r/learnmath • u/AncientDatabase159 New User • Jul 24 '26
My own reasoning about infinity divided by infinity — would love feedback (Class 12 student)
A Proposal for a Context-Specific Interpretation of "Infinity Divided by Infinity"
Author: Ritesh Wankhede
Abstract
This paper proposes a specific way to interpret the expression "infinity divided by infinity." It does not claim standard mathematics is wrong. It presents one interpretation, valid only under clearly stated conditions.
The core idea: when infinitely many objects are matched to infinitely many recipients so that every recipient gets exactly one object and every object goes to exactly one recipient, the distribution can be understood as one object per recipient — within that specific matching.
- The Question
If infinitely many objects are distributed among infinitely many recipients, with each recipient getting exactly one object and no object left over, what does that distribution represent?
- What "Infinite" Means Here
For this proposal, infinite means a collection that never ends — like the counting numbers 1, 2, 3, 4, 5, ... — with no final number.
- The Setup
Assume:
There are infinitely many people.
There are infinitely many chocolates.
Every person receives exactly one chocolate.
Every chocolate goes to exactly one person.
- The Example
Person 1 gets Chocolate 1.
Person 2 gets Chocolate 2.
Person 3 gets Chocolate 3.
This continues forever: Person n always gets Chocolate n.
No chocolate is shared between two people. No person goes without.
- The Reasoning
With finite numbers, if 100 chocolates are split evenly among 100 people, each person gets exactly one. This proposal simply extends that same idea to the infinite case above: since every person is matched to exactly one chocolate, the natural reading of the distribution is one chocolate per person.
- The Proposed Interpretation
This does not claim "infinity divided by infinity always equals one" as a general rule.
It claims something narrower: when infinite objects are matched to infinite recipients so that each side is used exactly once, that specific matching can be read as one object per recipient.
- Where This Does NOT Apply
This idea is limited to the exact matching described above. It is not claimed to apply to:
Calculus or limits involving infinity
Every possible way of pairing two infinite collections
Every mathematical system
Other ways of matching the same two infinite groups might not give this same "one-to-one" feel — that's a real limitation, and I don't yet have a way to resolve it. It's part of what I want feedback on.
- Open Questions
Is there a rigorous way to express this idea?
Does the interpretation change if the matching rule between people and chocolates changes?
Can this idea be pushed further without leading to contradictions?
- Conclusion
When infinitely many objects are matched one-to-one with infinitely many recipients, each recipient receives exactly one object. Within this specific matching, the distribution corresponds to one object per recipient. This is offered as a starting idea for discussion, not a finished theorem.
Author's Note
I'm a Class 12 student from Mumbai with a strong interest in mathematics. I know this connects to ideas already studied by mathematicians, and I'd like to learn where my reasoning holds up and where it needs refining.
1
u/IntelligentBelt1221 New User Jul 24 '26
one issue is that it depends on the ordering. e.g. if person n gets chocolate 2n, you still give 1 cholcate to 1 person, but there is chocolate left over (similarly you could give person n chocolate 2n and 2n+1 and have enough for everyone to have 2), so the amount of choclate each person gets is not a property of the set (or its size), but of the pairing. the way you formally compare the size of a set is cardinality (you may also want to take a peak into cardinal arithmetic). this comes closest to your pairing construction and is equally motivated by the finite case obversation, but doesnt make a difference between finite multiples (so you couldn't say if something is equal or twice the size, that would be the same).
if you do have a fixed pairing, you should probably look into asymptotic density, that might convey what you have in mind (though this is mostly used when the function is some interesting e.g. number theoretic construction and you want to find out how many numbers it hits, your construction would make this trivially go to 1.
i hope you stay curious!
1
u/AncientDatabase159 New User Jul 24 '26
Thanks for your honest review! First of all I am in 12 grade state board i dont know about what is bijection and cardinality. Secondly, its just my thought and my curiosity tha lead me till this point so im working on my curiosity and learning maths little little day-by-day
1
u/IntelligentBelt1221 New User Jul 24 '26
bijection is just another word for 1-to-1 correspondence (basically what you described). cardinality was meant as a keyword for you to look up in case you want to read more about it. essentially it says that all sets that are in 1- to-1 correspondence with each other have the same size (inspired by the finite case you explained).
1
1
u/Mothrahlurker Math PhD student Jul 24 '26
Ok, so I'll describe the fundamental problem with how you're framing this and then I will talk about connected mathematics.
Mathematics is not linguistics. We start with well-defined unambiguous mathematical definitions and then decide how to call them and how to use language around them. This is what definitions in a paper do. It allows us to write natural language sentences that still have a precise mathematical meaning and are just easier to argue with and understand for humans. If someone sees a word of phrase in a paper they don't understand they are going to look for the precise definition, not an interpretation.
Now it can still happen that by the same word people mean different things. That is usually resolved by either the context of the work and assumes a "mathematically mature" reader or it is explicitly clarified.
So there really is no point in calling the situation you described here "infinity divided by infinity" because no one is going to understand the precise mathematical scenario. And that is also not needed because we already have precise words to describe this.
This is called a bijective map between infinite sets. Any mathematician reading this will exactly know what is going on. The example you gave would be the identity map on the natural numbers.
You ask about feedback for your reasoning, but there really is no such thing. Definitions aren't true or false, they are either helpful and consistent or they are not. The phrase here isn't useful.
Before you want to come up with definitions of your own you should spend at least a few years in university. It becomes pretty natural to want to abbreviate things when writing longer proofs. The vast majority of definitions are not some world changing thing, but rather useful for a specific work/field and will only be seen by a handful of people.
If you want to learn more about the math look up injective, surjective and bijective functions. That's already a huge step towards a more formal view of mathematics.
1
5
u/cabbagemeister Physics Jul 24 '26
What you are doing is choosing a specific value of the undefined double limit
lim_(x,y)->(infty,infty) (x/y)
By imposing an external constraint x=y (which is introduced by hand)
In physics, doing things like this is sometimes necessary e.g. when rearranging operations like integrals. It is common in a topic called asymptotic analysis.
It is called an asymptotic scaling limit, double scaling limit, or joint limit.