One divided by Zero (1/0)

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exfret
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One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 2:46 am UTC

Hi! I believe 1/0=∞. Don't yell at me! I have my reasons. My own reasons. I'm not just circulating internet myths. In fact I've even written up a 'proof' with my ideas (it's probably not formal enough to be considered a true proof). It's copied and pasted, so please excuse anything out-of-context. You can view the discussion I copied-and-pasted it from here: I'll post the url when I get up to five posts (I can't post a link when I have less than five). It's quoted to separate it from the rest of this post. I've included comments in [brackets]. (If you don't want to see these comments, just read the original post in the above url). Well, here it is:

Defining inifinite [Bad spelling, yes, but I felt like preserving the original. I hope this doesn't spur any grammar-geeks...] as a number followed by an endless stream of zeroes before the decimal point. [Yes, that wasn't a complete sentence. Will you not go grammar-geek on me now that I've acknowledged that?] That is, infinite = 1 0 . 0 [There's supposed to be a bar over the first 0, (placed there using an unbreaking space) but my comments mess all this up] (the decimal point and the zero after it are just for demonstration and could be removed at your discretion, and the bar is signifying an endless number of that digit following the number to the left). So, with this definition, we can create this proof:
(Just a side note: I can define the number infinite, because there aren't any properties surrounding it, except that it is not smaller than any number, which my property satisfies, which is hard to show now, so I'll do it later on. [I'm sure I figured out a way somehow later on in that thread. I also got another way to arrive at 1/0=∞ in the threed (from now on, for ease of reference, the thread where this proof originates shall be called, the threed) later on, but it involved dividing by zero, so it was just a bit problematic...] Also, we cannot just create a definition for raising to the power of 1/2 that sets it equal to square-rooting, because that would violate its property that it is the inverse of squaring.) [The square root example comes from the fact that the post this came from also discussed raising to the power of a half] [Okay, I admit now that I can't just say one followed by infinite 0's is infinity, because I haven't proved that it satisfies that sole property of infinity, but 1/0 is at least some number, and I'm sure I could pull together a proof that this number is even ∞, but let's just start with saying that it at least isn't undefined]

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0 . 0 = 0 Adding zeroes, even an endless amount of zeroes, after the decimal point will not change the value of a number
_ _
0 . 0 1 = 0 . 0 *** Sorry, I don't know the name of the property. [This comes from the fact that 1=0.999999999..., so therefore 1-1=0 implies 1-0.99999999...=0, which you can 'simplify' to 0.00000000...1=0. I hope I'm not practically begging for this thread to be closed by discussing a topic closely related to 0/0 and using the fact that 1=0.99999999... in a proof]
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0 . 0 1 = 0 Substitution Property of Equality
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( 0 . 0 1 ) ^ ( - 1 ) = ( 0 ) ^ ( - 1 ) "Operation" Property of Equality (performing an operation [What I mean to say is "the same operation," but you get what I mean anyways, right?] to each side of an equation will preserve the equality)
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1 / 0 . 0 1 = 1 / 0 Property that states x^(-1)=1/x

***if you would like to see the proof for this, just ask, but I am assuming you already know because it is the foundation for turning fractions into decimals and vice versa [Oops, forgot a period there. Anyways, I guess I already showed you all the jist (Is that how you spell that word? Oh great, grammar-geeks are gonna spam this thread...) of the proof anyways]
_
Now, we must find out what 1 / 0 . 0 1 is. So, an interesting property is that 1/0.(n)1 = 1(n).0 * 10, where n is a number of zeroes. For example, 0.0001 has 3 zeroes, so 1/0.0001=1 followed by three zeroes times ten, or 1000.0*10. We can use this right now, and since we know that our number is followed by an infinite number of zeroes, 1/it will get a number followed by an infinite number of zeroes before the decimal point (hmmm... sounds familiar) times ten.
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1 0 . 0 * 1 0 = 1 / 0 Aforementioned property
_ _ _
But, wait, what about that nasty times ten in there? Hmmm... Well, since 0 . 0 1 = 0, and 10*0=0, then 1 0 * 0 . 0 1 = 0 = 0 . 0 1! <-Not factorial operator [Yes, my tone does get a little less 'official' around this point]
_ _
0 . 0 1 = 1 0 * 0 . 0 1 Multiplication property of zero
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1 / ( 0 . 0 1 * 1 0 ) = 1 / 0 Substitution Property of Equality
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1 / 0 . 0 1 / 1 0 = 1 / 0 Dividing by multiplying property, or whatever it's called
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1 0 . 0 * 1 0 / 1 0 = 1 / 0 Aforementioned property
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1 0 . 0 = 1 / 0 Inverse Property of Multiplication
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1 0 . 0 = infinite Given/Definition

infinite = 1 / 0 Substitution Property of Equality

1/0 = infinite Reflexive Property of Equality (Just to make it look nicer)

Now, this proof has some major implications. For starters, where k is any complex number (excluding infinite), k*infinite=infinite, because k*infinite=k*1/0=1/(1/k)*1/0=1/(1/k*0)=1/0, because
0*(1/k) = 0*(1/any complex number, given k != 0, because 1/0 is not a complex number according to our definition) = 0. In fact, 0*infinite is indeterminate, [Please just ignore this if you're out to have a fight about 0/0, or if you're a forum admin looking 24/7 to close any post with the term 'indeterminate' in it] because since k*0=0, 0*infinite=k*0*infinite=k*0*1/0, so since k*1/k=1 (Give me a proof showing this is untrue when it comes to infinite and 0 and I'll try to disprove it or I'll offer an explanation for why 1/0 still = infinite. I think proofs "demonstrating" this exception to be "true," thus showing this notion of infinite breaks the laws of complex numbers (even though infinite isn't even a complex number in some rights) and meaning that infinite can't be 1/0, are a large part of the reason why people don't believe 1/0=infinite) [this idea of me disproving your disproof is discussed more in the little bunch of text after this quote], 0*infinite=0*1/0=k*0*1/0=k*1=k, and since k is any complex number (it can be zero this time), infinite*0 ends up equalling indeterminate. In fact, 1/0 equalling infinite doesn't even violate either the property that k*0=0 or that 1/k*k=1, because 0*infinite=0 has an infinite number of solutions, one of them being 0, and another being 1. [This concept of getting multiple 'solutions' (yes, I use that word loosely here, but so many other people use it this way too that it's become math-slang) is a recurring idea in my mathematics of infinity. In fact, if you create a disproof of 1/0=∞, most likely it has something to do with multiple solutions that makes it invalid.] Moving on, we can see that infinite^l=infinite, [Why'd I use 'l'? It's such an irregular letter...] where l is a non-infinite rational number, because infinite^l=(1/0)^l=1^l/0^l=1/0. Now to show that my definition of infinite is not smaller than any number. Well, my proof for this isn't that concrete, but once you go infinitely far out from the number line, there's no way to measure how big a number is. [Technically, once you get an infinite distance from the number line, every number is infinitely large.] If you add 1 to the definition of my number, you get this:
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1 0 1 . 0 / 1 0 [The divided by ten is because I moved things to the left to make room for that one. I see a problem with this, though: I'm not adding 1, I'm adding 0.1. I can just say I'll do that ten times and it'll get adding one.]

If you keep on adding one to the point that you've added an infinite number of ones, you get:
_ _
1 0 . 0 + 1 0 . 0 [See, not quite the most concrete proof. But still a way to communicate the concept.]

Which equals the number before adding all those one. Of course this could be wrong, because we are only assuming that adding one infinitely should increase the number. [See? I told you it wasn't the most concrete proof.] The meaning behind the fact that you can't just add 1 to increase the size of the number is that you can't create a number greater than infinite. [That sentence seems out-of-place... Oh well, but it conveys an important point about my definition of ∞.] Oh, and how do we even know if infinite is positive? Well, it's both positive and negative in a way, because infinite=-1*infinite=-infinite. Strange. [It's like i: neither positive nor negative.] Well, we know that a positive over a positive is a positive. And, a negative over a negative is a positive and a negative over a positive is a negative and a positive over a negative is a negative, and then a zero over a positive is zero and a zero over a negative is zero and a zero over a zero is a positive,zero, or a negative, [Once more, since this is post just contains tiny references to 0/0, it's not bad. As long as no one starts an indeterminate internet post war. Then, the value of this discussion would become indeterminable.] but then when we divide a positive or negative by zero, what do we get? We get an infinite, [which is neither a negative nor a positive (nor zero) (in a way)] a now defined infinite, and a very strange one, too.


First of all, I would like to acknowledge that my way of proving it isn't the strongest, but most of my belief in this comes from the fact that the disproofs I've seen demonstrating the impossibility of 1/0 are wrong, and from the fact that 1/0=∞ makes for a more complete mathematical language. I can't create rebuttals in advance to every disproof all of you might come up with, so I'll just leave you all to post your own reasons for why I'm wrong. (Because you all totally weren't going to do that if I hadn't told you to). It would a good idea to read through the thread in the url above. It obviously wouldn't be worth your time to read through all of it (not only is it big, but it's math-y, too, even if it is mostly just space and quotes). Just skim through to get the general idea of what I mean. Of course, you don't have to (especially if you're already burned out from the quote part of my proof), but doing so would make you better-informed about my reasons for my opinion and the topic at hand. Well, anyways, have a Happy Friday (or Saturday, depending on your time zone)! Bye! :D

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gmalivuk
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Re: One divided by Zero (1/0)

Postby gmalivuk » Sat Feb 08, 2014 3:14 am UTC

exfret wrote:which you can 'simplify' to 0.00000000...1=0.
No, you can't. The thing on the left side isn't a number, because decimal expansions don't have an infinite number of digits followed by another digit. That's just now how decimal expansions work.

I approved this post in the (perhaps naive) hope that you'll accept this and other people's no-doubt forthcoming explanations of why your proof can't be right, but I will lock the hell out of it if you turn out to just be another math crackpot.
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exfret
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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 4:11 am UTC

gmalivuk wrote:I approved this post in the (perhaps naive) hope that you'll accept this and other people's no-doubt forthcoming explanations of why your proof can't be right, but I will lock the hell out of it if you turn out to just be another math crackpot.

I don't mean to be ranting... I just wanted to share my ideas. :( I recognize there must be lots of people who go on math rants and all, but how else should I get my ideas somewhere than to post on a forum? In fact, the reason I came to these fora was because my discussions about 1/0 on the one I'm from seemed to be going nowhere. Please, no matter how misguided I may appear to be, could you allow me to continue this thread?

I admit it: I'm biased towards my ideas. Even so, that doesn't mean that I'm just being a math conspiracy theorist if I may not always believe others who may disagree with me later on in this thread. I want to try to have a constructive conversation here, and I promise you I am completely ready to believe against what I have written. I know my proof is weak, but like I said at the end of my post, my strong arguments are how the current consensus is wrong. I mean, although I'm being a little liberal in going as far as saying 1/0 is ∞, our current explanation of it being 'undefined' is in the least shaky (like I said in the last part of my post, I can't disprove all proofs of the inexistence of 1/0 at once, so I was hoping to just disprove them as people posted them), and that seems to be in the least bit worth discussing. So, may I respond to your comment with my own thoughts, or do I have to just 'accept what you say as true'?

Also, thank you for the comment to my 'proof'. All comments are welcome. I also apologize for potentially seeming like a math crackpot. I was actually trying to avoid anyone coming on strongly to me by requesting "not to be yelled at," but I know that large forum must have lots of spam you have to deal with, meaning you have to be stern sometimes, but hopefully you'll see me as less spam-crackpot-y after reading this post. Anyways, I hope you have a nice day (or night). :)

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Dopefish
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Re: One divided by Zero (1/0)

Postby Dopefish » Sat Feb 08, 2014 4:45 am UTC

I feel like you've used a number of properties that when stated more carefully, explicitly exclude 0, and so aren't valid properties to be using (such as x^-1=1/x for x=0).

Even overlooking things like that, I'm fairly sure you could construct a near identical argument showing that 1/0 is negative infinity, which behaves somewhat differently from positive infinity, despite your claims to the contrary.

Besides, infinity isn't a real number so lots of manipulations involving it aren't valid, so you have to be extra careful when trying to treat it like one (which is the source of a number of your errors in the end).

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Re: One divided by Zero (1/0)

Postby notzeb » Sat Feb 08, 2014 5:47 am UTC

The OP might be interested in the real (or even complex) projective line. This can be thought of as the set of ordinary real numbers together with infinity, where infinity and -infinity are both treated as the same number (so if you were to draw this version of the number line, you would draw it as a circle, with 0 at the bottom, infinity at the top, 1 at the right, and -1 at the left).

In order to formalize this, mathematicians take a different approach than the OP's, since we don't want people to yell at us for dividing by 0. Here's how we do it: a point on the projective line is an equivalence class of ordered pairs (numerator, denominator) where not both of them are 0 (0 divided by 0 is bad no matter how you slice it). Two ordered pairs (a,b) and (c,d) are called equivalent when we can find a nonzero scale factor m such that ma = c and mb = d. We then name the equivalence class containing the ordered pair (a,b) as [a:b], and we think of it as corresponding to the ordinary number a/b whenever b is not 0.

So from this point of view, [0:1] corresponds to 0, [1:1] corresponds to 1, [-1:1] corresponds to -1, and [1:0] corresponds to infinity. What's so great about this? One thing we get out of this is a sensible way to plug infinity into algebraic expressions: suppose f(x) is a rational function of x, say f(x) = p(x)/q(x) where p(x) and q(x) are both polynomials with no common roots. Let d be the maximum degree out of the two polynomials p, q, and write p(x) = a0 + ... + adxd, q(x) = b0 + ... + bdxd. Then we extend the definition of f to the projective line by the following formula:

f([s:t]) = [a0td + ... + adsd : b0td + ... + bdsd].

So for instance when we plug in an ordinary number x, we get

f([x:1]) = [a0 + ... + adxd : b0 + ... + bdxd] = [p(x):q(x)],

and when we plug in infinity we get

f([1:0]) = [ad:bd].

If our function f is given by f(x) = 1/x, then we get f([s:t]) = [t:s], so for instance f([0:1]) = [1:0]. This is a rigorous sense in which we can say "1 divided by 0 is infinity (in the context of projective geometry)". (Of course in another context such as real analysis it may not be appropriate to say this.)

Edit: I can't stop myself from making one little additional comment. The projective line also works if you replace "real numbers" in the above by "remainders modulo 2". So the projective line mod 2 has just three numbers on it: [even:odd], which is called "0" or "2" or "even", [odd:odd], which is called "1" or "-1" or "odd", and [odd:even], which is called "1/2" or "infinity". You can define rational functions on the projective line mod 2 by the same exact formulas as above, and everything works out nicely.
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korona
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Re: One divided by Zero (1/0)

Postby korona » Sat Feb 08, 2014 11:37 am UTC

There is an elementary proof that division by zero must be undefined in every field. The object you obtain is not a field anymore. That's why it is not really interesting to use such definitions in many contexts.

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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 2:29 pm UTC

Thank you for all the comments! :)

Dopefish wrote:I feel like you've used a number of properties that when stated more carefully, explicitly exclude 0, and so aren't valid properties to be using (such as x^-1=1/x for x=0).

I was just raising to the negative one to make it clear that I was applying the same operation to each side. I could just place each statement under a "one divided by," which would be the same operation, so that would be okay as well.

Dopefish wrote:Even overlooking things like that, I'm fairly sure you could construct a near identical argument showing that 1/0 is negative infinity, which behaves somewhat differently from positive infinity, despite your claims to the contrary.

As I said in my proof, 1/0 is in fact negative infinity, as well as positive infinity. This doesn't necessarily mean that positive infinity equals negative infinity. 1/0 could just be giving more than one return (like how 1^0.5 returns two numbers, -1 and 1). Even so, I doubt negative infinity acts differently from positive infinity. Could you give me examples of this?

Dopefish wrote:Besides, infinity isn't a real number so lots of manipulations involving it aren't valid, so you have to be extra careful when trying to treat it like one (which is the source of a number of your errors in the end).

I've just have three questions for you:
1) How is it not a real number?
2) How does it not being a 'real number' make it so I can't use it like such?
3) How does this result in being the "source of my errors"?

korona wrote:There is an elementary proof that division by zero must be undefined in every field. The object you obtain is not a field anymore. That's why it is not really interesting to use such definitions in many contexts.

Could you state the said proof? Is it the one where you multiply by the inverse? If so, this is actually wrong, because multiplying 0 by infinity has multiple returns. One of them is one, and another is zero. Therefore, it satisfies both of these properties. (Besides, the proof I've seen that any number multiplied by zero is zero fails for infinity by assuming that n != n+1, which clearly fails for infinity due to its property that ∞ = ∞ + 1). This doesn't even lead to the statement that 0=1, just in the way that 1^0.5 equalling either one or negative one doesn't mean one equals negative one. Basically, 1/0 * 0 = {0,1 (and all other real numbers)}, and that means it's 0 or 1, not both at the same time, so you can't just state their equality.

notzeb wrote:The OP might be interested in the real (or even complex) projective line.

You're referring to the magistrate when you say OP, right? (Sorry, I was just a little confused by your capitalization of the word OP and by the relevance of the statement).

notzeb wrote:This can be thought of as the set of ordinary real numbers together with infinity, where infinity and -infinity are both treated as the same number (so if you were to draw this version of the number line, you would draw it as a circle, with 0 at the bottom, infinity at the top, 1 at the right, and -1 at the left).

Yes, I am familiar with the concept of number circles. In fact, I even thought of something like that myself after thinking of how infinity could equal negative infinity. Of course, the circle would have infinite radius (or all non-infinite numbers would be grouped at a single point), which would make it just a line exactly like our own number line. That is, until you reach infinity. Edit: I sound so stupid here. "Number Circles," really? You get what I mean by 'number circles', right?

notzeb wrote:In order to formalize this, mathematicians take a different approach than the OP's, since we don't want people to yell at us for dividing by 0.

I'm still a little confused about this.

Sorry for not responding to your whole post, notzeb. Lots of it was confusing me, so I'll just need some time to really think about it later on so that I can respond appropriately.

Have a good day!
Last edited by exfret on Sun Feb 09, 2014 5:12 am UTC, edited 5 times in total.

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Re: One divided by Zero (1/0)

Postby Qaanol » Sat Feb 08, 2014 3:42 pm UTC

So yeah, projective line, extended line, surreal numbers, all that jazz.

You might be interested to learn about absorbing elements in abstract algebra.
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Re: One divided by Zero (1/0)

Postby brenok » Sat Feb 08, 2014 4:15 pm UTC

exfret wrote:
Dopefish wrote:I feel like you've used a number of properties that when stated more carefully, explicitly exclude 0, and so aren't valid properties to be using (such as x^-1=1/x for x=0).

I was just raising to the negative one to make it clear that I was applying the same operation to each side. I could just place each statement under a "one divided by," which would be the same operation, so that would be okay as well.

That's the problem, independently of you called "one divided by zero", or "0^-1", anything that implies zero in the denominator is prohibited in the common real numbers.

Dopefish wrote:Even overlooking things like that, I'm fairly sure you could construct a near identical argument showing that 1/0 is negative infinity, which behaves somewhat differently from positive infinity, despite your claims to the contrary.

As I said in my proof, 1/0 is in fact negative infinity, as well as positive infinity. This doesn't necessarily mean that positive infinity equals negative infinity. 1/0 could just be giving more than one return (like how 1^0.5 returns two numbers, -1 and 1). Even so, I doubt negative infinity acts differently from positive infinity. Could you give me examples of this?

1) 1^0.5 is just 1, not -1.
2) 2 to the power of negative infinity is arguably 1 (using limits, etc). 2 to the power positive infinity is infinity.
3) Division on the reals always yields a single number. (And dividing by zero is, as I said, not allowed, but that is beside the point)

Dopefish wrote:Besides, infinity isn't a real number so lots of manipulations involving it aren't valid, so you have to be extra careful when trying to treat it like one (which is the source of a number of your errors in the end).

I've just have three questions for you:
1) How is it not a real number?
2) How does it not being a 'real number' make it so I can't use it like such?
3) How does this result in being the "source of my errors"?

1) The real numbers have many properties that don't work if you try to use infinity, for instance, how much is 2 times infinity? 2 infinity? or just infinity?
If it's the former, isn't infinity supposed to be the largest number? If it's the latter, does it means that 2 = 1?
2) If it's not a real number, you can't use real number properties, simple as that.

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Re: One divided by Zero (1/0)

Postby gmalivuk » Sat Feb 08, 2014 4:18 pm UTC

Yeah, basically there are ways to add an infinity to the set of real numbers, but then either infinity or the whole set has to stop acting like we expect real numbers to act, because one explicit part of how they act is that they don't include infinities and there's no defined result to dividing by zero.
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Re: One divided by Zero (1/0)

Postby alessandro95 » Sat Feb 08, 2014 4:33 pm UTC

exfret wrote:1) How is it not a real number?
2) How does it not being a 'real number' make it so I can't use it like such?

Have a good day!


1)∞ is not a real number because our definition of real numbers doesn't allow for it be inside the set of real, ∞+1 is an expression as meaningless as §+1, you're adding a number to a symbol that has no meaning in the number system in which you're working.

2)you cannot use something that isn't a real while working inside the field of reals because it simply is meaningless, it's just as pretending to evaluate 4+√3 inside the field of natural number, there are precise axioms that defines what's a natural number and what's a real number, ∞ is neither.

If we were to consider ∞ as a real than our number system simply won't work because (for example) subtracting the same quantity from both sides of an equation is definitely a valid operation, we defined it to be so.

That doesn't mean you cannot do operations involving ∞, but you need a number system inside which those operations involving ∞ are defined
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Re: One divided by Zero (1/0)

Postby Dopefish » Sat Feb 08, 2014 5:42 pm UTC

Most things have already been addressed, but for ways positive infinity behaves differently from negative infinity (in the context of real analysis at least) is that x< ∞ for all real x, but x>- ∞ for all real x.

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Re: One divided by Zero (1/0)

Postby Schrollini » Sat Feb 08, 2014 5:55 pm UTC

exfret wrote:
notzeb wrote:The OP might be interested in the real (or even complex) projective line.

You're referring to the magistrate when you say OP, right? (Sorry, I was just a little confused by your capitalization of the word OP and by the relevance of the statement).

The OP is the Original Poster, i.e. you. notzeb is suggesting that you take some time to learn about the real projective line, because that sounds like what you are describing. It allows 1/0 to be defined, but to get this, you have to give up a number of useful properties of the reals. Notably, the real projective line is not well-ordered; on it you cannot say whether 1 < 2 or 1 > 2.

Everyone else, may I suggest that instead of saying, "You can't do this because X," we say, "You could do this, but you have to give up X." The existence of projective spaces shows that you can construct a system that includes infinity, but doing so forces you to give up so many useful properties that it's generally not worth it.
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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 6:02 pm UTC

brenok wrote:That's the problem, independently of you called "one divided by zero", or "0^-1", anything that implies zero in the denominator is prohibited in the common real numbers.

I'm saying that it shouldn't be so math-taboo. There's no reason to exclude zero from being in the denominator, (if you really must, I guess you could exclude it from being real) and if there is, then please show me it.

brenok wrote:1) 1^0.5 is just 1, not -1.

Coincidentally, my post on 1/0 was also a post to prove 1^(1/2)=+/-1. I just thought everyone here would agree with this, so I wouldn't have to put that in there, too. Here's the proof of 1^(1/2):
Assuming that 1^(1/2)=sqrt(1)=1 [This is going to be a proof by contradiction, by the way.]

-1=-1 Reflexive property of equality
(-1)^1=-1 Property of raising to the power of one (I forget what it is called)
(-1)^(1/2*2)=-1 Inverse multiplication property (1/x*x=1)
((-1)^2)^(1/2)=-1 Power of a Power (converse)
(1)^(1/2)=-1 Simplifying the square
sqrt(1)=-1 Given/Assumption
1=-1 Simplifying the radical

Now, the end result is not equal, meaning our assumption is false. If we use what I believe is correct, then we get:

(1)^(1/2)=-1 Simplifying the square
+/- 1=-1 (My) Given/Assumption

And this is true, because plus or minus 1 does equal minus 1. [Think of it like "either positive one or negative one is negative one," which is true because negative one equals negative one, so one of those do in fact equal negative one] Now, we could also define raising to the power of 1/2 as the inverse of squaring, in which case it would have to be plus or minus, because there isn't a one-to-one relation in quadratic functions. In other words, flipping an upright parabola over the line y=x will give you +/-sqrt(x) and not just sqrt(x). It makes no sense at all for it to return positive values alone. [Yes, it won't be a function, but it doesn't have to be. It does have to be the inverse of x^2, which it wouldn't be if it were just the top of a sideways parabola.] I showed this last year to my math teacher though, and she somehow found out in an Algebra I book that the Power of a Power Property only works for integer numbers, even though we used it in class for solving radical equations, quadratics, and who knows what else, so this is completely controversial. I didn't quite respond well, so she still believes in what the state of Virginia wants her to teach, but I'm hoping to eventually change her mind. Any thoughts on how I could go about it? [Of course, I'm no longer looking for thoughts on how to convince my math teacher of this. After some discussion with another person on the forum this came from, I decided to just move on and not care whether my math teacher was teaching the wrong thing. They'll (the students being taught) get it right further down the math road in calculus anyways. (Hopefully)]

Now, was it just the raising to one half that you didn't agree with, or was it my whole thing about infinity in operations returning multiple possible values?

brenok wrote:2) 2 to the power of negative infinity is arguably 1 (using limits, etc). 2 to the power positive infinity is infinity.

Actually, raising to the power of infinity behaves differently than you might expect. Also, using limits, wouldn't 2 to the negative infinity be zero? How'd you get one? Anyways, this was a problem I came across when I first came up with my theory-thingy, but it turns out to be far from a nail-in-the coffin. In fact, there are multiple 'solutions' to this 'problem':
1) I'm not necessarily saying that ∞=-∞, just that 1/0={∞,-∞}
2) 2^∞ could have multiple returns, one of them being 1 and another being ∞
3) Limits might lead you to believe this, but do not be deceived, limits tell no truth of what actually happens in the realm of infinity. I think I had something saying 2^∞ was negative, but I can't seem to remember where I got it from, but anyways, you can't disprove me just by saying what it looks like. How do you get infinity when you actually raise 2^∞? Well, even though I don't have any proof of the value of 2^∞, I do have proof for the value of the summation of the infinite sequence 2^n. In fact, the good 'ol 1/(1-x) rule still works out, everywhere, no restrictions needed. The summation of 2^n is actually -1 (MinutePhysics did a video on this), even though limits might lead you to believe otherwise.

brenok wrote:3) Division on the reals always yields a single number.

Please show me your evidence of this. Yes I'm serious. No, I am not being a math crackpot. (Hopefully). I understand that it always turns out this way (except for possibly in the case of 1/0), but haven't we allowed for division of any two numbers except for in the case of a zero denominator anyways? An exception here would just seem natural, given the strangeness of zero.

brenok wrote:(And dividing by zero is, as I said, not allowed, but that is beside the point)

Well, considering that's exactly what I'm arguing against, it's rather that you shouldn't just say that as your evidence than that being beside the point.

brenok wrote:1) The real numbers have many properties that don't work if you try to use infinity, for instance, how much is 2 times infinity? 2 infinity? or just infinity?
If it's the former, isn't infinity supposed to be the largest number? If it's the latter, does it means that 2 = 1?

If it's the former, 2*∞ isn't necessarily larger than infinity, if it's the latter, then no, it does not mean 2=1. The reason for this is multiple returns. (They strike!)
∞=2*∞
Divide each side by infinity:
∞/∞=2*∞/∞
You must be itching to simplify this, but you can't assume ∞/∞ in this case equals 1, that's just one of it's returns. In fact, because ∞=2*∞, the infinity on the right side could be substituted as 2*∞, like this:
∞/∞=2*∞/(2*∞)
This is all mathematically sound. I'd like to draw an analogy of saying otherwise to this 'proof':
1=1
(1)^2=(-1)^2
((1)^2)^(1/2)=((-1)^2)^(1/2)
(1)^(2*1/2)=(-1)^(2*1/2)
1^1=(-1)^1
1=-1
Yes, this does look a lot like my power of a half proof, but it has a different meaning. Here, we are assuming that raising to the power of a half will not raise the need for a negative square root, even though it does in fact require a negative square root to be present on one (and only one) side of the equation. This assumption would be wrong, just like the assumption that dividing each side of the about equation by infinity does not require a division by 2*∞ on one side of the equation.

brenok wrote:2) If it's not a real number, you can't use real number properties, simple as that.

But, if it's not a real number, and it has no real number properties as you said, then I can just use it following its sole rule that it is the number that acts like all others except that ∞+1=∞.


Here's a proof I just remembered from ∞=-∞:
∞+1=∞
∞=∞-1
That's simple. From this, we can infer we cannot make a number smaller than infinity, since we used ∞=∞+1 to note we could not make a larger number.
Last edited by exfret on Sat Feb 08, 2014 6:17 pm UTC, edited 1 time in total.

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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 6:16 pm UTC

May I try to make this less messy by making a new post for each reply to another significant post? If not, I can just delete this one and edit the previous one to merge it.

Schrollini wrote:The OP is the Original Poster, i.e. you. notzeb is suggesting that you take some time to learn about the real projective line, because that sounds like what you are describing.

Oh, that's quite ironic given the way I responded.

Schrollini wrote:It allows 1/0 to be defined, but to get this, you have to give up a number of useful properties of the reals. Notably, the real projective line is not well-ordered; on it you cannot say whether 1 < 2 or 1 > 2.

But like I said, it would have infinite radius, so it would act exactly like our number line. Starting from zero, negative numbers are all numbers up to infinity to the left and positives are all numbers up to infinity on the right. There, no loss, right?

Schrollini wrote:Everyone else, may I suggest that instead of saying, "You can't do this because X," we say, "You could do this, but you have to give up X." The existence of projective spaces shows that you can construct a system that includes infinity, but doing so forces you to give up so many useful properties that it's generally not worth it.

It's good to know that what I'm doing has been done before. It makes me feel less guilty for possibly being a misguided mathematical conspiracy theorist. Also, I believe that our current mathematics actually allow for an infinity, so switching to a new mathematical system for the incorporation of infinities would not be needed.

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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 7:04 pm UTC

This post contains my replies to alessandro's post (in the middle) and to the smaller posts of gmalivuk (in the beginning) and to dopefish (at the end).


gmalivuk wrote:Yeah, basically there are ways to add an infinity to the set of real numbers, but then either infinity or the whole set has to stop acting like we expect real numbers to act, because one explicit part of how they act is that they don't include infinities and there's no defined result to dividing by zero.

How would this impact the whole set just to add in infinity? What properties would it disobey? How is the exclusion of infinities from our number system such a clear and defined thing? You can't possibly just say that it has to be undefined without making sure that it doesn't cause the system to fall apart, can you?

gmalivuk wrote:No, you can't. The thing on the left side isn't a number, because decimal expansions don't have an infinite number of digits followed by another digit. That's just now how decimal expansions work.

I assume that since you have allowed this topic to continue, and since it has been developing constructive replies (in my opinion), that you will continue allowing it to continue. In this case, my reply to you is, why can't I follow it by another digit? I thought I'd have a specific thing I could cite as evidence for why I can do it, but I can't seem to find it (I wouldn't be able to post the url anyways). The concept of it was that in Conway's Soldiers, the solution to going 5 out (in an infinite number of moves) has something to do with going infinite moves, and then doing a move (I think). I don't see why, then, it isn't possible to have a number exist that has an infinite number of zeroes and then a 1.


alessandro95 wrote:1)∞ is not a real number because our definition of real numbers doesn't allow for it be inside the set of real,

And what definition of real numbers would it violate?

alessandro95 wrote:∞+1 is an expression as meaningless as §+1, you're adding a number to a symbol that has no meaning in the number system in which you're working.

x+1 has meaning. It's a number with one added to it. I'm conveying the exact same meaning when I use ∞. I am using it as a number with one added to it, and then I'm saying that it equals itself.

alessandro95 wrote:2)you cannot use something that isn't a real while working inside the field of reals because it simply is meaningless,

Restrictions come to life because they are needed to keep a mathematical system from becoming contradictory, not "meaningless". Do not call this "meaningless" when the same could be said about all mathematics. Saying that my statement is "meaningless" is therefore meaningless itself. Also, How am I working in the field of reals? If infinity is not a real number, then wouldn't that cause this not to be in the field of real numbers?

alessandro95 wrote:it's just as pretending to evaluate 4+√3 inside the field of natural number, there are precise axioms that defines what's a natural number and what's a real number, ∞ is neither.

Could you demonstrate how ∞ violates such axioms? Also, how am I evaluating ∞ within the field of reals (which I assume you are implying)? Is it because I use 'real operations'?

alessandro95 wrote:If we were to consider ∞ as a real than our number system simply won't work because (for example) subtracting the same quantity from both sides of an equation is definitely a valid operation, we defined it to be so.

Yes, I agree with you. Subtracting the same quantity from both sides of an equation is a valid operation, but how does this mean ∞ can't be? Are you referring to subtracting ∞ from both sides of the equation ∞=∞+1? Even if you are not, I expect this will be a common thing for people to bring up, so, I'll just deal with it now. It actually involves multiple returns. Basically, when you're subtracting ∞ from each side of an equation, you can't just assume ∞-∞=0, it might also be equal to 1, because remember that ∞=∞+1, so ∞-∞ is the same as (∞+1)-∞, which you could evaluate to be 1. So in the equation:
∞ + 1 = ∞
(∞ + 1) - ∞ = ∞ - ∞
Now you could evaluate this to become this:
1=0
But that would be assuming that the operation of ∞-∞ returns the same thing on both side, which it doesn't because it can have multiple returns. There's a post above where I discuss about this more, and I even have a post in the threed (the thread where my original post in quotes comes from) that goes into deeper detail about this specific problem in case you are unsatisfied with this how it is.

alessandro95 wrote:That doesn't mean you cannot do operations involving ∞, but you need a number system inside which those operations involving ∞ are defined

What I don't understand is why this 'newfangled' number system should be so much more different from our own. What I see is that our mathematical system has limited itself in an area where it can expand to without having to redefine anything in the real system at all.


Dopefish wrote:Most things have already been addressed,

Do you mean most things in my proof, or most things that have been said. If the former, then my proof was weak anyways, and I think more constructive arguments would come from the topic in general, which seems to be the case for what is happening. If the latter, then I disagree with you completely. The whole idea of there being a limited number of things waiting to be said seems to be misguiding to me.

Dopefish wrote:but for ways positive infinity behaves differently from negative infinity (in the context of real analysis at least) is that x< ∞ for all real x, but x>- ∞ for all real x.

Note the wording: "∞ is the number for which there are no greater numbers." Therefore -∞ <= x, and x <= ∞ for all real x. Still, though, I can see many problems in my own statement. (Edit: Just ignore everything I said before this point, I take back what I said after rethinking it). My opinion on this is that the comparisons can act strangely once you get to infinity. For example, there is no reason x can be both less than and greater than infinity. Using a transitive inequality property to state this means infinity is less than and greater than itself at the same time (which would be doubly bad, because 1) nothing can be greater than infinity and 2) no single number can be both greater than itself and less than itself), but if you look at the number line like a circle with infinite radius, this inequality property could in fact be misleading at infinity. Even so, I must admit, you've got me there. I have to think about this one a little more. Still, there's much more than just this to discuss, and this doesn't mean the discussion has reached a conclusion. Good thought.

Edit: Okay, I thought about it a little more. I definitely think that ∞ would be both greater and lesser than a number, but that doesn't mean that it's both greater and lesser than itself. Also, just because it's lesser than the real numbers doesn't mean it isn't greater than all real numbers, which it still is, so it is still consistent with my definition of it. I like the number circle with infinite radius thing, because it allows for this to work out. Imagine infinite at the top half. From the bottom half, you can go left (towards numbers less than the point you started at) or right (towards numbers greater than the point you started at). If you continue this for infinity in either direction, you get to infinity, meaning that infinity is both greater than and lesser than all real numbers, but this doesn't mean it's greater than or lesser than itself, just in the same way you can go west or east to get to the International Date Line from the Prime Meridian, but that doesn't mean that the International Date Line is both west and east of itself. Of course, the Earth isn't an infinitely large sphere like the number circle in my analogy.


Well, as always, have a good Saturday (I assume most of the world is experiencing Saturday at this instant in time or at least has gone to bed by now). Thanks for all the comments! Hopefully I'll be able to respond to the majority of the ones you post later on. Bye! :D
Last edited by exfret on Sat Feb 08, 2014 7:56 pm UTC, edited 2 times in total.

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Re: One divided by Zero (1/0)

Postby gmalivuk » Sat Feb 08, 2014 7:19 pm UTC

exfret wrote:
gmalivuk wrote:Yeah, basically there are ways to add an infinity to the set of real numbers, but then either infinity or the whole set has to stop acting like we expect real numbers to act, because one explicit part of how they act is that they don't include infinities and there's no defined result to dividing by zero.
How would this impact the whole set just to add in infinity? What properties would it disobey? How is the exclusion of infinities from our number system such a clear and defined thing? You can't possibly just say that it has to be undefined without making sure that it doesn't cause the system to fall apart, can you?
As previously stated, it violates the field axioms for division by zero to be defined, and we kind of like having the Reals be a field. The exclusion of infinity from the standard definition of the real numbers is clear and defined because the real numbers are not defined as having an element at infinity.

As stated, there are things you can do with points at infinity, but those things are no longer the standard operations on the standard sets. If you're allowed to divide by zero and treat the result as a real number to be manipulated like any other, then you can manipulate your symbols to show 1=2, and all that implies. Where "all that implies" is everything.

Since we'd rather *not* have a system of mathematics where every statement has a valid proof, we don't define the reals as including infinity or a multiplicative inverse of 0.
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Re: One divided by Zero (1/0)

Postby brenok » Sat Feb 08, 2014 7:56 pm UTC

exfret wrote:
brenok wrote:1) The real numbers have many properties that don't work if you try to use infinity, for instance, how much is 2 times infinity? 2 infinity? or just infinity?
If it's the former, isn't infinity supposed to be the largest number? If it's the latter, does it means that 2 = 1?

If it's the former, 2*∞ isn't necessarily larger than infinity, if it's the latter, then no, it does not mean 2=1. The reason for this is multiple returns. (They strike!)
∞=2*∞
Divide each side by infinity:
∞/∞=2*∞/∞
You must be itching to simplify this, but you can't assume ∞/∞ in this case equals 1, that's just one of it's returns. In fact, because ∞=2*∞, the infinity on the right side could be substituted as 2*∞, like this:
∞/∞=2*∞/(2*∞)

The problem is that you're making your own numbers and definitions. 2A is always greater than A, when a is positive (and when the set we're using allows comparrison). A/A is always 1 for every nonzero number. Which is related to the next point:
brenok wrote:2) If it's not a real number, you can't use real number properties, simple as that.

But, if it's not a real number, and it has no real number properties as you said, then I can just use it following its sole rule that it is the number that acts like all others except that ∞+1=∞.
[/quote]
The number you're proposing doesn't fit the operations with real numbers. It's like comparing apples and oranges. You could set a number system where 1 + apple = orange. The could be technically correct, but also meaningless. Such a system have no relation with the mathematics used everywhere else.

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Re: One divided by Zero (1/0)

Postby Lopsidation » Sat Feb 08, 2014 8:05 pm UTC

One nice thing about math is, mathematicians can define things however we want to. If we all wanted to, then we could define x+y to equal 42 if x=y and the normal result of addition otherwise. Nobody defines x+y like that, because it's not useful or interesting at all.

Adding ∞ to the real numbers is interesting! We don't call the resulting system the real numbers anymore; as other posters said, we call it "the projective real line", "the end compactification of R", and probably some other names as well.

But most of the time, people work with the real numbers, i.e. without ∞. I don't think excluding ∞ holds us back at all. That's because, if you want ∞, you have to give up on some useful properties of the real numbers. For example, ∞-∞ is a problem. You suggested fixing this in your last post by saying "the operation of ∞-∞ ... can have multiple returns". But then, when you try to solve an equation like "x=a-b", you have to worry about a-b not being equal to a-b! You can try to fix this more, or fix it differently, but no matter what, you will have to give up on one of the properties that make the real numbers so nice.

And yet, adding ∞ is useful sometimes - topology, for example.

The point is, it doesn't make much sense to prove 1/0=∞. You can either (i) work with the axioms of real numbers, in which case ∞ is not a number, or (ii) explicitly define another system where 1/0=∞, like the projective real line, and figure out the interesting consequences you get from it.

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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 8:15 pm UTC

brenok wrote:The problem is that you're making your own numbers and definitions. 2A is always greater than A, when a is positive (and when the set we're using allows comparrison). A/A is always 1 for every nonzero number. Which is related to the next point:

I'm simply defining numbers that have been left 'undefined' until now, I'm not exactly creating them out of thin air. Also, ∞/∞ does not violate what I have said. ∞ isn't exactly positive, so you can't charge it with not being less than 2*∞, and ∞/∞ is one, it's just not always equal to one. It could be two as well. In fact, it's indeterminate (exactly like in the case of 0/0).

brenok wrote:The number you're proposing doesn't fit the operations with real numbers. It's like comparing apples and oranges. You could set a number system where 1 + apple = orange. The could be technically correct, but also meaningless. Such a system have no relation with the mathematics used everywhere else.

But it does fit the operations with real numbers. I'm defining it to do so. It follows all the normal operations in all the normal ways (well, maybe its ways aren't completely normal). Its only difference from being any number at all is that ∞=∞+1. Also, this does relate to our number system, because I'm setting it equal to 1/0, which is its relation to this number system. Because it has relations to current mathematics, it's not useless or meaningless, or it's at least as meaningless as mathematics itself, which would make its meaninglessness meaningless to state here, as it rather applies to all of mathematics and not just to 1/0.

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Re: One divided by Zero (1/0)

Postby Schrollini » Sat Feb 08, 2014 8:19 pm UTC

exfret wrote:
Schrollini wrote:It allows 1/0 to be defined, but to get this, you have to give up a number of useful properties of the reals. Notably, the real projective line is not well-ordered; on it you cannot say whether 1 < 2 or 1 > 2.

But like I said, it would have infinite radius, so it would act exactly like our number line. Starting from zero, negative numbers are all numbers up to infinity to the left and positives are all numbers up to infinity on the right. There, no loss, right?

I don't know what you mean by "radius" here. You can visualize the projective line as a circle, but that doesn't make it a circle. It's just the set R+{∞}, with certain rules about how ∞ operates in arithmatic expressions. It's most certainly not exactly like R (what I assume you mean by "our number line"); it has an additional element.

You can try to take the ordering of the reals, as you do by saying the positive numbers are to the right of 0, but you also have to define how ∞ is ordered. Some arguments suggest 0 < ∞; others 0 > ∞. Put together, we get 0 < ∞ < 0, which is a contradiction. Therefore, the projective line is not well-ordered.

This isn't necessarily a bad thing -- the complex numbers aren't well-ordered either. It's just one of the many things you have to give up to be able to treat ∞ as an element of your set. Others have pointed out that you also lose the field axioms. In most cases, the loss of these things outweighs the inclusion of ∞. (What good does it do to allow division by zero if division is no longer well-defined?)

I'm guessing from what you've written that you're a high-school student, and you haven't had any abstract algebra. If so, all our objects about groups and fields probably sound pointless and, well, abstract. Please rest assured that there are good reasons to care about these things, and if you keep studying math, you'll learn them soon. It's good to question the rules we use for mathematics, but it's better do to so for the purpose of learning why they were chosen. You'll get a more measured response if you post, "Why can't we treat ∞ as a real number," rather than, "Look at me! I've invented a new kind of mathematics where ∞ is a number!"
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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 8:50 pm UTC

Lopsidation wrote:One nice thing about math is, mathematicians can define things however we want to.

:) Exactly!

Lopsidation wrote:If we all wanted to, then we could define x+y to equal 42 if x=y and the normal result of addition otherwise. Nobody defines x+y like that, because it's not useful or interesting at all.

Well, if x and y were to stand for something, like the number of people posting on xkcd, y, per x seconds, then it would be useful, and people could define x+y like that. In pure mathematics, however, since since x and y are simply placeholder symbols and have no relation to the current mathematical system, relating them in this way would just be giving a new way to say the number 26. It wouldn't create any new relations, and mathematics would stay the same, so there's no reason for this to happen in pure mathematics.

Lopsidation wrote:Adding ∞ to the real numbers is interesting!

It also adds new relations to mathematics, meaning it actually changes something. It actually depends on who you are whether these mathematics are interesting. (I assume they are interesting to you if you are posting in this thread, though).

Lopsidation wrote:We don't call the resulting system the real numbers anymore;

You mean we don't call the resulting number a part of the set of real numbers anymore. The system includes all of mathematics, and that would still be the same system, except expanded to include infinity.

Lopsidation wrote:as other posters said, we call it "the projective real line", "the end compactification of R", and probably some other names as well.

This must be the "circle with infinity radius" that I speak of, right?

Lopsidation wrote:But most of the time, people work with the real numbers, i.e. without ∞. I don't think excluding ∞ holds us back at all.

I agree. I thought of this, then I was trying to think of a way it could expand mathematics. The only thing I could think of was that it was more of a neat thought. It does hold us back from discovering things with infinity, though, but like I said (try re-reading this sentence differently if the "though, but" sounds strange), it doesn't seem like that much of a loss. Even so, we could say that about pretty much any new, seemingly controversial idea, like maybe zero when it was first thought of.

Lopsidation wrote:That's because, if you want ∞, you have to give up on some useful properties of the real numbers.

People keep on saying this! Which properties? (I know, I know, my question's just about to be answered in your next sentence).

Lopsidation wrote:For example, ∞-∞ is a problem. You suggested fixing this in your last post by saying "the operation of ∞-∞ ... can have multiple returns". But then, when you try to solve an equation like "x=a-b", you have to worry about a-b not being equal to a-b! You can try to fix this more, or fix it differently, but no matter what, you will have to give up on one of the properties that make the real numbers so nice.

I'm not suggesting a solution. I'm just saying that's the way it is. In fact, it's already this way, such as with, as aforementioned, raising to the power of a half. We already have the problem of 1^(1/2) not equaling 1^(1/2), or maybe of +/-a not equalling +/-a. I'm not introducing a new problem to the real number system, I'm just allowing this problem to spread into arithmetic operations as well. And besides, you don't have to worry about this problem if you're not working with infinity, and if you are working with infinity, then you're using this system anyways, so there's no reason for there to be 2 separate systems, one with infinity included, and another without. I see no give and take in this, just something we've left undefined because people were either too afraid it would break something to define it or they weren't able to understand it fully enough. (Given the fact that, as you said, people have already come up with a projective real line, I'd go with the former).

Lopsidation wrote:And yet, adding ∞ is useful sometimes - topology, for example.

Ooh, what do topologists do with it?

Lopsidation wrote:The point is, it doesn't make much sense to prove 1/0=∞.

I don't know if I really proved that anyways. 1/0 is at least some number, be it infinity or not, and that number has extremely similar properties with infinity. 1/0 is not just "undefined," as the consensus in current mathematics says it is.

Lopsidation wrote:You can either (i) work with the axioms of real numbers, in which case ∞ is not a number,

What I don't understand is: Why would you just work in the real number system? Isn't that just a part of the complex number system, which is just a part of the complex number system plus infinity? If you're just working within the reals, you don't have to explicitly exclude other numbers that you're not working with. For example, it's like stating 1+1=2, and then saying that since you're only working with natural numbers in that case, every other number is "not a number."

Lopsidation wrote:or (ii) explicitly define another system where 1/0=∞, like the projective real line, and figure out the interesting consequences you get from it.

I think of it more like an 'expansion pack' rather than a 'sequel'. (Please excuse my analogy-like terminology). We don't have to create an all new system; I'm saying that we could just add on this expansion and everything would be as it was before. (Are there any grammar-geeks here to tell me how to correctly use a semi-colon, or did I horrify and scare them off by not changing my bad grammar earlier?) There's no need for a separate system. Adding on to the current one would be just fine.

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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 9:23 pm UTC

And yet, for the third time now, as I finish one post, another one comes up. Typical.

Schrollini wrote:I don't know what you mean by "radius" here. You can visualize the projective line as a circle, but that doesn't make it a circle. It's just the set R+{∞}, with certain rules about how ∞ operates in arithmatic expressions. It's most certainly not exactly like R (what I assume you mean by "our number line"); it has an additional element.

A circle with a radius of an infinite length is a line (in a degenerate way) when you zoom into its arc (just like a circle of 0 radius is a point (in a degenerate way)). I'm saying that this number line is exactly like our own system, except it has infinity, and that is the only difference.

Schrollini wrote:You can try to take the ordering of the reals, as you do by saying the positive numbers are to the right of 0, but you also have to define how ∞ is ordered. Some arguments suggest 0 < ∞; others 0 > ∞. Put together, we get 0 < ∞ < 0, which is a contradiction. Therefore, the projective line is not well-ordered.

I discussed that above in one of those posts in a reply to dopefish. Here, I'll copy and paste it so you don't have to go looking for it:
Dopefish wrote:but for ways positive infinity behaves differently from negative infinity (in the context of real analysis at least) is that x< ∞ for all real x, but x>- ∞ for all real x.
I think that ∞ would be both greater and lesser than a number, but that doesn't mean that it's both greater and lesser than itself. Also, just because it's lesser than the real numbers doesn't mean it isn't greater than all real numbers, which it still is, so it is still consistent with my definition of it. I like the number circle with infinite radius thing, because it allows for this to work out. Imagine infinite at the top half. From the bottom half, you can go left (towards numbers less than the point you started at) or right (towards numbers greater than the point you started at). If you continue this for infinity in either direction, you get to infinity, meaning that infinity is both greater than and lesser than all real numbers, but this doesn't mean it's greater than or lesser than itself, just in the same way you can go west or east to get to the International Date Line from the Prime Meridian, but that doesn't mean that the International Date Line is both west and east of itself. Of course, the Earth isn't an infinitely large sphere like the number circle in my analogy.

Basically, the projective line is only non-well ordered at infinity, because that's when it 'turns into a circle' and loops around, making infinity both greater and larger than the reals. Of course, I may be understanding the projective line completely wrongly as this is only an explanation of my infinite circle analogy.

Schrollini wrote:This isn't necessarily a bad thing -- the complex numbers aren't well-ordered either.

And just like complex numbers, adding it on to our current number system would do nothing to it, but rather expand our viewpoint.

Schrollini wrote:It's just one of the many things you have to give up to be able to treat ∞ as an element of your set. Others have pointed out that you also lose the field axioms. In most cases, the loss of these things outweighs the inclusion of ∞. (What good does it do to allow division by zero if division is no longer well-defined?)

How would division be no longer well-defined? Allowing division by zero would only make division by zero not well-defined, and if we leave it how it is, then it wouldn't be defined at all, which is even worse.

Schrollini wrote:I'm guessing from what you've written that you're a high-school student, and you haven't had any abstract algebra.

I feel like this conversation will degenerate if we start discussing each poster's level of education, but I'll tell you just to be honest, no, I haven't taken a formal class on abstract algebra, but that doesn't mean I can't understand it. I know it's a relevant question, but the tone that might result from each person's saying there education levels might be destructive to this discussion's constructive-ness.

Schrollini wrote:If so, all our objects about groups and fields probably sound pointless and, well, abstract.

Probably just the vocabulary. I can definitely see mathematics as a system where you can define your own things, and it sounds like this is the sort of abstract algebra you're talking about. Also, they don't seem pointless, but I don't exactly know what they mean. I am still capable of having a discussion about 1/0, though it may be limited a bit.

Schrollini wrote:Please rest assured that there are good reasons to care about these things, and if you keep studying math, you'll learn them soon.

1) Why would I believe against there being good reasons to care about these things?
2) Since when is "soon" a time years away? Yes, from a historical, perspective, I'll get to studying this math really soon, but this isn't historical perspective. In fact, 1/0 has bothered me since elementary school. It's wrong to just cut off a branch of a tree, however small it was. The branch was doing the tree no harm, so why cut it off in fear that it was?

Schrollini wrote:It's good to question the rules we use for mathematics, but it's better do to so for the purpose of learning why they were chosen.

I already feel like I know why they were chosen. Like I said, there are many reasons why people have decided to exclude 1/0, but they are all wrong. I just can list every reason because I don't what reasons people will come up with.

Schrollini wrote:You'll get a more measured response if you post, "Why can't we treat ∞ as a real number,"

I probably should have brought it up more like that. The way I did it got me yelled at by the magistrate, and no one wants that to happen from their first post/thread! (For some reason, this always happens to me whenever I join a large forum). I hope you all recognize that I'm not a math crackpot now.

Schrollini wrote:rather than, "Look at me! I've invented a new kind of mathematics where ∞ is a number!"

I'm not inventing a new system, it's only an expansion on the current system. Put another way, I'm trying to say "Hey look, there's no reason for us to exclude this part of mathematics" rather than, "Look, I invented a totally new type of mathematics independent of our current mathematics."
Last edited by exfret on Sat Feb 08, 2014 10:02 pm UTC, edited 1 time in total.

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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 9:37 pm UTC

Sorry for taking so long to reply.

gmalivuk wrote:As previously stated, it violates the field axioms for division by zero to be defined, and we kind of like having the Reals be a field. The exclusion of infinity from the standard definition of the real numbers is clear and defined because the real numbers are not defined as having an element at infinity.

As stated, there are things you can do with points at infinity, but those things are no longer the standard operations on the standard sets. If you're allowed to divide by zero and treat the result as a real number to be manipulated like any other, then you can manipulate your symbols to show 1=2, and all that implies. Where "all that implies" is everything.

Since we'd rather *not* have a system of mathematics where every statement has a valid proof, we don't define the reals as including infinity or a multiplicative inverse of 0.

Ah, I see. A field is just a set of numbers through which division, multiplication, addition, and subtraction by any two non-zero numbers within that field do not return numbers outside of the field. Also, how'd you get to 1=2 from my symbols? if you started with ∞=∞+1 and subtracted infinity or something of the sort, I already have an explanation of the sort which I will paste here:
alessandro95 wrote:If we were to consider ∞ as a real than our number system simply won't work because (for example) subtracting the same quantity from both sides of an equation is definitely a valid operation, we defined it to be so.
Yes, I agree with you. Subtracting the same quantity from both sides of an equation is a valid operation, but how does this mean ∞ can't be? Are you referring to subtracting ∞ from both sides of the equation ∞=∞+1? Even if you are not, I expect this will be a common thing for people to bring up, so, I'll just deal with it now. It actually involves multiple returns. Basically, when you're subtracting ∞ from each side of an equation, you can't just assume ∞-∞=0, it might also be equal to 1, because remember that ∞=∞+1, so ∞-∞ is the same as (∞+1)-∞, which you could evaluate to be 1. So in the equation:
∞ + 1 = ∞
(∞ + 1) - ∞ = ∞ - ∞
Now you could evaluate this to become this:
1=0
But that would be assuming that the operation of ∞-∞ returns the same thing on both side, which it doesn't because it can have multiple returns. There's a post above where I discuss about this more, and I even have a post in the threed (the thread where my original post in quotes comes from) that goes into deeper detail about this specific problem in case you are unsatisfied with this how it is.

Also, I do in fact know of the Principle of Explosion. Isn't there an xkcd comic on it? Also, I think that if it must, there is no reason for infinity not to be a nonreal number, but it should at least be considered a number.

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Re: One divided by Zero (1/0)

Postby brenok » Sat Feb 08, 2014 9:48 pm UTC

So, what exactly is infinity divided by infinity?

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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 9:55 pm UTC

Let me try to clarify how I see the current discussion before another postal wave hits me:
-Many of you are trying to say infinity is not a real number, to which I say:
Um... So? It's still a number. If it must not be 'real' then let's not include it in that domain of numbers. Isn't 'real' just a grouping we made to help classify a set of numbers that follow certain properties. If infinity doesn't follow those properties, then it shouldn't be apart of that set. It still exists, though.
-I see a lot of posts about the properties that ∞ would violate if it were a real number. My thoughts on this are as follows:
Why can't it just be a number, not a real number? It doesn't even violate many of these properties anyways.
-To those of you who think 1/0 != ∞, I answer with:
Well, fine, I admit it. I don't exactly think I've proved that the result is infinity, but it's at least a number, and it would be wrong (or at least ignorant) to leave it as undefined when it might be useful otherwise. In this way, it's almost like the pi versus tau argument, except arguing whether we should have pi or tau in mathematics at all or whether adding a circle constant would break stuff. (Edit: This part might sound arrogant. It's not meant to. Please, imagine a fluffy unicorn dancing through a rainbow on a sunny day and hopefully it'll sound less arrogant. Unless you're imagining Charlie the Unicorn, then that just makes it worse.)
-Anyone who thinks I can't use it like a real number should know my opinions on that are this:
Why not? I can perform any operations on anything. :mrgreen: + 1 is perfectly fine.
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Re: One divided by Zero (1/0)

Postby alessandro95 » Sat Feb 08, 2014 10:00 pm UTC

exfret wrote:Could you demonstrate how ∞ violates such axioms? Also, how am I evaluating ∞ within the field of reals (which I assume you are implying)? Is it because I use 'real operations'?


R is defined as the unique (up to isomorphism) archimedean, complete and ordered field.
For a field to be archimedean it must not contain neither infinity nor infinitesimal, this is why ∞ violates the axioms of the real numbers.

I assumed you were evaluating ∞ inside the reals, since you "proved" it should be a real.

But that doesn't mean 1/0 has to be undefined and ∞+1 to be wrong, it's just that you cannot do that while working with real numbers, but you can of course define such things in other sensible and even useful ways
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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 10:16 pm UTC

alessandro95 wrote:R is defined as the unique (up to isomorphism) archimedean, complete and ordered field.

The unique, etc. field? Doesn't the set of rational numbers follow those properties?

alessandro95 wrote:For a field to be archimedean it must not contain neither infinity nor infinitesimal, this is why ∞ violates the axioms of the real numbers.

Why do we have to have an axiom specifically excluding infinity? Wouldn't that be like specifically excluding 3 as a number? We'd obviously have to create all sorts of bizarre new rules if that were the case.

alessandro95 wrote:I assumed you were evaluating ∞ inside the reals, since you "proved" it should be a real.

Umm... Well, if division of a real by a real is defined to be real as an axiom somewhere, then I'm sure there's a side-note excluding 1/0 anyways, so I'm not exactly bent on it being real. I was arguing against some people's reasons of why it couldn't be real, though, because they were incorrect. This doesn't mean that it was my belief that infinity was real; I just wanted to correct those whose reasons Edit: seemed incorrect to me. (No grammar-geeks?)

alessandro95 wrote:But that doesn't mean 1/0 has to be undefined and ∞+1 to be wrong, it's just that you cannot do that while working with real numbers, but you can of course define such things in other sensible and even useful ways

And when would you have to exclude every number you're working with to being real? Besides, this doesn't lead to the conclusion that 1/0 is undefined. That, I believe, we can agree is in the Edit: Sorry, I totally didn't mean what I wrote here as I wrote it. When I said "ignorant," I didn't mean of you all being ignorant, I meant rather that mathematics would be ignoring infinity, and thus they would be being "ignorant." I really can't believe I'm wording my posts this way.
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Re: One divided by Zero (1/0)

Postby brenok » Sat Feb 08, 2014 10:18 pm UTC

exfret wrote:-Many of you are trying to say infinity is not a real number, to which I say:
Um... So? It's still a number. If it must not be 'real' then let's not include it in that domain of numbers. Isn't 'real' just a grouping we made to help classify a set of numbers that follow certain properties. If infinity doesn't follow those properties, then it shouldn't be apart of that set.

Are you sure that's what you meant? It seems contradictory
-I see a lot of posts about the properties that ∞ would violate if it were a real number. My thoughts on this are as follows:
Why can't it just be a number, not a real number? It doesn't even violate many of these properties anyways.

As far as I'm aware, there isn't a unique definition of what a number is, so, sure. Like I said, it's true, but not significant.
-To those of you who think 1/0 != ∞, I answer with:
Well, fine, I admit it. I don't exactly think I've proved that the result is infinity, but it's at least a number, and it would be wrong (or at least ignorant) to leave it as undefined when it might be useful otherwise.

If you say why is useful, and that usefulness outweight the reasons why it's undefined, people will probably listen.
-Anyone who thinks I can't use it like a real number should know my opinions on that are this:
Why not? I can perform any operations on anything. :mrgreen: + 1 is perfectly fine.

Isn't that exactly what I said here?
brenok wrote:The number you're proposing doesn't fit the operations with real numbers. It's like comparing apples and oranges. You could set a number system where 1 + apple = orange. The could be technically correct, but also meaningless. Such a system have no relation with the mathematics used everywhere else.

We usually expect some things from numerical operations. I expect, for instance, to do things like x - x , or x/x, and that it yields a single result.

EDIT:

Also:
exfret wrote:
alessandro95 wrote:R is defined as the unique (up to isomorphism) archimedean, complete and ordered field.

The unique, etc. field? Doesn't the set of rational numbers follow those properties?

The Rational, isn't complete, as it doesn't include the irrationals (pi, sqrt(2) etc)

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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 10:21 pm UTC

brenok wrote:So, what exactly is infinity divided by infinity?

It's value is indeterminate. (It has exactly the same value as 0/0).

Edit: It's exactly like 0/0. It won't always have the same value in the way the -/+ is exactly like +/-, but it never has the same value. (Of course, differing from that +/- example, 0/0 may in fact return the same value as ∞/∞ at times).
Last edited by exfret on Sun Feb 09, 2014 12:46 am UTC, edited 4 times in total.

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Re: One divided by Zero (1/0)

Postby alessandro95 » Sat Feb 08, 2014 10:34 pm UTC

exfret wrote:
alessandro95 wrote:R is defined as the unique (up to isomorphism) archimedean, complete and ordered field.

The unique, etc. field? Doesn't the set of rational numbers follow those properties?

No, the field of rationals is ordered and archimedean, but it isn't complete.

exfret wrote:
alessandro95 wrote:For a field to be archimedean it must not contain neither infinity nor infinitesimal, this is why ∞ violates the axioms of the real numbers.

Why do we have to have an axiom specifically excluding infinity? Wouldn't that be like specifically excluding 3 as a number? We'd obviously have to create all sorts of bizarre new rules if that were the case.

That axiom isn't there specifically to exclude ∞, it is there to state a very useful properties of real numbers without which they simply won't work as we wish them to and which happens to exclude infinitesimals and infinities from the field of reals

exfret wrote:
alessandro95 wrote:I assumed you were evaluating ∞ inside the reals, since you "proved" it should be a real.

Umm... Well, if division of a real by a real is defined to be real as an axiom somewhere, then I'm sure there's a side-note excluding 1/0 anyways, so I'm not exactly bent on it being real. I was arguing against some people's reasons of why it couldn't be real, though, because they were incorrect. This doesn't mean that it was my belief that infinity was real; I just wanted to correct those whose reasons were incorrect. (No grammar-geeks?)

R is defined as a field {R,+,*,<}.
Let's not worry about the order relation but only with addition and multiplication for a moment.
The definition of a field is a commutative ring (in this case {R,+,*}), which has a multiplicative inverse for every element but the additive identity, and R contains indeed a multiplicative inverse for number except 0, and it is not supposed to contain one for 0, which is why 1/0 is undefined

exfret wrote:
alessandro95 wrote:But that doesn't mean 1/0 has to be undefined and ∞+1 to be wrong, it's just that you cannot do that while working with real numbers, but you can of course define such things in other sensible and even useful ways

And when would you have to exclude every number you're working with to being real? Besides, this doesn't lead to the conclusion that 1/0 is undefined. That, I believe, we can agree is in the very least ignorant if not incorrect.

1/0 is undefined in the fields of reals for the fields axioms I quoted above
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Re: One divided by Zero (1/0)

Postby Schrollini » Sat Feb 08, 2014 11:12 pm UTC

exfret wrote:Basically, the projective line is only non-well ordered at infinity, because that's when it 'turns into a circle' and loops around, making infinity both greater and larger than the reals. Of course, I may be understanding the projective line completely wrongly as this is only an explanation of my infinite circle analogy.

A set S is well ordered by the operator < iff ∀xyS, x<y or x>y. The projective line does not satisfy that definition for any operator; therefore, it is not well-ordered. It can't be partially well-ordered or kinda well-ordered, because there ain't no such thing.

exfret wrote:
Schrollini wrote:I'm guessing from what you've written that you're a high-school student, and you haven't had any abstract algebra.

I feel like this conversation will degenerate if we start discussing each poster's level of education, but I'll tell you just to be honest, no, I haven't taken a formal class on abstract algebra, but that doesn't mean I can't understand it. I know it's a relevant question, but the tone that might result from each person's saying there education levels might be destructive to this discussion's constructive-ness.

I mention this not to belittle you in any way, but because it's fairly clear that you're unfamiliar with a number of important things that go into our decision not to include infinity in the reals. (For example, well-ordering and the field axioms.) There's no shame in this -- abstract algebra is typically a junior or senior level college class for math majors. It's not something most people know. (It's not something most people even know exists!)

But you are talking to people who do know abstract algebra, and know it well. (Not me, by the way. It was my worst class in college.) So when you say something like,
exfret wrote:I already feel like I know why they were chosen. Like I said, there are many reasons why people have decided to exclude 1/0, but they are all wrong.
it comes off as incredibly haughty and conceited. You may well be the smartest person here, but you're not the most educated. So why not take this chance to learn something, rather than railing about how we're fools for not agreeing with you? If someone tells you that your proposal is a problem because it isn't a field, ask, "What's a field? Why is it important to be a field?" You'll surely end up with some links to Wikipedia, but chances are good someone will take the time to explain it to you.
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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 11:23 pm UTC

brenok wrote:Are you sure that's what you meant? It seems contradictory

The one thing I am sure of is that the way I've said it, it surely seems like that wasn't what I meant. That is exactly why I posted that. Now, hopefully, that clears some things up.

brenok wrote:As far as I'm aware, there isn't a unique definition of what a number is, so, sure. Like I said, it's true, but not significant.

As far as I'm aware, there isn't a unique definition of what "not significant" means, so sure. Like I said, it's "not significant," but that's not significant. (I don't mean to be making fun of you, I just thought copying you would expose the irony of this from my viewpoint while adding light humor). (Edit: I saw a wikipedia article mention undefined having something to do with meaningless. Is this what you mean. If so, could you explain it a little more? I'm sorry for my inability to understand what you mean. :( )

brenok wrote:If you say why is useful, and that usefulness outweight the reasons why it's undefined, people will probably listen.

Please tell me "the reasons it's undefined." These reasons are what I'm arguing against, yet no one has told me them, so how could I possibly disprove them?! That was meant to be the major purpose of this thread! Edit: Oh my, oh my. I did not mean this the way I put it. I was expecting people to post of reasons I had already heard of or that went along the same lines. When you all posted more advanced reasons, I failed to recognize them, and I probably also failed to understand part of their meaning. I'm sorry. :(

brenok wrote:Isn't that exactly what I said here?
brenok wrote:The number you're proposing doesn't fit the operations with real numbers. It's like comparing apples and oranges. You could set a number system where 1 + apple = orange. The could be technically correct, but also meaningless. Such a system have no relation with the mathematics used everywhere else.

We usually expect some things from numerical operations. I expect, for instance, to do things like x - x , or x/x, and that it yields a single result.

Yes, and until you prove those things, it's perfectly fine for infinity to not return a single result. Edit: Does this sound arrogant? Please don't sound arrogant. Please don't sound arrogant. Please don't sound arrogant. :(

brenok wrote:The Rational, isn't complete, as it doesn't include the irrationals (pi, sqrt(2) etc)

Wait, but wouldn't that mean that the real aren't complete as they don't include the imaginary or non-real complex or infinity?
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Re: One divided by Zero (1/0)

Postby alessandro95 » Sat Feb 08, 2014 11:36 pm UTC

exfret wrote:Please tell me "the reasons it's undefined." These reasons are what I'm arguing against, yet no one has told me them, so how could I possibly disprove them?! That was meant to be the major purpose of this thread!

I did tell you, it is because of fields axioms

exfret wrote:Wait, but wouldn't that mean that the real aren't complete as they don't include the imaginary or non-real complex or infinity?

Complete doesn't mean that is has to contain everything you can think of, it simply means that every cauchy sequence of real numbers converges to a real limit, which is not the case in Q since there are sequences converging to irrational limits
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Re: One divided by Zero (1/0)

Postby exfret » Sat Feb 08, 2014 11:59 pm UTC

alessandro95 wrote:That axiom isn't there specifically to exclude ∞, it is there to state a very useful properties of real numbers without which they simply won't work as we wish them to and which happens to exclude infinitesimals and infinities from the field of reals

Okay, but this only means that infinity is only excluded because it's an oddball and not because and not a number, right?

alessandro95 wrote:R is defined as a field {R,+,*,<}.
Let's not worry about the order relation but only with addition and multiplication for a moment.
The definition of a field is a commutative ring (in this case {R,+,*}), which has a multiplicative inverse for every element but the additive identity, and R contains indeed a multiplicative inverse for number except 0, and it is not supposed to contain one for 0, which is why 1/0 is undefined

I can see that a field is defined for adding, multiplying, and order, (I overlooked that last one) but why do we have to specifically exclude the multiplicative inverse of the additive identity? Even if we should exclude it just so it doesn't violate the real number's properties, why does it have to be excluded from all fields? Also, why must it be undefined just because of this exclusion? It has a value that can be defined. Here: I define x such that x = 1/0. There. Call it meaningless, but it's not undefined. Edit: Call it meaningless but it's not undefined, right?

alessandro95 wrote:1/0 is undefined in the fields of reals for the fields axioms I quoted above

So it's undefined for the reals? But that doesn't make any sense. Edit: But that doesn't make any sense to me. "The Reals" are just an isolated part of mathematics that we created mostly because they can be ordered. 1/0 would be inexistent for the reals. Of course, I still don't understand why you'd have such a restriction to use explicitly only real numbers anyways.

alessandro95 wrote:I did tell you, it is because of fields axioms

Oh, sorry. I overlooked that because I was expecting different reasons (e.g. zero times any number is always zero). Anyways, the field of axioms would state that it's undefined for a field because it would be the multiplicative inverse of the additive identity (which is what I said myself), is that correct? But, why would it be 'undefined'? It's a number, a defined number, but just one outside of the specific field one might be focusing on. I find it as absurd as calling i undefined when working with reals. Edit: That's just an analogy, I'm not calling your idea absurd.
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Re: One divided by Zero (1/0)

Postby exfret » Sun Feb 09, 2014 12:42 am UTC

Schrollini wrote:A set S is well ordered by the operator < iff ∀xyS, x<y or x>y. The projective line does not satisfy that definition for any operator; therefore, it is not well-ordered. It can't be partially well-ordered or kinda well-ordered, because there ain't no such thing.

How about the thing we lose from adding infinity to our mathematics be the rule that we can't have only part of a set be ordered? Besides, there is a such thing as 'partially ordered' in a way. A subset of the reals and infinity (which would be just the reals) is ordered. Just because not everything in the set is completely ordered doesn't mean we lose something. We're only losing something by not including infinity. The complex numbers, for example, aren't ordered, but we don't lose order in the reals just from adding them. Just like with the complex numbers, including infinity would be including an non-ordered number, but infinity's going to be that way whether we add it to our number system or not, so we can either ignore it and have a less complete number system, or we can include it at no cost at all. Infinity's going to break those properties no matter what, ignoring it doesn't change anything.

Schrollini wrote:I mention this not to belittle you in any way, but because it's fairly clear that you're unfamiliar with a number of important things that go into our decision not to include infinity in the reals. (For example, well-ordering and the field axioms.) There's no shame in this -- abstract algebra is typically a junior or senior level college class for math majors. It's not something most people know. (It's not something most people even know exists!)

I did not think you were trying to belittle me. No, not at all! I just thought better of admitting to being inferior in my education to those I disagreed without. I thought it might make my ideas seem invalid just because I was the sole person who argued them, and I had less math experience than my 'opponents'. I didn't want this to become a discussion where I, the person who knows best the ideas I was trying to get across (as in my 1/0 proof, I would obviously be the person who knew what I meant by it, I don't mean this as in I know what I'm talking about the most), seemed to be inferior, thus making the ideas seem less valid. I wanted the debate of those ideas to exist whether I seemed like or am in reality a good mathematician or not, and it seemed like it would get to be more like people arguing than a clash of ideas if I didn't word it the way I did.

Schrollini wrote:But you are talking to people who do know abstract algebra, and know it well. (Not me, by the way. It was my worst class in college.) So when you say something like,
exfret wrote:I already feel like I know why they were chosen. Like I said, there are many reasons why people have decided to exclude 1/0, but they are all wrong.
it comes off as incredibly haughty and conceited.

Oh my! I... You're right. I didn't mean to write it that way. No wonder I angered the magistrate. :oops:
Not to be boastful, hypocratic, and contradictory, but I'm modest... There are so many things wrong with saying that, but I had to get my point across somehow. I hope you'll forgive me and believe me... So when I see that I've written something like that, it's just, so bad of me. Thank you for pointing this out. It would be extremely useful to me if you could point this out in the future, as I don't get the time to read over my posts with how frequently people post in here.

Schrollini wrote:You may well be the smartest person here, but you're not the most educated.

Yes, I am totally not the "most educated," and I wouldn't dare claim to be the smartest. I just wanted to, like I said, have a clash of ideas exist whether either side was good at math or not. I didn't want it to come down to a clash of either side, but rather a clash of their ideas, so I was trying to word my response in such a way that it would end up that way. I guess it failed in quite an ironic way, though.

Schrollini wrote:So why not take this chance to learn something,

I was actually hoping to see whether we could get down to whether or not 1/0 was undefined. Think of this as a way for searching for the truth with greater brainpower than just one person possesses. I just didn't know if the current consensus was right, and I can't just hold a discussion with numbers and text (unless of course, there's a person on the other side of those numbers and text), so posting in forums like these is the way I find out whether something's correct. It's actually taking too much time already to respond to you all's posts, and I've so far been able to get by understanding (hopefully, you never really know if what you interpreted was what the communicator meant) most of what people have been saying (except for notzeb's first post). I was kind of hoping we might even be able to reach an answer. So learning isn't exactly what I was hoping to get out of this thread, except for learning the "truth" or what "should" happen. Still, though, i have been learning. For example, I now know what fields are, and I've become the slightest bit more familiar with mathematical notation.

Schrollini wrote:If someone tells you that your proposal is a problem because it isn't a field, ask, "What's a field? Why is it important to be a field?" You'll surely end up with some links to Wikipedia, but chances are good someone will take the time to explain it to you.

I actually wiki'd it and it was pretty straightforward (in my opinion). Of course, if you're telling me this, it's probably because I messed up something and what I think a field is isn't exactly what a field is.

Schrollini wrote:rather than railing about how we're fools for not agreeing with you?

Oh noes. :o :( I appeared to be that way? No, no, no, no, no. I seem to be having difficulties communicating what I mean. In none of this do I believe any of you are fools. I am grateful for all of the responses you have all given me. Sometimes I wonder if I myself am a fool for my thoughts. Please, I'm sure you are all wonderful people, and I mean nothing to offend any of you in any sort of way. I'm sorry if I may in the future or may have in the past seemed to come across in such a way. :(

exfret
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I'm Sorry

Postby exfret » Sun Feb 09, 2014 1:34 am UTC

After Schrollini reminded me of how I was wording things, I just couldn't believe myself. Please forgive me for sounding so arrogant, etc. I'm surprised gmalivuk hasn't closed this thread already: I sound very much like a math crackpot with how I've been wording things. It's just that from my viewpoint, I've been labeling ideas as absurd, but I overlooked that it might be offensive to the creator of the ideas. Please, forgive me, for I've been the absurd one. I think a major reason my possibly insultive writing has passed my knowing is that I haven't had much time to read over it with all the posts you all have been posting, so I looked over some of my past posts and I've edited some of the areas that I thought might look arrogant, but it wasn't at all a very comprehensive search. (I didn't view as that important now that you've all read it anyways). Anyways, this possibly seems like a good time to summarize what this discussion has come to so far. In my eyes, it's kind of hazy what we're talking about right now, (I'm sorry, that's my fault) but it seems like it's come down to this:

-Me trying to repeat over and over that 1/0 isn't undefined
You all saying it's undefined for reasons that I either don't understand or that I'm trying to argue against

-Me trying to discuss something I'm hoping I got correct
You all discussing it back with a better understanding of the topic having to explain to me the mathematics I'm trying to use to argue against you

-Me not understanding what we lose from adding infinity to mathematics
You all either explaining what we lose poorly or giving me an explanation of what we lose that I'm poorly understanding (or it could just be the inherent noise of the English language, but it's usually just annoying and rarely this significant)

-Me talking about a number circle with infinite radius that might be the same thing as the projective line
You all talking about the projective line while getting confused when I mention a number circle of infinite radius when you were talking about the projective line

-Me, not understanding what it means for what I'm saying to be 'meaningless'
-(One of) You all saying infinity could be a number, but it's 'meaningless'

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Qaanol
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Re: One divided by Zero (1/0)

Postby Qaanol » Sun Feb 09, 2014 2:50 am UTC

The short answer is yes, you can define an object X with the property X = b/0 for any real number b, and it behaves pretty much as you expect. It absorbs addition and multiplication, while division and subtraction work for the most part except X-X and X/X are undefined.

The thing is, the word “infinity” and the symbol “∞” get used for many different things. There is the cardinality of the integers, ℵ0, called “countable”. There is the cardinality of the reals, 20, called “continuum”.

There is every other non-finite cardinality. There are the ends of the extended real line. The top of the projective line. The north pole of the Riemann sphere.

The question you want to ask is, “What problems does this object X solve?”
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notzeb
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Re: One divided by Zero (1/0)

Postby notzeb » Sun Feb 09, 2014 6:36 am UTC

Schrollini wrote:A set S is well ordered by the operator < iff ∀xyS, x<y or x>y. The projective line does not satisfy that definition for any operator; therefore, it is not well-ordered. It can't be partially well-ordered or kinda well-ordered, because there ain't no such thing.
What are they teaching in schools these days?

A set S is totally ordered by < if any two unequal elements of S are comparable. Well ordered means something completely different! It means, basically, that you can do induction with it (precise definition is that every subset of S has a least element). The reals are not well ordered by the standard ordering (for instance, there is no least positive real number).

Carry on.


For exfret: it seems that one of your main points here is that infinity is fine as long as we allow many of our basic operations to become multivalued. Many people will disagree with you about this, since multivalued functions introduce a whole slew of new problems and subtleties to worry about. For instance, when we extend the square root function to all of the complex numbers it becomes multivalued since there really is no good way to tell the "positive square root" from the "negative square root" of a number like -1+i without introducing branch cuts. This phenomenon leads to fake proofs that 1 = -1, such as the well known

1 = √1 = √(-1*-1) = √(-1) * √(-1) = i*i = -1.

The problem here is naively treating a multivalued function as an ordinary function. If one is careful and does everything algebraically (or introduces branch cuts, or talks about correspondences) then you can get around these difficulties and avoid contradictions of this nature. Since many people do not want to deal with being careful about multivalued functions, the way mathematics is standardized is generally that we simply make no definition for standard operations in the cases that they are multivalued. It makes it easier for the kids to learn about it and harder for them to get into trouble.

The point I am making is not that what you are doing is fundamentally wrong. Rather, it is that it will not be accepted as standard mathematics. You will not convince people to include infinity in the real numbers, but there are plenty of other number systems that can and do include infinity (which mathematicians introduce because they feel like it's convenient to treat infinity as a "number" in certain contexts). If you like you can feel free to introduce yet another number system including infinity, and no one will get mad at you so long as

1. You do not give it a name which can be mixed up with a number system people are already using. (So, don't call infinity a "real number". That has a standardized definition. Instead call it a "cool number" or a "multivalued number" or something. You get to make up whatever name you like, so long as it hasn't been taken yet.)
2. You are absolutely clear about which manipulations are allowed in this new number system. (For this, you may want to learn how to formalize multivalued functions. I suggest taking the time to read about relations and correspondences.)
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Schrollini
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Re: I'm Sorry

Postby Schrollini » Sun Feb 09, 2014 7:20 am UTC

exfret wrote:After Schrollini reminded me of how I was wording things, I just couldn't believe myself. Please forgive me for sounding so arrogant, etc. I'm surprised gmalivuk hasn't closed this thread already: I sound very much like a math crackpot with how I've been wording things.

No worries. If I had thought you were a crackpot, I wouldn't have bothered trying to warn you. (And gmalivuk would have locked the thread, I'm sure.)

One other piece of unsolicited advice -- don't feel that you have to respond quickly or to every individual point raised. We're dumping a whole lot of heavy mathematics on you; it takes some time to absorb. It's fine to read everyone's posts, go off and think about them for a few hours, or a few days, and then compose a single response to all of them. We aren't going anywhere.

exfret wrote:-Me trying to repeat over and over that 1/0 isn't undefined
You all saying it's undefined for reasons that I either don't understand or that I'm trying to argue against

Several people have said that 1/0 isn't defined in a field, but I don't think anyone has quite specified why. Here's my attempt to explain the point.

Essentially, you are asking for 0 to have a multiplicative inverse, 0-1, so that 1/0 = 1⋅0-1 = 0-1. This leads to problems with axiom of distributivity, which says a⋅(b+c) = ab + ac for all a,b,c in the field F.

Let us consider the case where c = 0. The left-hand side gives us a⋅(b+0) = ab, by the definition of the additive identity. The right-hand side is ab + a⋅0, so we have ab = ab + a⋅0. To each side, we add the additive inverse of ab, so we get 0 = 0 + a⋅0 = a⋅0, by the definition of the additive identity. Therefore, every element aF will satisfy a⋅0 = 0.

Thus, the putative element 0-1 must satisfy both 0-1⋅0 = 1 (definition of multiplicative inverse) and 0-1⋅0 = 0 (above theorem). This would be a contradiction, so we don't include an element 0-1 in a field.

Now the real numbers are defined a certain field. Or maybe they're defined and then shown to be a field. Either way, they can't contain 1/0, because fields don't contain the multiplicative inverse of additive identity.

There are number systems other than the reals, of course. There's the real projective line, the extended real line, the complex numbers, the quaternions, the surreal numbers, the hyperreal numbers, the Grassman numbers, and probably a gazillion others. When you're working on a problem, you should choose the one that best suits your needs. If you need 1/0 to be defined, choose one where it is. But as it happens, the reals (or perhaps the complex numbers, depending on your point of view) most often are the most useful. (It turns out that being a field is really handy.) So that's what we tend to use.

notzeb wrote:
Schrollini wrote:A set S is well ordered by the operator < iff ∀xyS, x<y or x>y. The projective line does not satisfy that definition for any operator; therefore, it is not well-ordered. It can't be partially well-ordered or kinda well-ordered, because there ain't no such thing.
What are they teaching in schools these days?


Schrollini wrote:But you are talking to people who do know abstract algebra, and know it well. (Not me, by the way. It was my worst class in college.)


notzeb wrote:A set S is totally ordered by < if any two unequal elements of S are comparable. Well ordered means something completely different! It means, basically, that you can do induction with it (precise definition is that every subset of S has a least element). The reals are not well ordered by the standard ordering (for instance, there is no least positive real number).

Thanks for straightening me out. I was actually puzzled by this as I wrote it, since I remembered the definition of well-ordering you give, but thought it was an implication, not a definition. But it obviously doesn't work with open sets, so I decided I must have remembered wrong. Should have checked Wikipedia.
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