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Wnderer wrote:1/-0 = -infinity and 1/+0 = +infinity. In a sense it indicates that it matters whether a point is being approached from above or below.
Chuff wrote:I write most of my letters from the bottom
Goldstein wrote:Wnderer wrote:1/-0 = -infinity and 1/+0 = +infinity. In a sense it indicates that it matters whether a point is being approached from above or below.
While it is meaningful in limit theory to talk about a value being approached, that's not what's going on here, as explained in Phlip's nice post on the previous page.
Also note that 0.999... is equal to 1, not 'slightly below', so the +0 and -0 you've defined are also equal to each other.
addams wrote:This forum has some very well educated people typing away in loops with Sourmilk. He is a lucky Sourmilk.
mike-l wrote:But that makes perfect sense even without making 89.9999.. a different number than 90... tan of both of them is 'infinity', and you can make this rigorous by working on the projective plane.
enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};
void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}
phlip wrote:But even if you did want to set up some kind of extended number system where each point had two "sides"... this wouldn't be the way to do it. For one, you can't do it to every number, only the rational numbers with terminating expansions (that is, the decadic rationals - ones where the denominator only has prime factors of 2 and 5). Non-terminating rationals and irrational numbers only have one decimal expansion. Which means that the set of points that's doubled up changes, depending on what base you're using (1/3 has one expansion in decimal, but two expansions in, say, base 12). A good number system behaves independently of its representation.
phlip wrote:But say you did try to set up a more complete version, where every number has two sides... R x {+,−}, say... so for every real number, you now have two: eg (2, +) and (2, −). But then how do you define arithmetic on that? Presumably (2,+) + (2,+) = (4,+) and (2,−) + (2,−) = (4,−), but what's (2,+) + (2,−)? If it's either of (4,+) or (4,−), then you can't subtract... same way you can't divide by 0. If (2,+) + (2,+) = (4,+) and (2,+) + (2,−) = (4,+), for example, then (4,+) − (2,+) isn't well defined, because both (2,+) and (2,−) would fit.
And on the eighth day God created Irony.
But on the ninth day Satan was all like, "Nuh uh!"
And ironically made Alanis Morrisette his minion.
math wrote:end its infinite string
enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};
void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}
And on the eighth day God created Irony.
But on the ninth day Satan was all like, "Nuh uh!"
And ironically made Alanis Morrisette his minion.
phlip wrote:Sigh...math wrote:end its infinite string
No. You fail forever. Read the thread again.
phlip wrote:Sigh...math wrote:end its infinite string
No. You fail forever. Read the thread again.
And on the eighth day God created Irony.
But on the ninth day Satan was all like, "Nuh uh!"
And ironically made Alanis Morrisette his minion.
kansasdave wrote:Of course, someone out there will say, "what if we make the separation an infinite number of zeroes followed by a 1?"
RebeccaRGB wrote:kansasdave wrote:Of course, someone out there will say, "what if we make the separation an infinite number of zeroes followed by a 1?"
Keep having zeroes forever, and they never end. There is absolutely no end to the zeroes. Then, after the zeroes end, which they don't, add a 1. No, that's silly.
lightvector wrote:Then, we can easily have infinite sequences, followed by values that come after that, followed by more infinite sequences, and more values afterwards, and so on. For instance, if we want an infinite number of zeros, followed by a 1, this is just the sequence where all terms are 0 except the \omegath term, which is 1. Of course, it might be hard to make the arithmetic have nice properties, but it *can* be done sensibly, if we are willing to step outside the real numbers.
xkcdfan wrote:lightvector wrote:Then, we can easily have infinite sequences, followed by values that come after that, followed by more infinite sequences, and more values afterwards, and so on. For instance, if we want an infinite number of zeros, followed by a 1, this is just the sequence where all terms are 0 except the \omegath term, which is 1. Of course, it might be hard to make the arithmetic have nice properties, but it *can* be done sensibly, if we are willing to step outside the real numbers.
No, that actually doesn't make sense.
Macbi wrote:xkcdfan wrote:lightvector wrote:Then, we can easily have infinite sequences, followed by values that come after that, followed by more infinite sequences, and more values afterwards, and so on. For instance, if we want an infinite number of zeros, followed by a 1, this is just the sequence where all terms are 0 except the \omegath term, which is 1. Of course, it might be hard to make the arithmetic have nice properties, but it *can* be done sensibly, if we are willing to step outside the real numbers.
No, that actually doesn't make sense.
Prove it.
t1mm01994 wrote:Macbi wrote:xkcdfan wrote:lightvector wrote:Then, we can easily have infinite sequences, followed by values that come after that, followed by more infinite sequences, and more values afterwards, and so on. For instance, if we want an infinite number of zeros, followed by a 1, this is just the sequence where all terms are 0 except the \omegath term, which is 1. Of course, it might be hard to make the arithmetic have nice properties, but it *can* be done sensibly, if we are willing to step outside the real numbers.
No, that actually doesn't make sense.
Prove it.
Only if you prove, without using the definition of 2, that 1+1=2.
enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};
void ┻━┻︵╰(ಠ_ಠ ⚠) {exit((int)⚠);}
And on the eighth day God created Irony.
But on the ninth day Satan was all like, "Nuh uh!"
And ironically made Alanis Morrisette his minion.
krucifi wrote:For all those who still don't believe the proof i hereby pose the ultimate question.
If 0.999... and 1 aren't equal then what number lies between them?
phlip wrote:the traditional addition algorithm, for instance, relies on the fact that every digit has a predecessor, for the "carry" step
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
addams wrote:This forum has some very well educated people typing away in loops with Sourmilk. He is a lucky Sourmilk.
snowyowl wrote:I still think it's convincing to just draw a line of length 1 and a line of length 0.999... and then ask people which one is which.
addams wrote:This forum has some very well educated people typing away in loops with Sourmilk. He is a lucky Sourmilk.
jestingrabbit wrote:krucifi wrote:For all those who still don't believe the proof i hereby pose the ultimate question.
If 0.999... and 1 aren't equal then what number lies between them?
Its not in the reals, but people have been coming up with some stuff that could be made to work.
And on the eighth day God created Irony.
But on the ninth day Satan was all like, "Nuh uh!"
And ironically made Alanis Morrisette his minion.
Turtlewing wrote:Oh boy a redicilous clallange, let me:
from the deffenition of addition
1+1=2
Wow, that was easy.
Seriously though, the above number system doesn't make any lese sence than say: complex numbers. However it doesn't on first glance have much usefulness outside allowing people to talk about a set of infinite series seperated by zero or more finite series as a single number, which outside making "the 1 at the end of the ininate 9's" make some form of sematic sence probably isn't overly beneficial (as evidenced by the fact that it isn't used even though it would be easy enough for a compotent mathamatician to think it up if they needed to).
xkcdfan wrote:Turtlewing wrote:Oh boy a redicilous clallange, let me:
from the deffenition of addition
1+1=2
Wow, that was easy.
Seriously though, the above number system doesn't make any lese sence than say: complex numbers. However it doesn't on first glance have much usefulness outside allowing people to talk about a set of infinite series seperated by zero or more finite series as a single number, which outside making "the 1 at the end of the ininate 9's" make some form of sematic sence probably isn't overly beneficial (as evidenced by the fact that it isn't used even though it would be easy enough for a compotent mathamatician to think it up if they needed to).
Oh boy a ridiculous challenge, let me:
from the definition of addition
1+1=2
Wow, that was easy. But wasn't actually a proof!
Seriously though, the above number system doesn't make any less sense than, say, complex numbers. However,it doesn'ton first glance it doesn't have much usefulnessoutside of allowing people to talk about a set of infinite series separated by zero or more finite series as a single number, which, outside of making "the 1 at the end of the infinite 9's" make some form of semantic sense, probably isn'toverlybeneficial (as evidenced by the fact that it isn't used, even though it would be easy enough for a competent mathematician to think it up if they needed to).
And on the eighth day God created Irony.
But on the ninth day Satan was all like, "Nuh uh!"
And ironically made Alanis Morrisette his minion.
xkcdfan wrote:Turtlewing wrote:Oh boy a redicilous clallange, let me:
from the deffenition of addition
1+1=2
Wow, that was easy.
Seriously though, the above number system doesn't make any lese sence than say: complex numbers. However it doesn't on first glance have much usefulness outside allowing people to talk about a set of infinite series seperated by zero or more finite series as a single number, which outside making "the 1 at the end of the ininate 9's" make some form of sematic sence probably isn't overly beneficial (as evidenced by the fact that it isn't used even though it would be easy enough for a compotent mathamatician to think it up if they needed to).
Oh boy a ridiculous challenge, let me:
from the definition of addition
1+1=2
Wow, that was easy. But wasn't actually a proof!
Seriously though, the above number system doesn't make any less sense than, say, complex numbers. However,it doesn'ton first glance it doesn't have much usefulnessoutside of allowing people to talk about a set of infinite series separated by zero or more finite series as a single number, which, outside of making "the 1 at the end of the infinite 9's" make some form of semantic sense, probably isn'toverlybeneficial (as evidenced by the fact that it isn't used, even though it would be easy enough for a competent mathematician to think it up if they needed to).
Turtlewing wrote:xkcdfan wrote:Turtlewing wrote:retarded comment
spell-checked version of quoted comment. fixing the spelling didn't make it any less retarded.
Thanks for the editing, as long as people like you exist I never have to do it myself
xkcdfan wrote:Turtlewing wrote:xkcdfan wrote:Turtlewing wrote:retarded comment
spell-checked version of quoted comment. fixing the spelling didn't make it any less retarded.
Thanks for the editing, as long as people like you exist I never have to do it myself
So do you like not understand what "prove" means or something
And on the eighth day God created Irony.
But on the ninth day Satan was all like, "Nuh uh!"
And ironically made Alanis Morrisette his minion.
xkcdfan wrote:Turtlewing wrote:xkcdfan wrote:Turtlewing wrote:retarded comment
spell-checked version of quoted comment. fixing the spelling didn't make it any less retarded.
Thanks for the editing, as long as people like you exist I never have to do it myself
So do you like not understand what "prove" means or something
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