0/0=?
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0/0=?
Oh God, I know this is going to be an evil thread...but quite frankly I want to see what all of you geniuses will do without any help other than the subject.
Just kidding, I will say that somebody said using circular logic you could prove 1+1=0/0, which I would very much like to see. Also, is this an uncountable uncountable uncountable...infinity sort of a problem? Another thing, is there a notation for such an infinity? Is the solution to 0/0 a plain old infinity? Is it 0? is it 1? Is it Slashdot.org? Why? Sorry, alot of questions, lets see where this goes...
Just kidding, I will say that somebody said using circular logic you could prove 1+1=0/0, which I would very much like to see. Also, is this an uncountable uncountable uncountable...infinity sort of a problem? Another thing, is there a notation for such an infinity? Is the solution to 0/0 a plain old infinity? Is it 0? is it 1? Is it Slashdot.org? Why? Sorry, alot of questions, lets see where this goes...
 Brooklynxman
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Re: 0/0=?
sje, and a stickied post warning not to make another thread. And this was made on mod madness day.
*sighs*
I can see where this is headed (its about to have its ass /0'ed hard)
*sighs*
I can see where this is headed (its about to have its ass /0'ed hard)
We figure out what all this means, then do something large and violent
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Re: 0/0=?
Sigh .. . .
division by zero is not defined. Have you ever had five zeroths of something? Now have you ever had zero zeroths of something?
division by zero is not defined. Have you ever had five zeroths of something? Now have you ever had zero zeroths of something?
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Re: 0/0=?
Oh, that's bad news. I thought limits were a fundamental principle of analysis. I guess you just rendered centuries of research void. I hope you have a bad feeling about this.
On a more serious note: You will be very hardpressed to give 0/0 any meaning that is not immediately rendered inconsistent even for a small subset of any problem you might have decided to try to resolve by giving a meaning to it. Algebraically the only sensible meaning it could have would be to equal 1, and then all hell breaks loose because that's obviously not compatible with, well, anything. (0/0 * 2/5 could be 1, or 2/5, you're losing stuff like associativity, commutativity and/or other fundamental properties of any reasonably useful algebraic structure that includes multiplication and a 0)
If you happen to run into 0/0 as an indeterminate form as the division of two limits it may very well have a meaning, but depending on the limits in question it might be equal to pretty much anything.
This has been discussed many times in many places. No need to start another topic about it that wouldn't contribute any new insights (because frankly, I doubt there is any more insight to be had).
On a more serious note: You will be very hardpressed to give 0/0 any meaning that is not immediately rendered inconsistent even for a small subset of any problem you might have decided to try to resolve by giving a meaning to it. Algebraically the only sensible meaning it could have would be to equal 1, and then all hell breaks loose because that's obviously not compatible with, well, anything. (0/0 * 2/5 could be 1, or 2/5, you're losing stuff like associativity, commutativity and/or other fundamental properties of any reasonably useful algebraic structure that includes multiplication and a 0)
If you happen to run into 0/0 as an indeterminate form as the division of two limits it may very well have a meaning, but depending on the limits in question it might be equal to pretty much anything.
This has been discussed many times in many places. No need to start another topic about it that wouldn't contribute any new insights (because frankly, I doubt there is any more insight to be had).

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Re: 0/0=?
If you define /0 as the multiplicative inverse of 0 you get exactly the issue I pointed out above. It's not compatible with some fundamental principles of algebraic strucutres (I don't feel like figuring out what exactly one would have to do away with to make it consistent, I bet other people already have), and as such your new structure that includes /0 has become a lot less useful than the ordinary Reals or Complex Numbers or whatever structure you decided to "enhance" with this. If you can think of some useful way to define /0 as something that isn't the multiplicative inverse of 0, please say so. Simply calling it a does not make it any useful at all.
Re: 0/0=?
Heh, math isn't meant to be useful. When are we gonna need uncountable infinites outside of math? Same thing with creating i, they had no clue itd be useful, which it hardly is anyways, just thinking it might be a useful function. I'll work on it and get back to u.
Re: 0/0=?
alayah64 wrote:Heh, math isn't meant to be useful. When are we gonna need uncountable infinites outside of math? Same thing with creating i, they had no clue itd be useful, which it hardly is anyways, just thinking it might be a useful function. I'll work on it and get back to u.
I don't know too much about imaginary numbers, but I think that it isn't necessary for real life things, for imrpoving the world directly, or whatever, but it is used to help out with math. The possibility of imaginary numbers makes it so you don't have to keep looking for zeroes, once you have enough imaginaries, you can stop.
So what I'm saying is that not only doesn't it fill a function for the real life world, but you can't do anything else with it (with defining dividing by zero). It doesn't help you with anything. It's arbitrary.
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 evilbeanfiend
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Re: 0/0=?
defining imaginary numbers helps in describing all manner of real life physics, and i'm sure has many pure maths uses as well, defining 0/0 doesn't appear to help with anything either practical or theoretical, but if you want to you could define it and call it nullity. don't we already have a thread on this btw?
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Re: 0/0=?
alayah64 wrote: which it [imaginary numbers] hardly is anyways
I'd say allowing perfect representation of A.C. currents, phase difference between voltage and current sinusoids, the resistance/impedance/reactance given to a current by a capacitor or inductor, is a seriously useful application of compex numbers for (electrical) engineering. Complex numbers and imaginary units aren't all that complex or imaginary, it's just when tied alongside the 'real' plane that they confuse people.

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Re: 0/0=?
Complex numbers describe nearly anything electrical engineering aspect all the way to bird flight.
Just because they're imaginary doesn't mean they don't occur in nature.
As for 0/0, it's annoying.
lim x/x = 1
x>0
0^0 = 1 (according to my calculator )
and power laws dictate x^0 = x/x.
Just because they're imaginary doesn't mean they don't occur in nature.
As for 0/0, it's annoying.
lim x/x = 1
x>0
0^0 = 1 (according to my calculator )
and power laws dictate x^0 = x/x.
Re: 0/0=?
It is an elementary 1st year proof in algebra that no field has a multiplicative inverse to the identity of addition.
sure you can go ahead and define such an element anyway, but then you lose the characteristics of a field. In other words the notion of defining an inverse to 0 is incompatible with the normal notions of addition and multiplication.
The resulting algebraic structure is most likely boring, pretty simple, and if its not it has long been defined in another elementary way that allows a much better way to think about it then going "Take the reals! Now throw away everything you know about the reals in order to introduce /0!!".
Also where oh where are the mods?
sure you can go ahead and define such an element anyway, but then you lose the characteristics of a field. In other words the notion of defining an inverse to 0 is incompatible with the normal notions of addition and multiplication.
The resulting algebraic structure is most likely boring, pretty simple, and if its not it has long been defined in another elementary way that allows a much better way to think about it then going "Take the reals! Now throw away everything you know about the reals in order to introduce /0!!".
Also where oh where are the mods?
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Re: 0/0=?
As far as limits go, this is answered rather simply: http://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule
tl;dr  I said nothing important.
Re: 0/0=?
The resulting algebraic structure is actually quite intereseting: http://www2.math.su.se/~jesper/research ... wheels.pdf
(the above was posted in a previous 0/0 thread, but since the above is the only constructive response to the 0/0 question that I have seen, here it goes again)
(the above was posted in a previous 0/0 thread, but since the above is the only constructive response to the 0/0 question that I have seen, here it goes again)
 qinwamascot
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Re: 0/0=?
Wheels are interesting in some ways, but many important properties of the real numbers break down if we define x/0.
Uncountable specifically refers to cardinal numbers, of which x/0 is not one. The real numbers were made to have the properties of commutative and associative addition and multiplication, which we lose if we define division by 0. In addition, several other nice properties no longer apply.
Think of the function 1/x as x>0. We'd like 1/0 to be the limit as x>0, but that limit doesn't exist. We could call it an unsigned infinity, because it goes to infinity on both sides, but it doesn't help us at all to do so. Letting it not exist is better if we want math to work the same way as it has for thousands of years.
alayah64 wrote:Just kidding, I will say that somebody said using circular logic you could prove 1+1=0/0, which I would very much like to see. Also, is this an uncountable uncountable uncountable...infinity sort of a problem? Another thing, is there a notation for such an infinity? Is the solution to 0/0 a plain old infinity? Is it 0? is it 1? Is it Slashdot.org? Why? Sorry, alot of questions, lets see where this goes...
Uncountable specifically refers to cardinal numbers, of which x/0 is not one. The real numbers were made to have the properties of commutative and associative addition and multiplication, which we lose if we define division by 0. In addition, several other nice properties no longer apply.
Think of the function 1/x as x>0. We'd like 1/0 to be the limit as x>0, but that limit doesn't exist. We could call it an unsigned infinity, because it goes to infinity on both sides, but it doesn't help us at all to do so. Letting it not exist is better if we want math to work the same way as it has for thousands of years.
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Re: 0/0=?
Certhas wrote:It is an elementary 1st year proof in algebra that no nontrivial field has a multiplicative inverse to the identity of addition.
Fix'd. A fairly necessary addition, since the usual proof goes that if we have a field with a zero inverse, then it must be trivial.
All posts are works in progress. If I posted something within the last hour, chances are I'm still editing it.
Re: 0/0=?
No, the zero ring is not usually considered to be a field, or even a ring with identity.Token wrote:Certhas wrote:It is an elementary 1st year proof in algebra that no nontrivial field has a multiplicative inverse to the identity of addition.
Fix'd. A fairly necessary addition, since the usual proof goes that if we have a field with a zero inverse, then it must be trivial.
Wikipedia wrote:For technical reasons, the additive identity and the multiplicative identity are required to be distinct.
Jerry Bona wrote:The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
Re: 0/0=?
OK, sheesh. To make explicit what was said before, let's take an ordered field and suppose that 0 has a multiplicative identity. That is, there exists some 0` such that 0*0`= 0`'*0 = 1.
Now, the field axioms give us that 0*a = a*0 = 0 for all a in the field. Thus 0` * 0 * a = 0` * 0 [imath]\Rightarrow[/imath] 1 * a = 1 [imath]\Rightarrow[/imath] a = 1. This applies for any in the field, so in particular, 0=1. But by the very definition of a field, 0!=1. So it's not a matter of whether or not we'd like to define a multiplicative inverse of zero, doing so causes the entire system to be contradictory. This is bad. Even if you dropped the requirement that the additive and multiplicative inverses be distinct, you'd have a degenerate field. This is not so bad, but a field with only one object is rather useless, not to mention uninteresting.
So if you insist on having the fabled demon of zero multiplicative inverses, you're essentially going to sacrifice any potential the system has of modeling what we think are numbers.
On a side note, with "circular logic", you can "prove" anything, since you suppose the consequent of what you're trying to prove anyway.
Now, the field axioms give us that 0*a = a*0 = 0 for all a in the field. Thus 0` * 0 * a = 0` * 0 [imath]\Rightarrow[/imath] 1 * a = 1 [imath]\Rightarrow[/imath] a = 1. This applies for any in the field, so in particular, 0=1. But by the very definition of a field, 0!=1. So it's not a matter of whether or not we'd like to define a multiplicative inverse of zero, doing so causes the entire system to be contradictory. This is bad. Even if you dropped the requirement that the additive and multiplicative inverses be distinct, you'd have a degenerate field. This is not so bad, but a field with only one object is rather useless, not to mention uninteresting.
So if you insist on having the fabled demon of zero multiplicative inverses, you're essentially going to sacrifice any potential the system has of modeling what we think are numbers.
On a side note, with "circular logic", you can "prove" anything, since you suppose the consequent of what you're trying to prove anyway.
What they (mathematicians) define as interesting depends on their particular field of study; mathematical anaylsts find pain and extreme confusion interesting, whereas geometers are interested in beauty.
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