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Discussion Interesting Math

Discussion in 'Off Topic' started by Xor, Jan 15, 2020.

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  1. Xor

    Xor @XorDev

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    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 likes this.
  2. dannyjenn

    dannyjenn Member

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    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...)
     
    Xor likes this.
  3. Mercerenies

    Mercerenies Member

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    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 ;)
     
    ElectroMan, Xor and dannyjenn like this.
  4. devKathy

    devKathy Member

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    Cpaz and Xor like this.
  5. Cupid Stunt

    Cupid Stunt Member

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    @Mercerenies
    The sum of all positive natural numbers approaches positive infinity. A sequence can never equal infinity.
     
  6. Xer0botXer0

    Xer0botXer0 Member

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    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.
     
  7. Mercerenies

    Mercerenies Member

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    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.
     
  8. Cupid Stunt

    Cupid Stunt Member

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    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.
     
  9. Cantavanda

    Cantavanda 〜Flower Prince〜

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    1+1=3
     
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  10. Xor

    Xor @XorDev

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    I like this answer! I would say the solution should be infinity, but there is some intriguing logic behind the -1/12 answer!
    Mathologer is the best!
     
    devKathy likes this.
  11. Cupid Stunt

    Cupid Stunt Member

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    delicious
    finally some good math
     
  12. Xor

    Xor @XorDev

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    It can be done (using a trick). Here's a proof of 1 = 2:


    Just add one to each side to get 1+1 = 3.
     
  13. Xer0botXer0

    Xer0botXer0 Member

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    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.
     
  14. Xor

    Xor @XorDev

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    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?
     
    RichHopefulComposer likes this.
  15. Xor

    Xor @XorDev

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    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.
     
  16. Mercerenies

    Mercerenies Member

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    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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  17. ElectroMan

    ElectroMan Jack of All Shades

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    You should be more worried of quantum computers for those kinds of things, really.
     
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  18. Cantavanda

    Cantavanda 〜Flower Prince〜

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    I agree!
     

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