math q


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Post Monday, 22nd August 2016, 18:30

Re: math q

dowan wrote: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.

I hope you are not serious. You can show real numbers on complex number plane.

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Post Monday, 22nd August 2016, 19:06

Re: math q

Derp, nevermind, it's been way too long since math class for me.

Still, if one can create a whole new number plane for i, why is it so hard to accept infinity. It's already on both number planes anyway, it's what those arrows are pointing toward!

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Post Monday, 22nd August 2016, 21:39

Re: math q

Where is it on those planes, exactly?

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Post Monday, 22nd August 2016, 22:05

Re: math q

Well, about infinity of the way down the line, I would think. Halfway to 2 infinity. About 1% of the way to 100 infinity.

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Post Monday, 22nd August 2016, 22:19

Re: math q

i is a number. it's part of the set of Complex Numbers, which has its own robust group of rigorously proven theorems, distinct from but somewhat similar to the set of all Real Numbers.

infinity is not a number. there is not a good analogy between inventing a value for the square root of -1 (which after it's invented, plays by all kinds of nice rules) and pretending infinity is a number** -- which generally doesn't play by nice rules.

**as Arrhythmia was explaining earlier, there are situations where we can pretend infinity is a number and get useful results, but these do not occur when working in the Real Numbers or the Complex Numbers

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Post Monday, 22nd August 2016, 22:59

Re: math q

goofygumdrops wrote:which generally doesn't play by nice rules.


As an example of some of these nice rules we lose, in the specific situation I gave, we lose cancellation laws and commutativity. Namely, if (a + b = c + b), we can't say that (a = c) like we can in R, and in general, (a + b) may not be (b + a). More specifically, 2 + ω = 1 + ω, but 2 != 1, and 1 + ω != ω + 1. Since these are absolutely fundamental results that pretty much all of analysis and calculus depend upon, we can't adjoin infinity to R in a manner similar to what we did for the ordinals and keep the results we like to have.

goodcoolguy wrote: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.


Fair enough. I'd still like to see an existence theorem for these numbers.

e: Really, I think the biggest problem is that Dowan doesn't actually understand what a decimal is.
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Post Monday, 22nd August 2016, 23:11

Re: math q

dowan wrote: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.


A "Grand Theory of Everything*" in mathematics is impossible, and has been proven to be impossible, by Goedel.

*An axiomatic system where every statement is provably true or provably false, and no statement is both true and false.
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Post Monday, 22nd August 2016, 23:21

Re: math q

Arrhythmia wrote:Really, I think the biggest problem is that Dowan doesn't actually understand what a decimal is.

Well, you don't need decimals if you live in the dungeon
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Post Tuesday, 23rd August 2016, 00:42

Re: math q

In the spirit of this thread:

Why isn't 1 prime? After all, its divisors are 1 and itself
1 = 1+(-1+1)+(-1+1)+(-1+1)+... = (1-1)+(1-1)+(1-1)+... = 0
If infinity isn't real, how does pi go on forever?
Does Banach-Tarski violate conservation of mass?
Does Godel's incompleteness theorem invalidate all of math?
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Post Tuesday, 23rd August 2016, 00:48

Re: math q

Category wrote:Why isn't 1 prime?

https://www.youtube.com/watch?v=IQofiPqhJ_s
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Post Tuesday, 23rd August 2016, 00:52

Re: math q

what do we know about extra dim... dimen...for fucks sake i`m having trouble saying it again, what do we know about 4d?
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Post Tuesday, 23rd August 2016, 00:54

Re: math q

twelwe wrote:what do we know about extra dim... dimen...for fucks sake i`m having trouble saying it again, what do we know about 4d?


It's the worst dimension, because you have enough space that fucky shit can happen, but not enough space that you can just move around the fucky shit.
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Post Tuesday, 23rd August 2016, 00:58

Re: math q

Category wrote:1 = 1+(-1+1)+(-1+1)+(-1+1)+... = (1-1)+(1-1)+(1-1)+... = 0

What does equality mean for an infinite sum?

If infinity isn't real, how does pi go on forever?

Different ideas; there's no real number which is infinity, but there is certainly the concept of "things which continue without end" which one could say to be infinite and also say to exist.

Does Banach-Tarski violate conservation of mass?

I don't know; does reality obey ZFC?

Does Godel's incompleteness theorem invalidate all of math?

No. It just means we'll never be able to prove everything, not that everything we've proven is wrong.

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Post Tuesday, 23rd August 2016, 01:47

Re: math q

Arrhythmia wrote:I don't know; does reality obey ZFC?

Aha I've got you now! Either reality obeys ZFC, in which case we have a contradiction with conservation of mass, or math isn't real! Now if you'll excuse me I have to go put an oscilloscope in this Hello Kitty TV case.

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Post Tuesday, 23rd August 2016, 03:02

Re: math q

phloomp wrote: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".


This is my number one advice for students when they're struggling to solve a math problem, at all levels. Before you can solve the problem you have to know what every word and symbol in it means. Not in a vague, everyday sense, but in the precise sense they're being used in the question.

dowan wrote: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.


Mathematicians don't have a problem with division by zero being undefined in arithmetic. If anything the current situation is ideal, because when division by zero occurs in a formula that applies to a physical situation it's a sign that something has broken. If you hacked your arithmetic rules and let people divide by zero to get some spurious 'infinity' value it might fool physicists into thinking they're doing legitimate calculations.

As for 0.99999. Let S = 0.999..., then 10S = 9.999..., so 10S -S = 9S = 9.999... - 0.999... = 9, and so S = 1. You could, if you wanted, define 0.999 to be equal to something else, but the rules governing the arithmetic operations would then be broken, and you'd have to special case them to avoid being able to prove that S = 1 in the way I demonstrated. It's not clear what benefit anyone would get by doing this. 0.999.. isn't a number, it's a symbolic way of writing a number. It turns out that if you want this symbol to obey the same manipulation rules as the terminating decimals then the number this symbol represents must be the same as the number commonly represented by 1.

As for infinitesimal values, it is possible to define them in a way that is consistent with the conventional arithmetic of real numbers. However, the existing numbers and the decimals representing them keep the same values and interpretations as they do in conventional models. They have to, as the whole point of these non-standard constructions is that they obey the same (first order) rules as the conventional numbers.
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Post Tuesday, 23rd August 2016, 03:04

Re: math q

it was explained to me that .99= 1 because of rounding up
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Post Tuesday, 23rd August 2016, 03:10

Re: math q

twelwe wrote:it was explained to me that .99= 1 because of rounding up


it's like that, but the rounding gets way harder and harder until there's no point to doing it anymore
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Post Tuesday, 23rd August 2016, 03:20

Re: math q

how can mirrors be real if our eyes aren't real

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Post Tuesday, 23rd August 2016, 03:42

Re: math q

Why do mirrors flip us left/right but not up/down?
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Post Tuesday, 23rd August 2016, 03:46

Re: math q

Hands wrote:Why do mirrors flip us left/right but not up/down?


They don't, they spin us around.
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Post Tuesday, 23rd August 2016, 12:47

Re: math q

Arrhythmia wrote:
goofygumdrops wrote:which generally doesn't play by nice rules.


As an example of some of these nice rules we lose, in the specific situation I gave, we lose cancellation laws and commutativity. Namely, if (a + b = c + b), we can't say that (a = c) like we can in R, and in general, (a + b) may not be (b + a). More specifically, 2 + ω = 1 + ω, but 2 != 1, and 1 + ω != ω + 1. Since these are absolutely fundamental results that pretty much all of analysis and calculus depend upon, we can't adjoin infinity to R in a manner similar to what we did for the ordinals and keep the results we like to have.

goodcoolguy wrote: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.


Fair enough. I'd still like to see an existence theorem for these numbers.

e: Really, I think the biggest problem is that Dowan doesn't actually understand what a decimal is.


Hey, I know what a decimal is! It's what goes before com in a website!

Yes, I agree my idea doesn't automatically make sense, or even perhaps make sense at all. Nevertheless, it is something I will continue to think about.
Because a line has infinite points on it, and we can pick one point on the line, that is, one out of infinite points. If we pick infinite points on the line, have we picked the entire line? Or are there still an infinite number of points left of the line to select? The odds of any given point being picked would be considered 0(1 out of infinity is as small a chance as it gets), yet a point will be picked, so it is clearly not exactly 0.

There just seems to be a relationship between 0, infinity, and 1/infinity. People have been trying to make 1/0 fit into that as well, but it doesn't quite seem to work. Yet at least recently we were considered to exist in a universe extending in all directions infinitely(due to homogenous background EM radiation in all directions), meaning an understanding of it would be essential to an understanding of our reality.
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Post Tuesday, 23rd August 2016, 12:52

Re: math q

I don't think anyone in a position to have a real opinion on the matter ever thought the universe is infinite.
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Post Tuesday, 23rd August 2016, 14:19

Re: math q

I sure hope not, because it would be problematic in many ways. But that's what I've absorbed from pop science, and the various physics texts I could get my hands on as a kid 10-20 years ago, on top of my occasional curiosity leading me to read some random wikipedia articles.

I realize an infinite universe doesn't really make sense if everything originated at the big bang and spread outward at the speed of light. Logically over a non-infinite amount of time that spread can't very well have been infinite...

However, I've read multiple times about the problem with seemingly homogeneous radiation in all directions, interpreted to mean either the universe is infinite, or we're close enough to the center for it to look the same in all directions, which seems quite farfetched. Maybe that's just outdated 80s stuff that's been disproven since then...

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Post Tuesday, 23rd August 2016, 14:26

Re: math q

I think in general trying to cobble together a real understanding of science and math from pop-science articles is a recipe for confusion. There's a reason people study full time for like 10 years before they get to a point where they can engage with the current level of knowledge on even a small part of it.

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Post Tuesday, 23rd August 2016, 14:29

Re: math q

Richard Feynman wrote:I think I can safely say that nobody understands quantum mechanics.


Currently accepted theory is that our universe is infinite and there can be infinite number of such universes, each of them was born as quantum effect.
Don't even try to imagine that, our brain can't.

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Post Tuesday, 23rd August 2016, 14:51

Re: math q

Well at least I'm not the only one under that impression sandman. I guess the take away is that nobody here knows what the hell they're talking about, myself least of all.

At least once my grand unifying theory of everything is complete and humans gain a complete understanding of everything in existence, creating a perfect utopia of awesomeness, you can all say you remember when I told you about it on the dungeon crawl forums. :P

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Post Tuesday, 23rd August 2016, 16:59

Re: math q

dowan wrote:However, I've read multiple times about the problem with seemingly homogeneous radiation in all directions, interpreted to mean either the universe is infinite, or we're close enough to the center for it to look the same in all directions, which seems quite farfetched. Maybe that's just outdated 80s stuff that's been disproven since then...

If you mean the cosmic microwave background (CMB), you should probably update your information on the subject...
DCSS: 77: ...SpWz{OgGlTrMo}{CeWnMfBeMiSk}DrEE{GrFiFoGl}DgEnFeNe{OpGlHuSu}DDAr
HaCKSpAEGrTmDgFEDsCjGhMoHuVM{HaAMBaEn}{HuMoHOWn}DsWzDDHu{DgWnGnBe}FeIE
{MiEnMfCj}SpNeBaEEGrFE{HaAKTrCK}DsFESpHu{FoArNaBe}FeEE{HOIEMiAE}GrGlHuWrGnWr
Bloat: 4: RaRoPrPh{GuStGnCa

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Post Wednesday, 24th August 2016, 12:46

Re: math q

Well, clearly much of my information is out of date. Eh, everyone knows science is cyclical anyway. I'll just wait until everything revolves around the earth again, then I'll be way ahead of the curve!

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Post Wednesday, 24th August 2016, 15:42

Re: math q

goodcoolguy wrote:I don't think anyone in a position to have a real opinion on the matter ever thought the universe is infinite.

Hmm, and I thought when we think about the universe these days we think it's expanding, and flat, and has an infinite space available to expand into. I guess you didn't say in which dimension you deny the infinity of the universe.

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Post Wednesday, 24th August 2016, 16:16

Re: math q

Sandman25 wrote:
Richard Feynman wrote:I think I can safely say that nobody understands quantum mechanics.


Currently accepted theory is that our universe is infinite and there can be infinite number of such universes, each of them was born as quantum effect.
Don't even try to imagine that, our brain can't.


Back in the days one theory was that antimatter particles travelling backwards in time colliding and creating normal particles (which in turn perhaps in the future will collide and create the aforementioned antimatter particles) could "explain" the birth of an universe. Is that the quantum effects you refer to?

I believe Frederick Pohl's Heechee science-fiction series used that idea, but I could be wrong...

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Post Wednesday, 24th August 2016, 18:54

Re: math q

Shard1697 wrote:how can mirrors be real if our eyes aren't real


https://www.youtube.com/watch?v=AjNa7nHFju4
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Post Thursday, 25th August 2016, 00:26

Re: math q

what kinds of number are there that science can`t explain in outer space?
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Post Thursday, 25th August 2016, 00:34

Re: math q

twelwe wrote:what kinds of number are there that science can`t explain in outer space?
69

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Post Thursday, 25th August 2016, 00:47

Re: math q

twelwe wrote:what kinds of number are there that science can`t explain in outer space?


Lots. goodcoolguy referenced it earlier, but most of the real numbers can't be described in a finite string of information.

I'd give you an example, but I can't! Sorry.

e: these can't be explained in space or on earth, btw.
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Post Thursday, 25th August 2016, 02:42

Re: math q

Gongclonker wrote: 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.)


Except things that originally seemed like mathematical curiosities (imaginary and complex numbers, for the easiest example), actually have real effects in terms of measuring the real world and creating objects that we use within it. There may be philosophical value to be had in a debate about the platonic reality of mathematics, but at some point you have to accept math as 'real' if you want to deal with things as they are in practical terms.

Or you can just vote Trump.

Edit: Someone beat me to the Trump joke, but if I spent time actually reading tavern threads then, well, you know what they say....

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Post Thursday, 25th August 2016, 03:01

Re: math q

prozacelf wrote:Edit: Someone beat me to the Trump joke


I think Trump did that.
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Post Thursday, 25th August 2016, 03:25

Re: math q

Arrhythmia wrote:
twelwe wrote:what kinds of number are there that science can`t explain in outer space?


Lots. goodcoolguy referenced it earlier, but most of the real numbers can't be described in a finite string of information.

I'd give you an example, but I can't! Sorry.

e: these can't be explained in space or on earth, btw.

chaitin constants, which is defined to be the proportion of strings that are programs that halt (for a giving programming language)

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Post Thursday, 25th August 2016, 14:04

Re: math q

duvessa wrote:
twelwe wrote:what kinds of number are there that science can`t explain in outer space?
69


Nice.

4Hooves2Appendages wrote:
goodcoolguy wrote:I don't think anyone in a position to have a real opinion on the matter ever thought the universe is infinite.

Hmm, and I thought when we think about the universe these days we think it's expanding, and flat, and has an infinite space available to expand into. I guess you didn't say in which dimension you deny the infinity of the universe.


I heard a number of years ago that some analysis of background radiation suggested that the universe has slightly positive sectional curvature. I have no idea if this finding was ever published. If this is true everywhere, not just near here, this means that either the universe either has edges you can reach in a finite amount of time (and/)or there is a finite collection of points such that every point in the universe is within some fixed distance to one of these points. In any case, being flat does not at all mean that it's infinite. A torus is also (potentially) flat.
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Post Thursday, 25th August 2016, 15:24

Re: math q

Of course being flat doesn't mean it's infinite. It just means that as it expands it doesn't in a weird way fold back onto itself. Given that the universe is expanding, and that many objects are moving away from us at relatively faster than the speed of light, it is actually not possible for us to reach the (current) edge, based on what we know now. We'd never catch up.

There really aren't any fixed distances as soon as we start thinking about stuff outside of our own solar system. Everything is moving, and most of it away from us (or we from it depending on your perspective).

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Post Thursday, 25th August 2016, 16:56

Re: math q

Isnt the universe 4 to 12d?
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Post Friday, 26th August 2016, 01:14

Re: math q

what would happen if poof, the moon fanished?
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Post Friday, 26th August 2016, 01:30

Re: math q

twelwe wrote:what would happen if poof, the moon fanished?

I think I read a anime about this once, have you considered asking the Japanese?

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Post Friday, 26th August 2016, 01:44

Re: math q

we'd have to fake mars landings instead

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Post Friday, 26th August 2016, 03:01

Re: math q

4Hooves2Appendages wrote:Of course being flat doesn't mean it's infinite. It just means that as it expands it doesn't in a weird way fold back onto itself.


Actually, it doesn't.

420Horse_ebooks wrote:There really aren't any fixed distances as soon as we start thinking about stuff outside of our own solar system. Everything is moving, and most of it away from us (or we from it depending on your perspective).


Actually, when people talk about this kind of thing, they're talking about a certain sort of slice and you can talk about distances.
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Post Friday, 26th August 2016, 09:05

Re: math q

goodcoolguy wrote:
4Hooves2Appendages wrote:Of course being flat doesn't mean it's infinite. It just means that as it expands it doesn't in a weird way fold back onto itself.

Actually, it doesn't.

OK, you've convinced me.

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Post Friday, 26th August 2016, 09:57

Re: math q

So a fair amount of this thread is over my head, but I wanted to respond to this post by dowan:

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)


Let me cut you off here. You already (correctly) explained that .9 repeating is exactly equal to 1. The word EXACTLY there is important; they are no different. You show 1 - 1 = 0, but then try to claim 1 - .9 repeating is not 0. It is zero. There is no .0 repeating followed by 1; repeating means "repeats forever". There can be nothing after it. A number that is .00000(lots more zeros)00001 does exist, but it has no relation to any 1/9 or "repeating" number.

Going way back to the start of the thread about why you can't divide by zero, I usually explain division (to small children or internet commentators, two groups which share a lot of similarities) as splitting something into groups. You have 10 M+M's (everyone loves candy), and have two children. They each get 5 M+M's. If there's only one person they keep all 10, if there's 20 people you're going to have to cut those M+M's in half if you want it to work. How can you split 10 M+M's into zero groups? You have the candy, the candy doesn't suddenly cease to exist. Eating it makes it a group in your stomach. Even if you sweep it all under the rug, it's still there in a group under the rug. There's no way to make it not a group and not more than one group.

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Post Friday, 26th August 2016, 14:46

Re: math q

tasonir wrote:Going way back to the start of the thread about why you can't divide by zero, I usually explain division (to small children or internet commentators, two groups which share a lot of similarities) as splitting something into groups. You have 10 M+M's (everyone loves candy), and have two children. They each get 5 M+M's. If there's only one person they keep all 10, if there's 20 people you're going to have to cut those M+M's in half if you want it to work. How can you split 10 M+M's into zero groups? You have the candy, the candy doesn't suddenly cease to exist. Eating it makes it a group in your stomach. Even if you sweep it all under the rug, it's still there in a group under the rug. There's no way to make it not a group and not more than one group.


The problem with this kind of argument is that according to 'common sense' reasoning about dividing up collections things like division by irrational numbers don't really make sense either. There is some mathematical sleight of hand going on that allows the 'common sense' basis for division to be extended to handle numbers that don't represent quantities. Why we can't do a similar thing for division by zero is a reasonable question. It's not a deep mystery of the universe, but understanding why, when and how to extend a basic concrete concept to a more abstract general one is an important part of developing mathematical sophistication. Trying to understand why extending arithmetic to irrational numbers is a 'good' generalization, but extending to include division by zero would be a 'bad' one is part of that.

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Post Friday, 26th August 2016, 16:36

Re: math q

Irrational numbers can represent quantities just fine though. You could have a jug with pi liters of water (or maybe some other less-popular irrational number that doesn't have its own symbol), mix in 5 grams of salt, then the amount of salt per liter is 5/pi grams.

e: although I guess you could argue that the water is made of discrete molecules so it's not really an irrational quantity, just a close approximation of one.

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Post Friday, 26th August 2016, 17:16

Re: math q

genericpseudonym wrote:Irrational numbers can represent quantities just fine though. You could have a jug with pi liters of water (or maybe some other less-popular irrational number that doesn't have its own symbol), mix in 5 grams of salt, then the amount of salt per liter is 5/pi grams.

e: although I guess you could argue that the water is made of discrete molecules so it's not really an irrational quantity, just a close approximation of one.


I think there's a significant conceptual step between dividing up sweets and dissolving things in hypothetical irrational volumes. In any case, we could make the same argument using something like irrational exponents. The point is, it's ok to do some things in arithmetic that don't make intuitive physical sense, so whatever the reason we don't define division by zero, it's not because it doesn't have an easy physical interpretation.

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Post Friday, 26th August 2016, 17:20

Re: math q

Shtopit wrote:
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.
twelwe wrote:It's like Blink, but you end up drowning.
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