Pandemonium Purger

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If it doesn't fit anywhere else, it belongs here. Also, come here if you just need to get hammered.

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Pandemonium Purger

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Tartarus Sorceror

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What's the largest prime number you can find in a 15 orb run?

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Halls Hopper

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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?

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?

And you think you're capable of finding the ORB? I had it, you know. Then I gave it away so I could steal it AGAIN.

Pandemonium Purger

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not sure what you mean by succ.. whatever it is you said there. couple big ones there.

Tartarus Sorceror

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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?

Let's be a little clear here, what do you mean by "the operation" in "when you can perform the operation an unlimited number of times in succession?". Do you mean division by an arbitrary number?

Last edited by Arrhythmia on Sunday, 21st August 2016, 22:49, edited 1 time in total.

take it easy

Halls Hopper

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I mean the operation of division itself. So, "yeah".

And you think you're capable of finding the ORB? I had it, you know. Then I gave it away so I could steal it AGAIN.

Tartarus Sorceror

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Gongclonker wrote:I mean the operation of division itself. So, "yeah".

Well, the first reason is, is that I can also divide an arbitrary number of times by negative numbers. If I divide like this, towards 0, one should expect to get negative infinity as an answer. So, when dividing by 0, I could get two answers, infinity or negative infinity. But you can't multiply or divide numbers and get two different solutions, so that's off the table.

However, if you were to do something sneakier and simply say, carte blanche, that "x divided by 0 is infinity", then you've got something more compelling, that a mathematician could actually study and do work in, but this environment is unsuitable for more pedestrian applications.

take it easy

Halls Hopper

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So it's again a matter of "our system works like this, we have decided thus". :p

Thanks for trying, though.

Thanks for trying, though.

And you think you're capable of finding the ORB? I had it, you know. Then I gave it away so I could steal it AGAIN.

Tartarus Sorceror

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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.

take it easy

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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?

Operations need to be reversible. If 5/0 = infinite, then infinite multiplied by 0 should be 5; but then infinite multiplied by 0 should give 5. However, by this same standard, 3/0 should also give infinite. Then infinite multiplied by 0 should give 3. This is impossible, because the same operation should give the same result. So infinite is the wrong result. If we arbitrarily decide that 1/0 = infinite and that 2/0 = 2infinite, we would apparently solve the previous problem, but we would also find ourselves contemplating multiplying infinity, which simply doesn't make sense: 2 infinite isn't larger or different than 1 infinite.

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Tartarus Sorceror

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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.

take it easy

Halls Hopper

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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.

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.)

Tartarus Sorceror

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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.)

Well, as I said before, the system's we're working in could be said to reflect different aspects of reality. For example, the real numbers, where you can't divide by 0, model distance very well. Multiplication represents taking a number of steps of certain length, addition represents stepping a certain length, etc. etc. Furthermore, over here, there really isn't a good idea of what dividing by 0 would represent. You take a distance and cut it into 0 pieces? Fooey, nonsense, poppycock, etc.

But let's shift over a moment to a place where we CAN divide by zero, for example, a real projective plane. Real projective planes are really good at describing pictures. For example, in the regular Euclidean plane, two parallel lines don't intersect. But in pictures of parallel lines, for example this one, parallel lines do intersect. And dividing by 0 in a real projective plane is a perfectly meaningful thing, it represents travelling from where we are to the "infinite point".

So, really, it's less that mathematics doesn't represent reality, but it doesn't represent it in toto. You need to decide what things you want represented.

e: Even "mathematical infinity" changes based on where you are. The concept of infinity behaves differently in the ordinals than it does in the cardinals or in calculus or...

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Tartarus Sorceror

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Or in the reverse, there are situations where normally accepted as good and fine operations suddenly aren't. For example, dividing by 7 is something no one usually has problems with, but it's impossible in the integers mod 14.

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Halls Hopper

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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

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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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

That's basically all math is. Axiom is the fancy way of saying "stuff that we think seems reasonable".

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Dungeon Master

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you can divide by zero, you can define what x/0 is for any x, the problem is that the actual act of assigning a value fucks up basic tenets of mathematics extremely hard. for example in current mathematics whenever you do something like this:

you implicitly assume a =/= 0, otherwise you can prove that 1 = 2. no matter how you define x/0, you will force at least one of + or x to have this property so now whenever you do math you'll have something like

so now you have a bunch of extra exceptions to worry about, but you didn't fix the original problem outlined above so you gain nothing while having to do a bunch of pointless paperwork that wouldn't exist if you didn't define x/0 in the first place.

some professor tried to do exactly this and it turns out, it was worthless!!!

if ab = ac, then b = c

you implicitly assume a =/= 0, otherwise you can prove that 1 = 2. no matter how you define x/0, you will force at least one of + or x to have this property so now whenever you do math you'll have something like

if a + b = a + c, then b = c except if a is equal to some weird value

so now you have a bunch of extra exceptions to worry about, but you didn't fix the original problem outlined above so you gain nothing while having to do a bunch of pointless paperwork that wouldn't exist if you didn't define x/0 in the first place.

some professor tried to do exactly this and it turns out, it was worthless!!!

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.

There are several reasons that defining division by 0 for real numbers doesn't work. Arrhythmia and Shtopit gave two that are basically about the same thing: in real arithmetic the division and multiplication operations are defined as functions that are inverses of each other, and 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).

But perhaps you're the type that says things like "I'm an engineer, I'm practical, why can't mathematicians just be practical instead of worrying about this imaginary theoretical stuff *honks clown horn and drives clown car into tree* excuse me while I put an oscilloscope in this Hello Kitty TV case".

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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.

I know. But treating the real numbers like the only, or at least the canonical mathematical system leads to a lot of the confusions we've seen in this very thread.

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Meta-observation:. I've TAed a bunch of math, and my biggest takeaway is that when you're confused, it is very important to ask yourself "what do these words even mean?" The reason people don't do this, and aren't taught to do this, is that the answer is often very difficult! Real numbers are quite intuitive, but what does "real number" even mean? If you haven't seen completions of metric spaces (or Dedekind cuts) you might enjoy to read about them, if only to see how much work needs to be done. Same goes for "infinity" or even "divide".

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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)...

But this IS practical and makes perfect sense. With that requirement/definition, I begin to see reasons for others' treatment of infinity. It's just that you're the first to mention it this way.

It seems the predictability of the system is the basis of 'usefulness', and with that in mind, I can dig it. (But now I'm playing devil's advocate. I tend not to like practical notions in general. Gimme useless mongrel-stuff with personality any day.)

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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.

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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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.

What bugs me is that opponents of n/0=∞ often extrapolate from math canon as if it represents things that are external to it, and use this as the entire basis for their reasoning. I wanted to know why, and why they're so stuck on the issue that they refused to discuss it beyond "It's not useful" - but I found some insight in this thread.

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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I think the confusion stems from thinking your idea automatically makes sense. You don't say what infinity is, what infinities there are, or which infinity you're talking about. In fact, as Arrhy points out, there are notions of infinity where one talks about division by 0 giving 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)

Well your question depends pretty heavily on the preconception that infinity is a specific number that things can be "equal" to.

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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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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".

Unless the question is "why did we chose these axioms", in which case the answer is "because they let us prove the things that we think should be provable".

Anyway, as a general reply to Gongclonker, a significant part of the story of mathematics in the first half of the 20th century concerns how mathematicians moved away from the idea that mathematics represents physical reality in a strict sense. Mathematicians collectively are very aware of the limitations of formal systems, and there are a variety of rigorously proved results about them.

Since infinity does not, as far as we are aware, occur in nature, how you deal with infinity is entirely dependent on what formal properties you want it to have, which will depend on the context in which it appears. You can define an extra point called infinity in your arithmetic system, and you can set division by zero to equal infinity. However, if you do this your definition of multiplication will have to be special cased for infinity to cover the fact that multiplying infinity by zero will now give results inconsistent with how division and multiplication usually work. Not being compelled either way by reality, mathematicians prefer to keep division by zero undefined rather than define it at the cost of a bunch of special cases for the arithmetic operations.

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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.

Okay. You'll probably need the basics of set theory for this one, if you don't know about that just say so and I'll write up another post on it.

You've probably heard, at some point, that we can build almost* all of mathematics using set theory. I'm going to start this post by actually doing some of the groundwork for that, because the processes we use at the start of this construction are also the processes used to construct the ordinals.

So, the first step in this challenge, to build math from set theory, is going to be to decide what set 0 is. The best candidate for this, is the unique set with 0 elements, the empty set, {}. Next up is what's 1? It would be nice if, like before, the set representing 1 also had exactly 1 element. However, unlike with 0, where there's only one set with zero elements, there are Too Many** sets with just one element. However, if we restrict our attention to numbers we've already constructed, we see that we've already constructed one set. So, we'll chose that set to be the element belonging to the set representing 1, and we get 1 = {0}.

In a similar way, we can construct every natural number, and we get

0 = {}

1 = {0}

2 = {0, 1}

3 = {0, 1, 2}

4 = {0, 1, 2, 3}

...

n + 1 = {0, 1 , 2, ..., n}

...

So, I've now constructed a set for every single natural number, which is pretty cool. One last step is to construct a set of all natural numbers, which we can*** do. This set I'm going to call ω****. Now that I've constructed ω, one could step away pretty satisfied, as now they can construct the integers, and the rationals, and the reals, and do all the math they'd normally hope to do. However, I'm not going to do that. Rather, I'm going to take the processes we did before, and just sort of continue doing them.

So, now our list of numbers looks like this

0 = {}

1 = {0}

2 = {0, 1}

3 = {0, 1, 2}

4 = {0, 1, 2, 3}

...

n + 1 = {0, 1 , 2, ..., n}

...

ω = {0, 1, 2, ...}

Well, one way to look at the addition above is that x + 1 = x \union {x], that is, we take all the elements in x, add x to this set, and call this new set x + 1. There's no reason we can't do this to ω, and we get

ω + 1 = {0, 1, 2, ..., ω}

And there's no reason why we can't stop this process either!

ω + 2 = {0, 1, 2, ..., ω, ω + 1}

ω + 3 = {0, 1, 2, ..., ω, ω + 1, ω + 2}

ω + 4 = {0, 1, 2, ..., ω, ω + 1, ω + 2, ω + 3}

...

As before, when we went from our list of natural numbers to the set of natural numbers ω, we can***** go from this list of "numbers after ω", to the set of "numbers after ω". that is, we get

2ω = {0, 1, 2, ..., ω, ω + 1, ω + 2, ω + 3, ...}

Likewise we can do similar processes to get 3ω, 4ω, 5ω, so on and so on. These numbers are the infinities that I referred to before: ω is infinity, 2ω is 2infinity, and this post is already getting really long and wordy so you'll just have to take my word****** that we can define addition in a reasonable way such that ω + 2ω = 3ω.

Parts of this post that I sort of handwaved away (such as going form the list of numbers ω + n to 2ω) can be done perfectly rigorously, but they require machinery (such as transitive sets ordered by membership or canonical well orderings) that I didn't want to develop here.

* The exceptions here generally being other axiomatic systems more powerful than ZFC (such as ZFC + GCH or V = L).

** A non-technical term, but a very technical idea.

*** We actually can't, and need to make another axiom for this. Still, this axiom is pretty much universally accepted.

**** If this didn't render, or rendered poorly, it's a lowercase omega.

***** Unlike before, we can actually do this without invoking a new axiom.

****** This isn't to say I wouldn't write another post on ordinal arithmetic if someone asked; this one's just gotten too long for that.

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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...

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...

Blades Runner

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Tartarus Sorceror

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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...

Some questions to ask yourself:

What is a decimal? 0.9 is actually 9/10, 0.09 is 9/100, and .99 is actually 9/10 + 9/100. In light of this, how could you reasonably say a decimal terminates with a number?

Why would "True 0 multiplied by infinity is still 0. But true 0 is equal to any infinitesimally small number."? What's an infinitesimally small number? Could you name one?

take it easy

Blades Runner

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But Arrhythmia, most real numbers do not have a description in terms of a finite amount of data. If he can't give an explicit description of any of these "infinitesimal numbers," this doesn't mean they don't exist.

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Ziggurat Zagger

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I am lost, are we talking about rational vs irrational numbers or what?

Feeling DCSS too easy/hard? Try it with custom difficulty.

Full Windows 64 bit version can be downloaded at https://www.dropbox.com/s/7ufdv1a2qbn6b ... 9.zip?dl=0

I play offline.

Full Windows 64 bit version can be downloaded at https://www.dropbox.com/s/7ufdv1a2qbn6b ... 9.zip?dl=0

I play offline.

Tartarus Sorceror

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An infinitesimally small number would be the difference between 0.333333333333 repeating * 3 vs 1/3*3.

In other words, 1 - 0.9999999 repeating.

It's equivalent to 0, clearly, but equally clearly it's not quite the same. The difference between those two numbers is 0.0000 repeating, followed by 1. That number is "one infinitith" or the result of 1 divided by infinity.

Let's say you have a line extending to infinity in both directions. Clearly there are infinite points on this line (2 * infinite points, if we consider + and - directions). Lets say you pick a point on this line at random. What are your odds of any given point being chosen? It's 1 out of infinity. Yet, it's not quite 0, because a point will be picked, and any given point has a chance of being picked.

No, you can't write this number, you can't represent it just like you can't write the number that i represents, nor can you write the number pi represents for that matter. The closest you can come to writing it is:

But that' not considered a valid number, real or unreal, rational or irrational. Yet it still seems to exist.

In other words, 1 - 0.9999999 repeating.

It's equivalent to 0, clearly, but equally clearly it's not quite the same. The difference between those two numbers is 0.0000 repeating, followed by 1. That number is "one infinitith" or the result of 1 divided by infinity.

Let's say you have a line extending to infinity in both directions. Clearly there are infinite points on this line (2 * infinite points, if we consider + and - directions). Lets say you pick a point on this line at random. What are your odds of any given point being chosen? It's 1 out of infinity. Yet, it's not quite 0, because a point will be picked, and any given point has a chance of being picked.

No, you can't write this number, you can't represent it just like you can't write the number that i represents, nor can you write the number pi represents for that matter. The closest you can come to writing it is:

But that' not considered a valid number, real or unreal, rational or irrational. Yet it still seems to exist.

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Tartarus Sorceror

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Well... kinda. if 3/3 = 0.999 repeating = 1 then 0 = 0.000 repeating 1.

It makes no difference at all until you try to multiply by infinity, then the difference becomes important. The problem is, how would you go about determining which value to use when you do multiply by infinity?

All that is just a roundabout way of getting to 1 infinitith, so I should probably stick with the point on a line example though, the whole 1/3 thing just causes a bunch of unnecessary confusion and makes math teachers angry.

It makes no difference at all until you try to multiply by infinity, then the difference becomes important. The problem is, how would you go about determining which value to use when you do multiply by infinity?

All that is just a roundabout way of getting to 1 infinitith, so I should probably stick with the point on a line example though, the whole 1/3 thing just causes a bunch of unnecessary confusion and makes math teachers angry.

Vaults Vanquisher

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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...

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.

Blades Runner

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Guys, the problem is you just don't know what numbers are.

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goodcoolguy wrote:Guys, the problem is you just don't know what numbers are.

I blame this game.

You shall never see my color again.

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Crypt Cleanser

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What is reality, actually. Really makes u think...

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this topic is why crawl doesn't display damage numbers

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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.

the problem isn't whether infinity is a number (it's certainly not a real number), the problem is 1/0. When mathematics runs into this, it breaks the formula. It's one of the big obstacles in the way of a Grand theory of everything.

The square root of -1 is also not a number, and you can't get it by multiplication or division (of numbers). Luckily, someone didn't let this get in their way, which is why we have i.

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You can show i on a number plane, therefore it is a number. But you can't show infinity.

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Your number plane has i as the numbers on the axis. I could make an identical plane with infinity swapped out for i and it would be the same thing. Positive infinities on the top part, negative infinities on the bottom. 0 being in the centre, counting up from 1 infinity to infinity infinity, counting down from -1 infinity to -infinity infinity.

You can't show a real number on that same axis, just as you can't show i on a real number axis.

You can't show a real number on that same axis, just as you can't show i on a real number axis.

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