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Gongclonker wrote:Why does dividing by zero not lead to infinity when you can perform the operation an unlimited number of times in succession?
I'm actually interested in whether anyone has an answer for this. I've yet to see any other than "Oh, that's not useful to us as mathematicians". But I care about the truth, not the usefulness. Any insight, twelwe?
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Gongclonker wrote:I mean the operation of division itself. So, "yeah".
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Gongclonker wrote:So it's again a matter of "our system works like this, we have decided thus". :p
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Gongclonker wrote:Why does dividing by zero not lead to infinity when you can perform the operation an unlimited number of times in succession?
I'm actually interested in whether anyone has an answer for this. I've yet to see any other than "Oh, that's not useful to us as mathematicians". But I care about the truth, not the usefulness. Any insight, twelwe?
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Shtopit wrote:which simply doesn't make sense: 2 infinite isn't larger or different than 1 infinite.
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Arrhythmia wrote:Gongclonker wrote:So it's again a matter of "our system works like this, we have decided thus". :p
That's not what I said at all. One thing you seem to be confused about, and that I'll admit mathematicians haven't been the best at explaining to the public, is the idea of there being some "canonical system of mathematics". There are perfectly good, reasonable, valid situations where one can divide by zero and get a meaningful answer; these just aren't the situations most people have in mind.
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Gongclonker wrote:Sure - infinity and mathematical infinity are different. What I don't see is how your assertions should make a person want to accept the current mathematical treatment of the issue. Mathematicians talk as if the math canon and associated knowledge represent an understanding of external reality. If they were good sports and admitted that math isn't necessarily about reality, I'd take it. But they don't. (My point stands, by the way - but serious props to you for not evading it.)
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Gongclonker wrote:After some thought, I have a partial answer to my question. n/0 would not equal ∞ in any system wherein there was no progression. So, for a finite series, operators with ∞ can loop, effectively terminating the mathematical progression. (Iterations still go on forever, though.)
Anyway, you've made some excellent points, but they still highlight that ∞ works the way it does because decided-upon matters have dictated it and not because mathematicians pursue the truth in earnest. If you want to change my view on that, maybe we should talk on IRC. :p
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if ab = ac, then b = c
if a + b = a + c, then b = c except if a is equal to some weird value
Arrhythmia wrote:Shtopit wrote:which simply doesn't make sense: 2 infinite isn't larger or different than 1 infinite.
I'd like to point out, this isn't true, as I mentioned in my previous post. For example, in the ordinals, there exists 2infinity and 1infinity and 2infinity is bigger than 1infinity, and 2infinity + 1infinty = 3infinity.
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duvessa wrote:It's pretty obvious that Gongclonker and Shtopit's posts are talking about only the real numbers and infinity (which, I must remind several people here, is not a real number).Arrhythmia wrote:Shtopit wrote:which simply doesn't make sense: 2 infinite isn't larger or different than 1 infinite.
I'd like to point out, this isn't true, as I mentioned in my previous post. For example, in the ordinals, there exists 2infinity and 1infinity and 2infinity is bigger than 1infinity, and 2infinity + 1infinty = 3infinity.
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duvessa wrote:you can't define x/0 as anything because that would require that either division or multiplication violate the definition of a function (one input always gives the same output)...
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Gongclonker wrote: Anyway, you've made some excellent points, but they still highlight that ∞ works the way it does because decided-upon matters have dictated it and not because mathematicians pursue the truth in earnest.
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Category wrote:Gongclonker wrote: Anyway, you've made some excellent points, but they still highlight that ∞ works the way it does because decided-upon matters have dictated it and not because mathematicians pursue the truth in earnest.
I think your dissatisfaction stems simply from a confusion about how math works. If you ask "why" enough times, the explanation for any mathematical fact becomes "because we decided that these axioms are true". Here, the simple answer for why dividing by zero does not give infinity in the real numbers is that division by zero is simply not a thing the rules allow. There are many responses here already about why this rule is in place. Arrhythmia then pointed out that division by zero is allowed in certain systems with different rules (systems, I may add, that arise naturally in studying "real" things).
You seem to hold the opinion that "infinity" is a concept that exists in reality, that all these mathematical systems are trying to model. That's fine. But when you ask "why doesn't n/0 = infinity", you can no longer distance yourself from mathematics, since the question only makes sense if you are in a mathematical system that supports concepts like 0, division, and infinity.
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Gongclonker wrote:It's possible to pose my original question without any particular preconceptions about infinity. I don't hold opinions, I need my hands for holding my brain. (I am a brainholder, it's like a beholder but retarded)
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BabyRage wrote:Zero is not even a number, how can you divide by it?
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Category wrote:Gongclonker wrote: Anyway, you've made some excellent points, but they still highlight that ∞ works the way it does because decided-upon matters have dictated it and not because mathematicians pursue the truth in earnest.
I think your dissatisfaction stems simply from a confusion about how math works. If you ask "why" enough times, the explanation for any mathematical fact becomes "because we decided that these axioms are true".
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Shtopit wrote:BabyRage wrote:Zero is not even a number, how can you divide by it?
Interestingly the word zero comes from cifr, which is still used in various versions in languages outside English alongside zero, but meaning "number" instead. While I have found nulla written in tables of moon phases from the middle ages, and it was written in the place of 0, and somehow null now means zero in German.
Now I am actually curious about double infinity. Care to share some info about it, maybe make some example? I have studied no mathematics in high school, so that's to keep in mind, and all my formation has been humanities based.
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dowan wrote:I've spent a fair amount of thought on this idea:
1/3 = 0.3333 repeating
2/3 = 0.6666 repeating
3/3 = 0.9999 repeating = 1
1-1 = 0
1-0.999 repeating = 0.0 repeating 1 (that is to say, infinity 0s, terminated with a 1).
0.0 repeating 1 = 0 (We learn this in math class to prevent our heads from exploding)
Here's where this well known piece of math voodoo might mean something:
1/0.0 repeating 1 would be 1 followed by infinity 0s. That represents 1 infinity pretty well.
With that logic, 1/0 = infinity! But there's still a huge problem. 5/0 should be 5 infinities, assuming my 0 is in fact = 0.0000 repeating 1. But what if this 0 doesn't terminate in a one, because someone divided 5 by infinity?
That means it's 0.000 repeating then 5. Dividing that by 0 gives 5 infinities.
The problem is, at some point you have to arbitrarily decide the 'value' of your 0. True 0 multiplied by infinity is still 0. But true 0 is equal to any infinitesimally small number. So what is (1-1)/0? Is it ERROR (that's what my calculator tells me). Is it infinity? 10 infinities?
I know infinity isn't really a number, but then, neither is i(The square root of -1), and that's very useful in mathematics. There's no number that becomes negative when you square it, but because we have formulas that want square roots of inputs, even negative ones, we just made up i to deal with it. And it works perfectly well, and gives correct answers, even though it's not real and doesn't describe anything that exists.
So by that logic, I believe we could have a way to divide and multiply by 0 without destroying the information, although I certainly haven't figured out how. Anything I come up with runs into the huge fundamental problem of how the hell do you decide the value of existing 0s not created through multiplication by 0 or by division by infinity? And even then, we need to know the value of the 0 we multiplied by, or the infinity we divided by...
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dowan wrote:I've spent a fair amount of thought on this idea:
1/3 = 0.3333 repeating
2/3 = 0.6666 repeating
3/3 = 0.9999 repeating = 1
1-1 = 0
1-0.999 repeating = 0.0 repeating 1 (that is to say, infinity 0s, terminated with a 1).
0.0 repeating 1 = 0 (We learn this in math class to prevent our heads from exploding)
Here's where this well known piece of math voodoo might mean something:
1/0.0 repeating 1 would be 1 followed by infinity 0s. That represents 1 infinity pretty well.
With that logic, 1/0 = infinity! But there's still a huge problem. 5/0 should be 5 infinities, assuming my 0 is in fact = 0.0000 repeating 1. But what if this 0 doesn't terminate in a one, because someone divided 5 by infinity?
That means it's 0.000 repeating then 5. Dividing that by 0 gives 5 infinities.
The problem is, at some point you have to arbitrarily decide the 'value' of your 0. True 0 multiplied by infinity is still 0. But true 0 is equal to any infinitesimally small number. So what is (1-1)/0? Is it ERROR (that's what my calculator tells me). Is it infinity? 10 infinities?
I know infinity isn't really a number, but then, neither is i(The square root of -1), and that's very useful in mathematics. There's no number that becomes negative when you square it, but because we have formulas that want square roots of inputs, even negative ones, we just made up i to deal with it. And it works perfectly well, and gives correct answers, even though it's not real and doesn't describe anything that exists.
So by that logic, I believe we could have a way to divide and multiply by 0 without destroying the information, although I certainly haven't figured out how. Anything I come up with runs into the huge fundamental problem of how the hell do you decide the value of existing 0s not created through multiplication by 0 or by division by infinity? And even then, we need to know the value of the 0 we multiplied by, or the infinity we divided by...
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goodcoolguy wrote:Guys, the problem is you just don't know what numbers are.
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BabyRage wrote:dowan wrote:I've spent a fair amount of thought on this idea:
1/3 = 0.3333 repeating
2/3 = 0.6666 repeating
3/3 = 0.9999 repeating = 1
1-1 = 0
1-0.999 repeating = 0.0 repeating 1 (that is to say, infinity 0s, terminated with a 1).
0.0 repeating 1 = 0 (We learn this in math class to prevent our heads from exploding)
Here's where this well known piece of math voodoo might mean something:
1/0.0 repeating 1 would be 1 followed by infinity 0s. That represents 1 infinity pretty well.
With that logic, 1/0 = infinity! But there's still a huge problem. 5/0 should be 5 infinities, assuming my 0 is in fact = 0.0000 repeating 1. But what if this 0 doesn't terminate in a one, because someone divided 5 by infinity?
That means it's 0.000 repeating then 5. Dividing that by 0 gives 5 infinities.
The problem is, at some point you have to arbitrarily decide the 'value' of your 0. True 0 multiplied by infinity is still 0. But true 0 is equal to any infinitesimally small number. So what is (1-1)/0? Is it ERROR (that's what my calculator tells me). Is it infinity? 10 infinities?
I know infinity isn't really a number, but then, neither is i(The square root of -1), and that's very useful in mathematics. There's no number that becomes negative when you square it, but because we have formulas that want square roots of inputs, even negative ones, we just made up i to deal with it. And it works perfectly well, and gives correct answers, even though it's not real and doesn't describe anything that exists.
So by that logic, I believe we could have a way to divide and multiply by 0 without destroying the information, although I certainly haven't figured out how. Anything I come up with runs into the huge fundamental problem of how the hell do you decide the value of existing 0s not created through multiplication by 0 or by division by infinity? And even then, we need to know the value of the 0 we multiplied by, or the infinity we divided by...
Maybe your problem is that you try to treat infinity like it's a number. Infinity is not a number and you can't get it by division or multiplication.
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