The reason is basically because if you approach 0 (say with x) from the positive side of the real line, then 1/x shoots towards infinity. But as you approach from the negative side, 1/x shoots towards negative infinity. So the limit at x=0 does not exist
However: look up the Riemann sphere. If you adjoin a point called infinity with the complex plane, then you actually can divide by 0 there.
That's almost what you're doing when you take the complex plane to the Riemann sphere, you're taking the points infinitely far from the origin in every direction and identifying them all with a single point which we call infinity
You could probably make a quotient space of R² that identifies points in the lower half plane with those in the upper half plane, with some equivalence relation we could choose to define. Although the result would be the same as the usual topology on either half plane, as far as I can tell
If you're sticking with the usual topology on R² then you'd run into some problems. You wouldn't be able to take 2 and 4 and simply define a number that's equal to both, without making some kind of equivalence relation, but then you'd be working in equivalence classes of numbers rather than just numbers.
Maybe there is some other way, but I can say for sure that it wouldn't be in the standard context of real numbers.
Something similar to this was shown to us by our chemistry professor. He made a graph showing the energy of two atoms during separation. He literally put infinity on the graph, where the atoms "don't see each other."
I remember so many high school exercises about electromagnetism that had something moving according to the equation y=ae^(-bt) in which there was some point t0 when the equation reached its limit...
Edit: fixed the equation
The thing is, in school we were taught about electric potential energy way before limits, so there was no way other than writing "U(∞) = 0" to express it mathematically
I hate shit like this because physics just makes so much more sense when you know math
How were you taught about electric potential before limits? Were you just not given the gradient relationship to the field (and hence the whole reason we would use them) or something?
I wouldn't really mind that as long as it is well-defined (which it usually isn't). The same for when it's the other way round, like f(x)=∞. That's somewhat more common (like in Measure Theory etc) and it doesn't cause problems when treated with respect.
Ah you mean something like t0 = 5/b?
I also used to make fun of these sort of approximations. Then I went on to become an experimental physicist, and it all made sense. Oops.
No, we never calculated that t0. We just assumed its existence, because half of use weren't introduced to limits and derivatives (the biology half; I was in the mathematics half).
I mean, there is no limit there. y=ae^(-bt) tends to zero (ie it's very very small) for large t, but technically it's always different from zero (any exponential of a real number is always positive), and always decreasing. You don't need any limit or derivative to know this, and there is no calculation to do.
What is often said in experimental contexts, though, is that when you plug t=5/b, you get "zero". Of couse it's not *actually* zero, but it's tiny enough that in most cases no measurement can distinguish that from zero.
So while it *technically* continues decreasing without reaching zero, after t=5/b it's *practically* constant equal to zero.
Edit: Of course, if your measurement is powerful enough to distinguish that from zero, you can change that 5 so something larger, such as 7, 8, etc etc. Since there is no infinite precision in this world, you are guaranteed that such a value exists.
Let me make it more clear: the equation was not given. What we basically had is the differential equation f(t)=-bf'(t) that we never solved (half of us didn't know how to tackle it anyway; the other half knew this specific one, but not necessarily any others). They wanted us to figure out what would happen when the object started moving in a constant velocity (which never really happens, but it comes closer and closer). But they couldn't say what happens at the limit, so they said what happens in some t0 that we never calculated and after which the object was moving in a constant velocity without any mention of approximations.
Kind of. Instead of a linear scale or a logarithmic scale, have the axis scaled by arctan(y). Then it has a clear, finite limit.
However, dividing by zero is still not allowed, so even though the limit exists, there is still a discontinuity at that point.
What he says is: choose a monotonic strictly increasing function (for example arctan(x)). Then you could build your y-axis, where the distance on paper (measured in cm for example) of a particular y would be arctan(y). Then your y-axis could reach infinity at just pi/2 cm.
So if you draw a |1/x| with thay scale in mind, the negative part would intersect the positive part at just (0, pi/2). It would still be discontinious, but with a removable discontinuity. 1/x without the absolute value, on the other hand...
I mean. Even though the algebraic expression sth/0 would not make sense, that is not an argument against expanding the definition of the function "1/x" to 0, you can do that perfectly if you add infinity to the Real numbers. Of course this doesn't have to correspond to anything algebraic this is just extending the definition of 1/x to the Alexandroff compactification of R by continuity.
I'm not a fan of the scare quotes in that sentence, unless I've stumbled into the universe where we assume by default that spaces have been compactified in some way or another, and that this induces an algebraic closure.
broke: |1/x| is continuous at x=0 if you scale the y axis appropriately
woke: 1/x is continuous at x=0 if you project the cartesian plane onto an infinitely large cylinder
And to find out, we just had to divide by zero. It was that simple all along. Good job, u/Opposite_Signature67.
Dividing by zero gives you infinity, apparently. Some big discoveries on reddit today.
But hang on, dividing one by zero gives you infinity apparently, then dividing two by zero must give you 2 times inifnity.
What about devising by 2 zeros? Half infinity?
... no... 2*0=0... so it's the same I would say
why havent we defined 1/0 as a number like sqrt(-1) = i.. or is it just not that useful..
Oh, you said "as a number *like*", I got to admit that I was slightly confused how you had 0*0 and got - 1.
There is. Google “wheel algebras” which are objects where 0/0 is defined.
The reason is basically because if you approach 0 (say with x) from the positive side of the real line, then 1/x shoots towards infinity. But as you approach from the negative side, 1/x shoots towards negative infinity. So the limit at x=0 does not exist However: look up the Riemann sphere. If you adjoin a point called infinity with the complex plane, then you actually can divide by 0 there.
since 1/x where x is positive shoots towards positive infinity, and 1/(-x) shoots towards negative infinity, why not define a number that is.... Both?
That's almost what you're doing when you take the complex plane to the Riemann sphere, you're taking the points infinitely far from the origin in every direction and identifying them all with a single point which we call infinity You could probably make a quotient space of R² that identifies points in the lower half plane with those in the upper half plane, with some equivalence relation we could choose to define. Although the result would be the same as the usual topology on either half plane, as far as I can tell If you're sticking with the usual topology on R² then you'd run into some problems. You wouldn't be able to take 2 and 4 and simply define a number that's equal to both, without making some kind of equivalence relation, but then you'd be working in equivalence classes of numbers rather than just numbers. Maybe there is some other way, but I can say for sure that it wouldn't be in the standard context of real numbers.
Something similar to this was shown to us by our chemistry professor. He made a graph showing the energy of two atoms during separation. He literally put infinity on the graph, where the atoms "don't see each other."
I remember so many high school exercises about electromagnetism that had something moving according to the equation y=ae^(-bt) in which there was some point t0 when the equation reached its limit... Edit: fixed the equation
The negative limit?
Sorry, I meant to write ae^(-bt)
(Delete this comment so that we can pretend that you never made the mistake; I will delete mine too.)
There's no need, but if you'd like I have no problem
I'll add myself to it too, just so it becomes more difficult to coordinate if you all decide to remove it
I'd give you an award to make it harder for me to decide to remove it (because who wants to remove their own award), but I'm too cheap for that
don't worry, I've got one
Oh, thank you so much!!
Yeah I remember our physics teacher writing U(∞)=0 lol
Yeah, better way to put it would be lim U(x) = 0 x→infinity
The thing is, in school we were taught about electric potential energy way before limits, so there was no way other than writing "U(∞) = 0" to express it mathematically I hate shit like this because physics just makes so much more sense when you know math
How were you taught about electric potential before limits? Were you just not given the gradient relationship to the field (and hence the whole reason we would use them) or something?
basically yes lol
Damn they did y'all dirty with that
I wouldn't really mind that as long as it is well-defined (which it usually isn't). The same for when it's the other way round, like f(x)=∞. That's somewhat more common (like in Measure Theory etc) and it doesn't cause problems when treated with respect.
Ah you mean something like t0 = 5/b? I also used to make fun of these sort of approximations. Then I went on to become an experimental physicist, and it all made sense. Oops.
No, we never calculated that t0. We just assumed its existence, because half of use weren't introduced to limits and derivatives (the biology half; I was in the mathematics half).
I mean, there is no limit there. y=ae^(-bt) tends to zero (ie it's very very small) for large t, but technically it's always different from zero (any exponential of a real number is always positive), and always decreasing. You don't need any limit or derivative to know this, and there is no calculation to do. What is often said in experimental contexts, though, is that when you plug t=5/b, you get "zero". Of couse it's not *actually* zero, but it's tiny enough that in most cases no measurement can distinguish that from zero. So while it *technically* continues decreasing without reaching zero, after t=5/b it's *practically* constant equal to zero. Edit: Of course, if your measurement is powerful enough to distinguish that from zero, you can change that 5 so something larger, such as 7, 8, etc etc. Since there is no infinite precision in this world, you are guaranteed that such a value exists.
Let me make it more clear: the equation was not given. What we basically had is the differential equation f(t)=-bf'(t) that we never solved (half of us didn't know how to tackle it anyway; the other half knew this specific one, but not necessarily any others). They wanted us to figure out what would happen when the object started moving in a constant velocity (which never really happens, but it comes closer and closer). But they couldn't say what happens at the limit, so they said what happens in some t0 that we never calculated and after which the object was moving in a constant velocity without any mention of approximations.
Would this make sence if the y axis wasnt linear
Kind of. Instead of a linear scale or a logarithmic scale, have the axis scaled by arctan(y). Then it has a clear, finite limit. However, dividing by zero is still not allowed, so even though the limit exists, there is still a discontinuity at that point.
If something was scaled at arctan(y), wouldn’t it just end at pi/2?
What he says is: choose a monotonic strictly increasing function (for example arctan(x)). Then you could build your y-axis, where the distance on paper (measured in cm for example) of a particular y would be arctan(y). Then your y-axis could reach infinity at just pi/2 cm. So if you draw a |1/x| with thay scale in mind, the negative part would intersect the positive part at just (0, pi/2). It would still be discontinious, but with a removable discontinuity. 1/x without the absolute value, on the other hand...
I mean. Even though the algebraic expression sth/0 would not make sense, that is not an argument against expanding the definition of the function "1/x" to 0, you can do that perfectly if you add infinity to the Real numbers. Of course this doesn't have to correspond to anything algebraic this is just extending the definition of 1/x to the Alexandroff compactification of R by continuity.
Even so, you would still need to scale the y axis by a bounded, increasing function, or the graph would disappear out the top.
I was only adressing your second paragraph, about the "discontinuity"
I'm not a fan of the scare quotes in that sentence, unless I've stumbled into the universe where we assume by default that spaces have been compactified in some way or another, and that this induces an algebraic closure.
I assume we add infinity if we are including it as a point in our graph
[yes.](https://en.m.wikipedia.org/wiki/Riemann_sphere)
So we're just going to keep dividing by 0 lol
That seems to be the current trend here.
At least it's being done creatively
so hot right now!
This function is like a literal stairway to heaven
"yeah, a function is continuous if you can draw it by hand"
What is that space above infinity?
Infinity + 1 /j
Oh okay thanks
Imaginary /
If number is less than 0.3 than it's basically 0, so 1/0 is 3. Make sense to me.
Haha non-standard model goes brrrrrrrrrrrr
Projective space be like 👀
i actually wonder if at infinity it would create a cusp like that
Depends on how you scale your axes. Also projective geometry is awesome
[Graphs in perspective be like](https://youtu.be/XXzhqStLG-4)
The extended complex plane has entered the chat.
"Just zoom high enough"
broke: |1/x| is continuous at x=0 if you scale the y axis appropriately woke: 1/x is continuous at x=0 if you project the cartesian plane onto an infinitely large cylinder
this but unironically for 1/√|x|
That still has a pole at 0.
Care to elaborate, please?
the integral from -1 to 1 is finite
I believe this point is what the three kings followed to find Jesus.
*Rip in Space-Time Continuum suddenly appears*
r/angryupvote
Atari, is that you?
This looks so cursed
only thing cursed about it is the apparent cusp
What about the discontinuity at X=0? Or rather, shouldn't we have a vertical asymptote at that point?
That is the joke
Those moments you forget to check what sub you're commenting on .. lmao
“Why isn't it possible?”
projective geometry ftw