hold true for 0, but that exception doesn't make 0 'undefined' or shunned like has been all these years. This especially holds true when you look at proofs that zero multiplied by any number is 0. They may assume that for any n, n+1 != n, but this obviously isn't true for ∞, so you can't argue this when 1/0 may in fact be equal to ∞. Also, something you're forgetting is that 1/0 is an operation, and operations can return multiple values (just like you stated with square roots), so 1/0*0 may in fact be 1 and/or 0.

Well, it's a single number, it's define-able, and it's a definite number of steps away from zero, right? Well how many steps away from zero is zero? Just think about that. So, doesn't it seem like a circular argument to defend zero's number-ness? You see, we invented numbers, so one more definition won't mess anything up.

Zero is the same numbers away from zero as one is away from one, or any number is away from itself. That is the definition of zero.

"The same amount of steps aways as any number is from itself" doesn't quite seem like a clear, definite number of steps away from zero to me. I meant that it was 0 steps away from zero, and if zero was to be debated as a number, it would be hard to prove that it followed the given definition of a number without a circular proof.

Are you talking about the complex number plane? That thing was only invented to help us imagine the complex numbers. It doesn't represent their actual distances from zero. Sure, their absolute values are some distance away from zero, but the actual distance they are from zero on a complex plane is measured inches (or some other measurement of length, but you get my point). The "definite number of steps" i is away from zero is i, regardless of how close you place it to zero on the complex plane or what operations you put on it (like absolute value) to get something that's positive.

I'm not exactly sure what to say about this one.

It was part of Andy's evidence anyways. You don't necessarily have to defend what he said.

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Andy, may I ask what inspired you to create such a beautiful creation as the triple dash?

My counter-proof, above, shows it is indeed not a number.

I would like to stress once more how silly this 'not a number' thing seems to be to me. Could you explain what's so important about something 'being a number'? Also, tell me why I must use only 'numbers' following the vague definition of being a number given to me by Andy? (I proved it to be vague with my explanation of i being a number and my discussion with 0 as well). And,:
My counter-counter-proof, above, shows it is indeed not necessarily 'not a number'.

That series actually converges towards infinity, because, for n terms in that sequence you sum, you get the number:

You mean it diverges towards ∞.

2^(n)-1
So if you sum only the first term, you get 1
Sum the first two, and you get 1+2 = 3
First five and you get 31

Sorry, I didn't include the original video. Here:
http://www.youtube.com/watch?v=kIq5CZlg8Rg
The only thing I see with it that might be wrong is that he didn't align his terms, so as I saw in one of the comments, as you go higher, you should get 2x-x, which the comment said was 2*∞-∞, which 'obviously' equalled ∞. Even so, 2*∞=∞, as I discuss in the following reply-paragraph thing, and therefore, 2*∞-∞=∞-∞=0, but since ∞=∞+1 (because no number can be greater than ∞), ∞-∞=∞-(∞+1)=∞-∞-1=0-1=-1, so his proof still isn't contradictory. If you're confused about this, remember ∞-∞ is an operation, so it can have multiple possible answers. The fact that it may be 0 doesn't make it 0, and since the MinutePhysics guy's proof can be used to show that it's -1, that information is enough to know which 'solution' of ∞-∞ to use. Also, the fact this follows the formula for convergent series, a/(1-r), isn't just a coincidence. a in this case is 1, because the formula for the nth sequence in this case is 1*2^n, and r=2, because it's the exponent, so the sum of all the sequences for the formula 2^n is 1/(1-2)=1/(-1)=-1. You see, we seem to think there's an exception for this formula because we can't wrap our minds around how this sum can be a finite number, but there is no such exception in the proof of this formula, meaning no such exception should exist at all, and it doesn't. It's so amazing, and it seems so false, but you needn't restrict this formula at all to numbers between 0 and 1 (just like we needn't restrict division in the case of 1/0). r can be 2, -1.4, or any real number. In fact, why restrict it to real numbers at all? This formula should work for complex numbers, too, but I do admit I present this claim with a lack of evidence (it would only add to the length of this post to explain it anyways). You see, it seems like it would be ∞, yet it isn't, which is exactly why limits are deceiving. Besides, I was just trying to defend the fact that I didn't use limits in my proof to Andy. It seemed like he was discrediting my proof by making a claim of 'not being a number.' And, sorry Andy, but I just feel like this statement seems so silly and meaningless. I mean, exactly what are you proving by trying to show 'isn't a number' and what's so significant about this that it discredits what I've stated? He also seemed to be criticizing my lack of limits, which I defended by showing that limits can make your proofs less reliable.
Also, a note to Andy:

Andy,
You say "do that at your peril," while using limits is the action that you should do at your peril, as I have shown. The fact that I am investigating ∞ just means that there will be many strange, unintuitive answers that I should be careful of, but I believe I have reached those answers, and the fact that they're unintuitive doesn't make them incorrect (just like with Cantor's work, which I like to compare to my own, even though mine isn't really 'work,' and he made more advancements and his advancements were more important, plus mine aren't official (yet) and may very well be completely wrong, but it's comparabola to Cantor's nonetheless). Don't view what I have done as "playing fast-and-loose with infinity" unless you have legitimate reasons for thinking I'm "playing fast-and-loose with infinity." If you do, then I would I appreciate it if you stated them. I've only tried to clear up a subject I've viewed as extremely misconceived. What about that makes you add such an 'ignorant teenager' connotation (please excuse me for that comparison, but it was the only one that I could think of that truly expressed my thoughts) to your reply? What's so wrong with using infinity instead of limits approaching it?
[Insert Salutation (or is the salutation in the beginning?) Here],
exfret

P.S. I apologize if I was ever 'forceful', 'overly-criticizing', or hurtful with anything I said or will say.

What that proof forgets is that, while there is an infinite number of numbers to sum (?), some infinities are greater than other infinities. The number of odd numbers isn't the same as the number of integers, despite the fact that for every integer you have an odd number and vice versa. In the end, the difference between odd numbers and integers is infinity, or the number of even numbers.

Woah, hold on there, now your even going against what the current consensus in mathematics is. Did I include my reference to Cantor in my reply, or did it delete that as well? Anyways, there was this guy named Cantor, and he basically completely disagreed with every single sentence in your whole last paragraph, and his work is mostly accepted nowadays. In fact, he even proved that there's the same number of rational numbers as natural numbers, and the same number of numbers from 0 to 1 exclusive as there are (points) in the whole cartesian coordinate plane and even in the whole Universe, even if it's infinite. You should read up on or watch some videos about him. You don't need to read/watch a lot, I just think you should get the general idea of what he did. You already summed up his reasons: there exists a one-to-one ration between odd integers and integers. Here's some resources:
http://www.youtube.com/watch?v=elvOZm0d4H0: Numberphile.
http://rjlipton.wordpress.com/2011/10/2 ... -is-wrong/: This discusses Cantor's work, even if on a slightly different subject than we are discussing here. Too bad I couldn't find the original videos I found about Cantor with that guy drinking a glass of water.
Anyways, since I know numberphile doesn't understand the reasons behind things in mathematics that well (at all), I'll give one (an explanation, that is). Basically, I proved in my proof (my first post) that ∞*n=∞, where n is any real number. Well, in reality, I proved that * n = for any n where n is a real number, but the same goes for ∞ as well, because no number can be greater than ∞, and that goes for 2*∞ as well as well (the fact that there's a number violating this property doesn't mean this isn't true, but rather that there may be something we're missing, but my proof of * n = is at least more concrete), and these similarities between and ∞ are actually what caused me to believe that = ∞ and therefore 1/0 = ∞, too. Getting off that tangent, because 2*∞=∞, and since there are ∞ odd numbers and comparatively 2*∞ integers, there are the same number of odds as there are integers. Also, you see how I had to use the word comparatively? Well, you could also say that there are ∞ integers, and therefore there must be 1/2*∞ odd numbers. You see, the number of odd numbers can be changed depending on which set of numbers you view as being ∞ first. Therefore, since you can view there being ∞ integers and ∞ odds without violating anything, there's the same number of odds as there are integers.

You don't necessarily need approximations. There are ways.
For instance, I know that sum(4^n/9^n) starting at zero growing towards infinity equals 9/5, and this isn't an approximation. If you watch vihart you know what I'm talking about. If not... Who am I kidding, you have.

Yes, I have, but your example is only an example, and thus proves nothing. My example with the sum of 2^n is a counterexample and thus disproves everything. Sorry for the vague language, but I just thought that would sound really cool. (Wow, that is a really bad symbol for that smily). Basically, that is an example of only one case where limits may be accurate (I could argue against it's validity, but I choose not to nitpick at it's flaws here). My example showed that there are cases where limits aren't correct, so you may not necessarily always need to attack a problem directly, but limits don't always work either, and even worse, there's not really any way to prove whether the limit worked or not without using a non-limit method (I admit this is an assumption, but my point is that limits aren't trustworthy, and this statement is supported).

That's the thing, they give you what happens at a particular spot.
I seriously recommend you look up something on calculus, it's quite interesting. It also helps you fend off those who don't understand the beauty of infinitely repeating nines.

You don't need Calculus for repeating nine's. I thought of that proof after being taught something by my 6th grade math teacher. I didn't even need algebra. I also appreciate your signature in this thread, because the thing I was taught by my 6th grade math teacher that inspired this proof was that 0.999999...=1. Oh, and the whole idea behind limits is that they get closer and closer to what happens at a particular spot instead of giving you that spot's actual value. You don't use limits to find the value at that spot, just like you don't take Riemann sums to find an integral. Instead, you use the concept of limits and work around the problem with some elegant moves. So, no, your wrong, they don't give you what happens at a particular spot.

Square roots (Took me a while to spell that right, darn your cute accents!) are simple:
sqrt(a) = b
b^2 = a
Of course, due to -(-x) = x, when you square something, you drop the plus sign. Which is why every number has two square roots.

Sorry, I don't really understand what you're saying here, but my real problem is that my math teachers are teaching that any number has one square root. In fact, this is taken straight from the VA Standards's of Learning:
"write radical expressions as expressions containing rational exponents and vice versa" (http://www.doe.virginia.gov/testing/sol ... matics.pdf, page 181, mathematics SOL c)
You interpret the meaning of this, but to me, it simply means that rational exponents are the same as radicals, but sqrt(x) is NOT equal to x^(1/2), because x^(1/2) must return both the positive and negative expressions, but it is well recognized that sqrt(x) returns only the principle square root. Technically, this just means it isn't always equal to x^(1/2), but many teachers and countless students at my school alone are being mistaught. I mean, the SOL people have screwed up this thing, and it's not even a prevalent myth! (Like the myth that people thought the Earth was flat in the Medieval Ages/Columbus' time). (It will be if they keep it up, though). I'm just asking what I should do about this.

Das basically everything I've an answer to. Your move.

My move was changing your name to robly19.

[robly18]

Man, you sure have great skill at making walls of text my brain hurts when reading.
As for the not a number thing, here's why it isn't a number:
The axioms apply for everything. They're the basis of mathematics. They are literally what defines our system of counting, adding, subtracting... If something doesn't obey these axioms, then it isn't really a thing.
As for the "how can it not be a number thing" let me make an analogy.

Say, n = n + 1. Is this possible?
As my good old friend Jim would say, this would mean 0 = 1. It's not possible as you can see.
Jim isn't a real person. Anymore, at least.

Anyway, as for the n = 1/0, here's why it's wrong:
Multiply both sides by zero
0n = 1
Use axiom proofs to figure out this:
0n = 0
And then ,you get 0 = 1.
This is... not really possible.
Therefore, the existence of a number that is equal to one over zero is impossible, just like the existence of a number equal to itself plus one. They do not obey the axioms and cause inconsistencies. It may sound like a case of "it doesn't obey what we know, so sweep it under the rug", but that's the thing. Their existence would make mathematics inconsistent. We call the undefined because, with our set of axioms, they cannot be defined.

[exfret]

Man, you sure have great skill at making walls of text my brain hurts when reading.

Why is your brain being mean to my poor posts?

As for the not a number thing, here's why it isn't a number:

Ah, now I see. So you're nailing it down for not following the basic axioms of mathematics. Just saying it wasn't a number wasn't a very precise definition (*cough* Andy *cough*) or very valid argument, but saying it can't be because it violates basic properties of arithmetic seems like a more solid argument. It's just that words like 'number' can be varied in their meaning, and it isn't very obvious or provable (at least I haven't seen it proven) what 'not being a number' would even mean, much less why it would imply the impossibility of taking an operation. Besides, i technically isn't a 'real' number, and you could define it as 'not a number' at all, so that would mean you could never take sqrt(-1), but i still follows basic laws of arithmetic. Saying an operation is impossible due to its violation of the axioms that are the foundations of mathematics makes much more sense.

The axioms apply for everything. They're the basis of mathematics. They are literally what defines our system of counting, adding, subtracting... If something doesn't obey these axioms, then it isn't really a thing.

Once again, may I say you offer a very clear explanation, but as I hope to do the same, you should hopefully see that the logic of these axioms needs to be revised when discussing ∞, and that ∞ actually follows these axioms anyways.

As for the "how can it not be a number thing" let me make an analogy.

Say, n = n + 1. Is this possible?
As my good old friend Jim would say, this would mean 0 = 1. It's not possible as you can see.

Actually, n=n+1->0=1 is not exactly correct. You see, when you write the whole thing out, something pops up:

  n=n+1
-n  -n
n-n=n-n+1

Now, you might say: "Hey! n-n=0, right?" Well, that's right, but remember, we're taking an operation here, and operations can return multiple values. This may hurt your head, but it happens a lot when it comes to ∞, and it happens all the time with square roots, too, so it isn't an entirely new concept. Now, remember, since n=n+1, we can plug in n+1 in as n, and we get this:

n-n=n-(n+1)+1
0=n-n-1+1
0=0+0
0=0

Now, before you start telling me how this violates the basic laws of mathematics, I never said n-n wasn't equal to 0. It just could be equal to other things, too. This doesn't mean 0 isn't one of its 'solutions'. You have to remember that some operations have multiple values, but only one of them is the 'right' one (remember that discussion about 0/0?), and sometimes we can even deduce which one is right, but you can't just pick out one of those values and plug it in, because it'll get contradictory things, like you've shown. In this case, n-n actually can't be 0 for both sides of the equation. It's kind of like derivatives. A derivative is actually just 0/0, but by doing some cool tricks, you can find out which of the values returned by the operation 0/0 was right. n-n is the same way: it's indeterminate, and that means you can't just assume one of it's values is correct.

Jim isn't a real person. Anymore, at least.

Anyway, as for the n = 1/0, here's why it's wrong:
Multiply both sides by zero
0n = 1
Use axiom proofs to figure out this:
0n = 0
And then ,you get 0 = 1.

Remember, 1/0 can be multiple values, and 0*1/0 can be multiple values, too. Think of it this way: You take both square roots of 1, and then you set it equal to both the square roots of 1. -1 is a square root of one, right? So is 1. So shouldn't that mean that -1=1 because they both equal 1^(1/2)? The thing that's wrong here is that you're picking out one value equal to 1^(1/2) and setting it equal to another value equal to 1^(1/2) and saying they're equal, which they aren't. The exact same thing is happening here. Yes, 1/0*0=1, yes 1/0*0=0, but both these equations aren't completely correct. A better answer would be 1/0*0={0,1}. You can then choose an answer based on context (that is, if you have context). Even so, this wouldn't be the best answer. In fact, the best answer is indeterminate, because 1/0*0=0*1/0=0/0*1=0/0, so 1/0*0 is actually 0/0. You can argue 0/0 isn't a number either, because n/n=1, but because it has a 0 in the numerator, it's the same as 0*1/0, and therefore should be 0 because 0*n=0, but we call it indeterminate, NOT undefined, and it's because it actually equals both/either of these numbers, and we should at least give some such name to 1/0, be it (which could even stand for a set of numbers if 1/0 in fact returns more than one value) or whatever.

Therefore, the existence of a number that is equal to one over zero is impossible, just like the existence of a number equal to itself plus one. They do not obey the axioms and cause inconsistencies.

As I have shown, there may not be a number equal to 1/0, but that doesn't mean there are no numbers equal to 1/0. Also, your disproof of a number equal to itself plus one overlooked the possibility that n-n could have more than one possible value. Finally, I would like to introduce the notion that the axioms aren't even extendable to this realm. The notion that 0 of anything is 0 creates an exception. Just make up a number, , where 0 * = 1, and there's no reason why this number can't be. It would be like saying there's no number that is the sqrt(-1), because it just can't be. Well, you know what we did? We created a number that was the sqrt(-1), and the applications of the number i have not ceased to amaze us. You see, this type of thinking limits us, and as I have shown, limits are deceiving. Axioms are more of a general rule. You can always make up your own exceptions. The logic that created them always has loopholes. Like the 0*n=0 logic forgets that infinite is so large that one thing out of it is literally nothing compared to infinite, so therefore, that one thing would be 0 of infinite, which although it 'violates' a 'fundamental axiom of mathematics,' has a logic of its own. And logic is what mathematics was based on, not some set of axioms created by mathematicians centuries ago.

It may sound like a case of "it doesn't obey what we know, so sweep it under the rug", but that's the thing.

You know what we've been doing all these years? Sweeping it under the rug, and now it's time for it to come out.

Their existence would make mathematics inconsistent.

So the solutions to inconsistency is to create inconsistency?

We call the undefined because, with our set of axioms, they cannot be defined.

But with logic, it can.