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[Schmaven]

I heard some discussion on the radio about the number 0 and everyone seemed to agree that dividing by 0 was impossible.  I think they're all wrong about that.

We already created imaginary numbers to define the square roots of negative numbers; 2i being the square root of -4. 

And infinities are not all equal.  Some infinities are bigger than others.  For example, as x goes to infinity, comparing: A) y = 2x^2  and B) y = 6x^22,  y is much larger in case B than in case A.  Vastly larger for any given value of x.  (don't be distracted by x=0). 

Infinity is a direction, not an actual number.  Because there are no ends to the series of numbers.  But looking at any number, there is always a value that is 5 more than that.  C) y = x;   D) y = x + 5. 

So why not compare various types of infinity?  inf_2 being equal to 2 ÷ 0 yet much smaller at every point than inf_102 being equal to 102 ÷ 0.  And inf+5 being the infinity that is greater than whatever number you can think of, by exactly 5. 

Essentially, just maintaining the same mathematical notation in any equation, but adding the notation to compare infinities when for example, dividing by 0.  Then dividing by 0 becomes possible, though admittedly, not very consequential.

Maybe, by definition, dividing by 0 is the final number of infinity.  The final number, of the endless series of numbers.  Whatever it is, you could add 5 to it.  There are some equations the resolve toward limits as x goes to infinity.  E) y = 7 - 2/x  practically equals 7 as x goes to infinity, despite never actually reaching 7.  But goes to -inf as x goes to 0.  D) y = 8 - 5/x  however descends toward -inf at a much quicker rate and y does in E.  That's just comparing the slope of the curve.  Calculus does this.  What use does talking about the infinity concept bring us that cannot already be explained by calculus? 

Infinity is the process.  The endless progression.  No end to the counting.  A loop with no end point.  There is no final number of infinity.

Is it worth thinking about dividing by 0?  It seems like there could be some utility to doing so, but not at the moment.  Maybe some extra-dimensional alien technology might be the use-case for it.

Anyway, the radio is interesting sometimes.

[McTraveller]

Dividing any nonzero number by zero is properly undefined; it's trying to answer a question "what number times zero is two?"  The answer to that is "there is no such number."  Trying to say "infinity" is categorically wrong for such an equation - I wish IEEE reported NaN as a result of divide by zero instead of ±INF.

The more interesting case is zero divided by zero, because asking "what number times zero is zero" has the answer "any number."  This isn't so much "undefined" as it is "you aren't asking a specific enough question."

Comparing the "size" of infinities already has a branch of mathematics, a good place to start is to look up 'cardinality'. For example, there are an infinite number of both integers and real numbers, but there are infinitely more real numbers than integers (real numbers are 'countably infinite' but real numbers are much "bigger" - you can't count them).

Also please please please remember that "infinity" is not a number.  It's a concept, and is more aptly a "progression" or description than a number.

[Ziusudra]

Dividing by is only undefined when what is being counted is undefined or when it makes no sense for what is being counted.

Consider Newton's second law of motion: acceleration = force / mass, for light mass is 0 so it instantaneously starts at c, this is infinite acceleration.

Consider 12 spoons placed evenly in some number of cups:

• 2 cups get 6 spoons each, 12 / 2
• 1 cup gets 12 spoons, 12 / 1
• 0 cups get either 0 spoons or the idea of 12 / 0 is nonsense

If IEEE just gav the result as NaN then you couldn't decide what to do in the case of infinity.

[Maximum Spin]

Dividing any nonzero number by zero is properly undefined; it's trying to answer a question "what number times zero is two?"  The answer to that is "there is no such number."  Trying to say "infinity" is categorically wrong for such an equation - I wish IEEE reported NaN as a result of divide by zero instead of ±INF.

This is not how division is defined. It's just an interpretation of the concept of division. It is possible to choose a set of axioms that makes division by zero defined, and there's no rule against it - just like "5 ÷ 2" or "7 - 12" are undefined in the natural numbers, but can be defined with simple extensions.

Also please please please remember that "infinity" is not a number.  It's a concept, and is more aptly a "progression" or description than a number.

All numbers are concepts, but the more important point is that math is just symbol manipulation according to rules. Choose the right rules, and you can treat infinity as a number. There are whole fields of math dedicated to this.

[McTraveller]

Sorry got to put on my pedantry hat: Newton’s law isn’t acceleration = force/mass; it is “force is change in momentum with respect to time.” Light has momentum even though it has no mass; it’s the frequency and/or direction of light that changes due to force, not its speed.

Also yes numbers are concepts, and infinity is a concept, but they are different classes. You know what I meant! 

[Maximum Spin]

Light has mass, it just doesn't have rest mass. Trying to apply Newton's laws in the relativistic domain is illegal for good reason.

[Eric Blank]

This thread title brought my mind back to 2007 memes

[Ziusudra]

And as usual we side tracked with tangential pedantry - the fucking point is that are domains in which it makes sense for division by 0 to result in infinity and domains in which it doesn't.

[Devastator]

And the ability to set up vastly different answers to the same question shows that there is no general answer to division by zero.  You can set up formulae where the limit as the divisor approaches zero is finite and defined..  at any arbitrary number.

For instance, the limit of 2x/x as x approaches zero is zero.  The limit of (2x/x)+2 as x approaches zero is 2.

[anewaname]

So, getting particles down to near-0 kelvin seems impossible due to Heisenberg uncertainty of the electrons... Just pointing this out because you can't get infinity if your divisor refuses to reach 0. Or, could we measure the Kelvin measurement at 0 if we could measure it while zooming by at near-c?

Spoiler (click to show/hide)

Haven't decided if I'm serious about this question, because I keep getting different answers when I ask myself.

[Maximum Spin]

The trick here is understanding that math is a set of axioms invented by humans. It relates to reality only insofar as we started out by basing those axioms on the things we could observe in the real world, like counting the number of sheep you have.

There's no specific real-world meaning to the idea of infinity, because it's inherently placed, by its definition, outside the bounds of finite human experience. An infinite process will never be completed. But, we can define, for the sake of symbol manipulation, a rule for assuming what that completion would look like, and then notate it with a little looping scribble, and then push that scribble around on the page with the same rules we use for all the other scribbles we pretend are numbers, and in some cases people will basically agree that the results of this seem to make sense, even though there's nothing physical we can point to that says what they should be. It's not inherently different in kind, just in scale, from saying that "6" means the idea of having one of something, and then another one, and then another one, and then another one, and then another one, and then another one, each of which you can point to all in order.

Is this useful? Does this matter? Well, no, but neither does most of what mankind does.

So, getting particles down to near-0 kelvin seems impossible due to Heisenberg uncertainty of the electrons... Just pointing this out because you can't get infinity if your divisor refuses to reach 0. Or, could we measure the Kelvin measurement at 0 if we could measure it while zooming by at near-c?

Spoiler (click to show/hide)

Haven't decided if I'm serious about this question, because I keep getting different answers when I ask myself.

No, that's not how temperature works - even at absolute zero, particles (any particles, not just electrons) still have quantum of action, but that isn't related to the temperature, the velocity of the observer doesn't affect it, and you can't reach zero K in practice for totally unrelated reasons.

[Schmaven]

So, getting particles down to near-0 kelvin seems impossible due to Heisenberg uncertainty of the electrons... Just pointing this out because you can't get infinity if your divisor refuses to reach 0. Or, could we measure the Kelvin measurement at 0 if we could measure it while zooming by at near-c?

Spoiler (click to show/hide)

Haven't decided if I'm serious about this question, because I keep getting different answers when I ask myself.

With temperature being defined as molecular motion, and motion being a change in position over some period of time, moving at light speed would have the relativistic effect of time stopping for whatever the speeding entity observes.  And if time does not pass, there can be no motion.  So from that reference point, everything passed by would be at absolute zero, frozen in time.  The challenge then becomes how each of those reference frames could interact with one another while maintaining light speed.


Edit: Dividing by zero as it relates to slices of pie:

If I have 1 pie, and divide it by 8, I get 0.125 of the pie per slice.  Dividing that pie by 10, and I get 0.1 units of pie per slice, approaching no pie at all per slice when it is divided into infinite slices.  In fact, far before splitting a pie into infinite slices, there would only be quarks, electrons, muons, and bits like that per "slice".  And it would be a bit of a stretch to call anything without at least 1 in tact sugar molecule a piece of a pie. 

But by decreasing the denominator, dividing the pie by 1/2 gives us 2 pies.  Which is a bit odd.  Kind of a different question really once the denominator becomes smaller than 1.  Instead of "how many units of pie do we get per x slices?", the question becomes, "how much pie would there have to be somewhere for this pie to be x slices of pie?"  Imagining 1 pie to be 1/2 a slice of pie makes sense when you consider Doug.  Doug will eat a whole pie in 1 sitting, from the middle out, and then go back for more.  So perhaps servings is better than slices.

So while decreasing the denominator from 1 to 0, the question is, "how much pie would there have to be for this 1 pie in front of us to be x servings?"  1 ÷ 1/4 gives us 4 pies for 1 pie to only be 1/4 of a serving.  1 ÷ 0.01 would require 100 pies for that 1 pie to be 0.01 of a serving.  As the denominator approaches 0, we need more and more pie.  Enough pie to fill the entire galaxy at a certain point.  But what happens at 0?  When 1 pie is 0 servings of pie.  Is there no longer any pie at all?  Or does everything have to become pie?  Zero in terms of units of pie refers to absolutely no pie at all.  So it seems like if 1 pie is exactly 0 servings of pie, then that would define the pie as not being a pie.  1 pie = absence of pie.  That doesn't make sense.  Pie is pie.  The flavor and type of pie is secondary here.  It could be apple pie, cherry pie, or strawberry rhubarb.  So we're left with the other option, that everything would have to become pie. 

I'm well over my head at this point, and not sure I can continue with the analogy.

But if all things everywhere were composed 100% of pie, then 1 pie out of the infinite expanse of space and time would be inconsequential.  Something would have to not be a pie in order for "pie" to have any meaning.  Literally all of the ingredients of the pie would have to be pies.  We would have an infinite fractal of pies spiraling all the way down such that even quarks become pie. 

So it does seem to break down when we consider dividing by 0.  At least in the sense of the observable universe maintaining the current laws of physics.  Not that we really know what quarks and muons are anyway.  Maybe they are pies?  Cobblers?

[Putnam]

Temperature in the thermodynamic sense is usually defined as the derivative of energy wrt entropy, i.e. "how much energy changes if you increase the entropy". In this sense, absolute zero is much more intuitively impossible: there's an actual division by zero, since the units are in energy/entropy, which you can represent as, say, (boltzmann constant)*joules/nat.

Thermodynamic beta is the inverse of temperature, entropy/change-in-energy, often represented in gigabytes per nanojoule, and this, too, is pretty obviously impossible to reach absolute zero in--here, absolute zero is at infinity, and the "zero" on the scale is "infinite temperature".

[zhijinghaofromchina]

For instance, the limit of 2x/x as x approaches zero is zero.  The limit of (2x/x)+2 as x approaches zero is 2.

Sorry devatator , I'm not intended to offend you, but if the x approaches zero, the limit of 2x/x might be 2 , accordingly the limit of (2x/x)+2 as x appoaches zero is 4.

I learnt about the limit during this term in the advanced mathematics class , I almostly forgot how to explain it since we don't need maths knowledge any more , but you can solve it with the Robita's Law.

[McTraveller]

Is Robita’s Law a transliteration of L’Hôpital’s Rule?  I’ve not heard of Robita.

Yes the T = dQ/dS definition is fun - it gives rise to negative temperature in some scenarios xD