Discussion Interesting Math

Xor

@XorDev
A couple days ago I asked the question "If there was a clock that halves it's speed every 12 hours, when would it reach midnight?". This spurred a long discussion about different ways to interpret the problem and a variety of answers.
Which brings to my attention that there are several people here who, like me, are interested in mathematics discussions so I decided to create a topic for this. Let's talk about interesting math problems and solutions!

I'll start us off by asking, what do you think the sum of natural numbers (1, 2, 3, etc) is equal to? -1/12?
 

dannyjenn

Member
I'll start us off by asking, what do you think the sum of natural numbers (1, 2, 3, etc) is equal to? -1/12?
No! That -1/12 thing is deceptive. But the fact is, you simply can't sum all the natural numbers together in the first place. (Yet their partial sums tend towards positive infinity...)
 
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Mercerenies

Member
The sum of the natural numbers is equal to infinity (in the standard Euclidean metric on the extended real line using a constant, discrete unbounded measure). We could change those constraints. For example, if we wanted the sum to actually be -1/12, we could take the one-point compactification of the real number line (which is a circle, mind you), then identify the "point at infinity" with -1/12 to get a quotient topology on the reals. In this topology, the sum actually converges to -1/12. We could also be sillier and use, say, the finite complement topology or the indiscrete topology. Both of those are non-Hausdorff, which means limits may not be unique, and indeed in both of those topologies, the sum of all natural numbers is everything. Literally, any number you could tell me would be the right answer. On the other hand, we could leave the Euclidean metric alone and mess with the measure. If we take Lebesgue measure (the measure we usually work with when taking integrals), we would turn the sum into an integral over the natural numbers. The set of natural numbers has measure zero, so the sum (integral) in this case is zero. We could take a finite measure where each natural number is weighted differently, but then all you're doing is finding the total probability of an event, and at that point you're just doing statistics. And if we're doing probability, the probability of a 0 outcome may be nonzero, so we need to agree on whether or not 0 is a natural number (it is; it categorically is; some wrong people like to disagree with me on that). So, I argue the answer to your question "what is the sum of the natural numbers" is: "define every word in that sentence".

And that's what mathematics is, kids. Changing the rules like a toddler until you get the answer you want ;)
 

devKathy

Member

Xer0botXer0

Member
I formulated a plan today, using math.

It would be nice if I can see reason to apply what I'm learning to other parts of my life.
 

Mercerenies

Member
@Mercerenies
The sum of all positive natural numbers approaches positive infinity. A sequence can never equal infinity.
(in the standard Euclidean metric on the extended real line using a constant, discrete unbounded measure)
Extended real line includes two infinities as genuine first-class elements. I'm not contending that it approaches infinity. I'm contending that the limit is infinity, a specific, well-defined element of the set of extended reals.
 

Cupid Stunt

Member
Extended real line includes two infinities as genuine first-class elements. I'm not contending that it approaches infinity. I'm contending that the limit is infinity, a specific, well-defined element of the set of extended reals.
There are two schools of thought on that. While I get what you are saying and I know that it's provable, my belief which is contrary is also provable. In this case I believe we must agree to disagree.
 

Xor

@XorDev
The sum of the natural numbers is equal to infinity (in the standard Euclidean metric on the extended real line using a constant, discrete unbounded measure). We could change those constraints. For example, if we wanted the sum to actually be -1/12, we could take the one-point compactification of the real number line (which is a circle, mind you), then identify the "point at infinity" with -1/12 to get a quotient topology on the reals. In this topology, the sum actually converges to -1/12. We could also be sillier and use, say, the finite complement topology or the indiscrete topology. Both of those are non-Hausdorff, which means limits may not be unique, and indeed in both of those topologies, the sum of all natural numbers is everything. Literally, any number you could tell me would be the right answer. On the other hand, we could leave the Euclidean metric alone and mess with the measure. If we take Lebesgue measure (the measure we usually work with when taking integrals), we would turn the sum into an integral over the natural numbers. The set of natural numbers has measure zero, so the sum (integral) in this case is zero. We could take a finite measure where each natural number is weighted differently, but then all you're doing is finding the total probability of an event, and at that point you're just doing statistics. And if we're doing probability, the probability of a 0 outcome may be nonzero, so we need to agree on whether or not 0 is a natural number (it is; it categorically is; some wrong people like to disagree with me on that). So, I argue the answer to your question "what is the sum of the natural numbers" is: "define every word in that sentence".

And that's what mathematics is, kids. Changing the rules like a toddler until you get the answer you want ;)
I like this answer! I would say the solution should be infinity, but there is some intriguing logic behind the -1/12 answer!
I'm not a math major or anything, but this video makes sense to me:


It's long so grab a snack and a drink if you're going to watch it. Basically there are special classes of sums for which the statement is true. But those types of sums shouldn't be expected to behave like normal sums... i.e., they are completely different. But they have cool names!

https://en.wikipedia.org/wiki/Cesàro_summation

https://en.wikipedia.org/wiki/Hölder_summation
Mathologer is the best!
 

Xer0botXer0

Member
Thank you for the post,

The last time I posted I knew less of what I know know, and so this cycle will repeat. Hopefully one day I won't sound stupid when it comes to math.

But I do have a puzzle for you all.
If you have two taps, one running hot water that can fill a tub in an hour, while the cold water tap can fill it in half that time, how long will it take the tub to fill were both taps running?
There are no stupid answers. But do consider the scope of the question, when you add variables such as water pressure then it does get complicated, for now let's leave it uncomplicated.
 

Xor

@XorDev
Thank you for the post,

The last time I posted I knew less of what I know now, and so this cycle will repeat. Hopefully one day I won't sound stupid when it comes to math.

But I do have a puzzle for you all.
Try to keep math fun and interesting to you. I find learning it to solve programming problems is the most fun and shaders are full of such problems!
As for your problem: You have a hot water tap that fills 1 tub/hour and cold which fills 2 tubs/hour so running both would be 3 tubs/hour and you could fill a tub in 20 minutes, right?
 

Xor

@XorDev
I just watched this video on encryption:

Do you guys think our encryption will fail one day due to finding an effective way to factor large numbers? Also, I'm curious where you guys stand on P vs. NP. I think we probably all agree on the distinction, but I wonder if anyone has anything interesting to add.
 

Mercerenies

Member
Do you guys think our encryption will fail one day due to finding an effective way to factor large numbers? Also, I'm curious where you guys stand on P vs. NP. I think we probably all agree on the distinction, but I wonder if anyone has anything interesting to add.
Just want to chime in to say that solving P=NP in the positive is not the same thing as breaking encryption. We could potentially come up with an existence proof reducing an NP-complete problem to P but not have an actual algorithm to do so. Even if we do have such an algorithm, it's also possible the asymptotics on it could be bad enough that we can't implement it on any real computer. I see us solving (or refuting) P=NP sometime in the foreseeable future. I can't pass judgment on whether it'll be a positive or negative result. But even a positive result won't change the world in the ways some people claim it will. Things will keep moving, crypto will be fine, and we'll all survive.
 
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