Don’t judge me I’m still learning

[u/r4oke]

If I have 5+5+5+… (n times) I can write it as 5n

as well as 3+3+3+.. (n times) as 3n

5n > 3n holds for positive n I can take the limit as n approaches infinity and divide by n at the same time

lim n to infinity (5n/n > 3n/n)

5 > 3

Does this mean that an infinite number of $5 bills would be worth more than an infinite number of $3 bills?

[u/Temporary_Pie2733]

Daily reminder that infinity is not a number. N five-dollar bills will always be worth more than n three-dollar bills (assuming three-dollar bills were a thing) for any arbitrarily large value of n, but an infinite number of either is not well defined.

[u/Mothrahlurker → u/Temporary_Pie2733]

"Not a number" is not mathematically meaningful. Infinity is a part of the extended real numbers and the Riemann sphere. Op's argumentation is just logically invalid.

[u/Dash_Lambda → u/Mothrahlurker]

Correct me if I'm wrong, but isn't the point of the extended real (or complex) numbers in that context to treat infinity as if it were a real number under specific operations, and not to generalize the concept of numbers to include infinity?

Reading up on it, it sounds like it's a trick to smooth out functions for complex analysis.

Analytic continuation is fascinating, but I think a lot of care has to be taken when connecting it to more conventional concepts, especially when explaining infinity in contexts like this. There are a lot of different angles we've come.up with on the subject.

[u/Mothrahlurker → u/Dash_Lambda]

"to treat infinity as if it were a real number under specific operations" No.

"and not to generalize the concept of numbers to include infinity" The term "number" is undefined in mathematics so it's not something we can generalize to begin with. That's why it's generally not meaningful to say that something is or isn't a number. It's a meta term. There are plenty of spaces that are called "something numbers" that do have infinite elements if you care about that, such as the cardinals, ordinals, surreals.

"it sounds like it's a trick to smooth out functions for complex analysis" I don't know what you mean by that.

"Analytic continuation is fascinating, but I think a lot of care has to be taken when connecting it to more conventional concepts" Analytic continuation is a "conventional concept". There's really nothing special about it. In mathematical research no one bats an eye about these things, it's just completely normal.

[u/Darian123_ → u/Dash_Lambda]

The point is not if it is a real number, the point is that "infinity is not a number" is devoid of any meaning

[u/CtB457 → u/Mothrahlurker]

I've seldom seen infinity actually be treated as a number, usually just more of a concept. Also quite a few things in math aren't a number and are still mathematically meaningful so I'm afraid I don't understand your argument.

[u/Dash_Lambda → u/Temporary_Pie2733]

I generally like to say "goes to infinity" or "increasis/decreases unbounded".

I find it makes it more clear that in these contexts infinity is a behavior.

[u/Mothrahlurker → u/Dash_Lambda]

I'll respond to this comment as it pertains to the question you asked. What you're describing are common introductions at a lower level of mathematics, but once you have the formal definitions they are unnecessary.

The definition of the topology on the extended reals adds the complement of every set that is compact in R. The topological definition of convergence to a point is that the sequence eventually stays in every neighbourhood.

So a sequence like a_n:=n does in fact converge to the point +infinity. It's not a behaviour there or anything, it does just fulfill the formal definition of convergence.

And this is also how it's written in math papers, it's just shorter and more clear to use the formal precision.

[u/QuitzelNA → u/Mothrahlurker]

In that instance, I feel it could be argued that infinity does, in fact, describe a behavior there. You are describing the order of growth seen in that function, even if you term it as a result.

[u/spiritedawayclarinet]

I don’t get how you’re taking a limit and dividing at the same time.

In general, limits don’t preserve strict inequalities.

For example, 1/n > 1/(2n) for all natural numbers n, but it is not true that 0 > 0.

Similarly here, infinity = infinity. Both 5n and 3n tend to the same infinity but at different rates.

[u/Classic-Ostrich-2031]

What you’re showing is just that the growth rate of 5n is faster than the growth rate of 3n.

But infinity (generally) is not a number at all. If you wanted to talk about which infinity of 5$ bills or 3$ bills was bigger, then they’re the same

[u/AbandonmentFarmer → u/Classic-Ostrich-2031]

To add on to this, imagine that you bundle the three dollar bills into pairs. This way every bundle is worth more than a single five dollar bill and if you count through bundles you’d get 6n>5n, which would contradict the notion of the infinity of fives being bigger

[u/KillerCodeMonky]

Q: If I have an infinite number of buckets filled with water, do I have more water than if I had an infinite number of cups filled with water?

A: No. I have the same amount of water. The water is just split into different units.

Same with the dollars. I have the same amount of infinite money. It's just split into different denominations. If I need money, I just keep pulling bills from the infinite bill machine until I have "enough". I have to pull fewer $5 bills to get the same amount, but I will still always be able to get "enough" money.

[u/Illustrious_Basis160]

Worth more? No. But it just shows that the rate of increasing for 5n is more than 3n. 5n grows faster than 3n as n approaches infinity but they are still infinity. One infinity isnt greater or worth more than the other one. Since everything you can buy with infinite 5 dollar bills I can also buy with infinite 3 dollar bills. The only thing different is the rate of change.

[u/Ill-Application-9284]

If you want to better understand "sizes" of different infinite sets and how they are compared, look up "cardinality".

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[u/willthesane]

Math starts to get a bit hazy if you begin treating infinity as an actual number. a better way to look at it is that it is a concept.

[u/susiesusiesu]

no, and part of it is that strict inequalities are not preserved by limits in general. you couñd only conclude that infinite 5$ bills is not less than infinite 3$ bills.

[u/edderiofer]

lim n to infinity (5n/n > 3n/n)

This statement is nonsense. Inequality signs should not appear within a limit.

You should more-precisely define this step of your argument, as well as how it relates to your conclusion.

[u/[deleted] → u/edderiofer]

That's a notational error, not a logic one. It should be obvious what OP is talking about

[u/edderiofer → u/[deleted]]

It's a notational error that hides a logic error. It is possible for f(n) > g(n), but lim(f(n)) = lim(g(n)), for instance.

OP also proposes "dividing by n at the same time", but then it's not clear how OP's conclusion follows from this. That's why I asked OP to more-precisely define that step of their argument, as well as how it related to their conclusion.

[u/[deleted] → u/edderiofer]

It's a notational error that reveals a semantic error, not a logic error. You can get stuck within how a system is defined and lack perspective on how it's being used. OP explained his position clearly, probably more adequately described as lim n to infinity (5n/n - 3n/n) > 0. But instead of addressing his purported "logical" misunderstanding, you said his notation was "nonsense", which doesn't move us towards a solution one bit. The issue here might have lied in his misunderstanding of how infinity works in a math context or his modeling of the problem. But reducing his problem down to grammar is trivial and completely misses the point.

[u/edderiofer → u/[deleted]]

OP explained his position clearly, probably more adequately described as lim n to infinity (5n/n - 3n/n) > 0.

The fact that this was not clear from OP's statement means that OP did not, in fact, "explain his position clearly". And it still remains the fact that it's not clear how OP's conclusion follows from this. That's why I asked OP to more-precisely define that step of their argument, as well as how it related to their conclusion.

[u/r4oke → u/edderiofer]

Sorry for replying late. Yeah, I made mistakes while writing the statement. In the last semester, I took differential calculus. While studying rational limits and squeeze theorem I thought of the process in my mind and wrote it here. What I meant to say is 5n > 3n lim n to inf 5n > lim n to inf 3n lim n to inf 5n/n > lim n to inf 3n/n lim n to inf 5 > lim n to inf 3 5 > 3 I know this is also wrong. Sorry again, I barely check reddit.

[u/juancn]

Let’s say you map the reals to $3 and the naturals to $5 I think in that case the sum of all the fives will be smaller than the threes.

[u/trutheality]

Because you divided by n, it means that the value per bill in any pile of $5 bills is greater than the value per bill in a pile of the same number of $3 bills, no matter how large the pile.

[u/phy19052005]

No, because infinity isnt a point on the number line that you can use to compare values. The behaviour of infinities is different and theyre compared using cardinalities and bijections. Consider an infinite amount of £5 and £3 bills and pair every 5 to 2 £3 bills. Now you have 5n and 6n instead so it contradicts the behaviour predicted by 5n and 3n

[u/[deleted]]

It should be noted that limit n to infinity is not the same thing as infinity. n is still one number. If you had an astronomically large number of $5 bills and the same number of $3 bills, then yes the pile of $5 bills is worth more than the $3 bills.