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The April Fool's joke was not the two elaborate posts (which appear to fool one into believing that the sum of all integers is -1/12), but the final sentence ("Happy April Fool's Day" in the spoiler), which I intended to fool you into thinking that you were fooled, because for math purposes (in assigning a value to an infinite series) and physics, the answer actually is correct; the sum of all positive integers is -1/12.  This is seen in physics in the Casimir force, as well as in bosonic string theory and in many ways in math (we keep getting the same result with different methods).

 

Numberphile has done a few videos on this series:

 

https://www.youtube.com/watch?v=w-I6XTVZXww

 

https://www.youtube.com/watch?v=E-d9mgo8FGk

 

https://www.youtube.com/watch?v=0Oazb7IWzbA

 

No math trickery was used.  The only thing questionable was Lisa's equation of the Zeta of minus one.  And Bart was right with his correction; it is valid only for s = >1.  However, Lisa's answer, and similar ones, are effectively correct, since they do work as useful sums of series in math and physics.  We could write:

 

1st.jpg

 

That is not, technically, correct, but as Lisa said, "It still works."

 

In order to legally evaluate the Zeta function with real part less than positive one, we need to utilize analytic continuation.  This can be a bit involved, and I will not go into it right now.  But in 1859, Bernhard Riemann did this to the Zeta function, and allowed us to use the function everywhere on the complex plane except for the specific value where s = positive one.  Below is an example of a legal way to use the Zeta function for the real part less than one, and is known as the reflection formula:

 

ref1.jpg

 

Now, that looks kind of scary, but we will get through it.  Let us go factor by factor.

 

We have zeta of s = -1.  The first term is 2 to the s, or 2^(-1), which is 1/2.  Next is pi to s - 1, which is pi to minus 2, and that is one over pi squared.  Next is sine of pi times s over 2.  That is the sine of minus pi over 2.  Looking at a plot of the sine, we see that that is minus 1.  Then we come to the Gamma function.  The Gamma function can be tricky, but fortunately for us, the Gamma function for positive integers is simply the factorial of the integer minus one.  That is:

 

4th.jpg

 

And for s = -1, we need the Gamma of 2, which is 1!, and that is simply 1.

 

Lastly, we have the Zeta of 2, and I have already mentioned that that is pi squared, over 6.

 

We have:

 

5th.jpg

 

Taking another look at the plot of the real zeta function from 0 to -2:

 

zeta.png

 

We see that Steven would have been better off had he waited until the bartender from the other night poured him half of a beer instead of 1/12.  He would have been one-half of a beer in deficit.  We see that the value of zeta at 0 is minus one-half.  With "s" in the zeta function being zero, we have:

 

6th.jpg

 

Steven would only need to serve each mathematician a single beer, and then when the infinity-ith one was served, he would have minus one-half of a beer and that would match his deficit with the other bartender.

 

In the Wiki page of 1 + 1 + 1 + 1 + , we read:

 

Emilio Elizalde presents an anecdote on attitudes toward the series:

 

In a short period of less than a year, two distinguished physicists, A. Slavnov and F. Yndurain, gave seminars in Barcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressed the audience with these words: 'As everybody knows, 1 + 1 + 1 + = −1/2.' Implying maybe: If you do not know this, it is no use to continue listening.

 

Besides the zeta function, it is easy to see that an infinity of ones equals minus one-half from the Ramanujan summation:

 

RSum.jpg

 

Our function is simply "1", and in the first term, the f(0) is one because there is nothing to evaluate at zero.  This makes the first term -1/2 times 1, or -1/2.  The second part, with the Bernoulli numbers and derivatives, are all zeros, since the first derivative and above are all zeros, so the Ramanujan summation is -1/2.

 

Steven could have been in worse condition if he had no beer deficit at all.  We see that the zeta function at minus two is zero.  So, in order for him to serve all of the mathematicians to infinity in order to sum to zero beers, he would have to serve:

 

7th.jpg

 

Again, one can get this result from the Ramanujan Summation as well as the Reflection Formula for the Zeta, since the sine of (-2 pi)/2 = 0.

 

He would have needed to have served each mathematician the square of their number in beers in order to recoup, and that would have been infinitely worse.

 

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On 2017-03-30 at 2:37 AM, Emery. said:

 

Mathematicians solve only trivial problems, as RichardFeynmann said ;) 

 

Me too. *adds to the "do it when you have time" list*

 

Heart, you're an admin! Congrats :)

Thanks :D

 

On 2017-03-31 at 4:45 AM, Kimmie. said:

Yeah math has never been my thing. Had to retake matematik 1 (the basic math course in gymnasiet our high school) to finnsh it. by the way the word High school has always confused me at first because if you translate it to swedish it is: Högskola which is our Uni.

 

But science is cool do.

Please don't feel dumb. Math and physics isn't everyone's forte, which is actually a really good thing. After all, we need all sorts of people in order to become a better society. Math and physics tell us about the world around us, but they don't tell us anything at all about how to live in it. For that, we need the humanities to help us figure out ethics and morality. Math and physics teach us nothing about compassion, or about hard work. At the end of the day, no one should feel bad for not being a "sciencey" or "mathy" person. Everyone has their own gift to give to the world. After all, if we were all good at the same thing, then we probably wouldn't do so well as a species, eh? ;)

 

On 2017-03-31 at 9:15 AM, Kelly said:

This might be the last one for a little while.  Basically, all of the infinite mathematician puzzles posted here are what I am using in a recreational math book that I am writing, and each is used to introduce a new concept.  And I am pretty much caught up at this time.  This one brings up ideas that at first seem preposterous, but in reality bring up very important concepts that are key in the book.

 

An infinite number of mathematicians walk into a bar.

 

But the bartender (Steven is working this day) is in trouble.  His beer keg is empty.  Indeed, last night (Thursday), he drank away his troubles at another bar.  And he had just ordered his third beer before he discovered that he had just spent his last dime.  He told the other barkeep to stop pouring, but he already had one ounce (one-twelfth of a beer) in the mug by that time.  Rather than pour it out, he let Steven drink that one ounce, and owe the barkeep an ounce of beer, to be paid as soon as he can.

 

It is Friday, and the infinite number of mathematicians walk into the bar at the Hilbert Hotel, and each one is thirstier then the previous one.

 

Steven has no beer, and he is in debt; he cannot buy a new keg.  What shall he do?

You're writing a book?!?!? You must tell me when it's done and when I can buy it. Especially if it's full of these sorts of story riddles. I enjoy them so much in this thread, thinking about a whole book of them makes me think I've accidentally stumbled into Heaven.

 

For the record, you have inspired me Kelly to try my hand at a riddle-story or two myself. I have one in mind, which is not about infinity in any way, or sums. But I promise it'll be fun, and probably solve-able even by those with high school math. That does not however mean that it will be easy. No, it will not be easy ;)

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It becomes not so bad after getting used to the zeta function and its mysteries interesting properties.

 

I had thought that there was a Riemann Zeta for Dummies book.  Perhaps someone should write one.

 

I was interested in fractals and iterative functions as an undergrad, but did not get started on exploring the weird stuff until grad school (when the professors brought up things that I had never imagined). 

 

amr33mo_700b.jpg

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  • 3 weeks later...

Cassini photographs the earth and moon between the rings of Saturn:

 

earth-moon-saturn-rings-April-2017-cp-e1

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On 5-4-2017 at 7:17 PM, Kelly said:

It becomes not so bad after getting used to the zeta function and its mysteries interesting properties.

 

I had thought that there was a Riemann Zeta for Dummies book.  Perhaps someone should write one.

 

I was interested in fractals and iterative functions as an undergrad, but did not get started on exploring the weird stuff until grad school (when the professors brought up things that I had never imagined). 

 

amr33mo_700b.jpg

 

46 minutes ago, Kelly said:

Cassini photographs the earth and moon between the rings of Saturn:

 

earth-moon-saturn-rings-April-2017-cp-e1

hqdefault.jpg

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11 minutes ago, Jayce said:

 

hqdefault.jpg

Yup that. 

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  • 1 month later...
  • 2 weeks later...
On 2017-06-10 at 4:29 PM, Kelly said:

I've always loved that story! It's one of those moments when you realize that physicists are just humans too in the end ;)

 

Have you heard the story of why penguin diagrams are called after penguins?

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:)

 

Yep. They are a branch of Feynman Diagrams regarding quarks and they do look like penguins.

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They didn't have to though. There's a story there ;)

 

One of my old professors back at UBC made a bet with the person who coined the term. Whoever lost the bet, had to use the word "penguin" in their next publication. He lost, so he had to find a way to sneak the word penguin in there and get it past reviewers! So he wrote the Feynman diagrams to look like penguins, specifically so he could call them penguin diagrams ^_^

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Way cool. No, I didn't know. That sounds like a fun bet. Perhaps we should make a bet and end up introducing "platypus" to physics terminology.

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6 hours ago, Kelly said:

Way cool. No, I didn't know. That sounds like a fun bet. Perhaps we should make a bet and end up introducing "platypus" to physics terminology.

Haha. Perhaps. The next time we meet, we can have a challenge or a bet, and make the one who loses find a way to get platypus into their next publication! *is already contemplating what new mechanism I could discover and call the Platypus Mechanism*

 

5 hours ago, Deus Ex Zero said:

Hey there! Welcome on board! Nice to meet you :D

Nice to meet you too :D Or have we met before? I'm not always good with memory :P

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I think that I could sneak "platypus" into my next publication. :twisted:

 

Can you? :twisted:

 

:D

 

Theirs was a totes awesome bet. :cake: 

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Dodecahedron314
21 hours ago, Heart said:

They didn't have to though. There's a story there ;)

 

One of my old professors back at UBC made a bet with the person who coined the term. Whoever lost the bet, had to use the word "penguin" in their next publication. He lost, so he had to find a way to sneak the word penguin in there and get it past reviewers! So he wrote the Feynman diagrams to look like penguins, specifically so he could call them penguin diagrams ^_^

I'd heard about this in the past, but I didn't know this was one of your old profs, Heart! Wow, science really is a small world sometimes.

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Guest Deus Ex Infinity
On 21.6.2017 at 11:53 PM, Heart said:

Haha. Perhaps. The next time we meet, we can have a challenge or a bet, and make the one who loses find a way to get platypus into their next publication! *is already contemplating what new mechanism I could discover and call the Platypus Mechanism*

 

Nice to meet you too :D Or have we met before? I'm not always good with memory :P

I don't think so but it's never too late as they say :D

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On 2017-06-22 at 0:04 AM, Kelly said:

I think that I could sneak "platypus" into my next publication. :twisted:

 

Can you? :twisted:

 

:D

 

Theirs was a totes awesome bet. :cake: 

... I dunno. You are indeed one of my role models, and much more established in your field. I'd have to pass it by the whole collaboration after all :P

 

But I'm down for a challenge. If we ever did do a bet though, I'd have to ask it to be "use platypus in a publication" instead of the next publication. After all, no one really takes students seriously to start with; I think I'd need to become a researcher in my own right, like you, before I could get that past reviewers ;)

 

2 hours ago, Deus Ex Zero said:

I don't think so but it's never too late as they say :D

Well then, nice to meet you :D

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Guest Deus Ex Infinity
3 minutes ago, Heart said:

... I dunno. You are indeed one of my role models, and much more established in your field. I'd have to pass it by the whole collaboration after all :P

 

But I'm down for a challenge. If we ever did do a bet though, I'd have to ask it to be "use platypus in a publication" instead of the next publication. After all, no one really takes students seriously to start with; I think I'd need to become a researcher in my own right, like you, before I could get that past reviewers ;)

 

Well then, nice to meet you :D

Same here - it's my pleasure :)

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That might be a deal, Heart.

 

My upcoming publication is not, however, a physics paper. It is a novel, out on Amazon on 14 July. I am writing its sequel presently (it would be easy to insert "platypus" into a novel :twisted: ). And I still need to finish my book on the Riemann Hypothesis. Perhaps I could have some math function act as a platypus.

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1 hour ago, Kelly said:

Perhaps I could have some math function act as a platypus.

Maybe one that appears to made of random parts of other equations, or appears to be sort of an "odd one out" in its group? :)

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The 14th of July?!? You have to promise me you'll send me a link so I can order it!! I'm so excited!!!

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Calligraphette_Coe

The Platypus Conjecture: a theory on how to cause magnetic dipoles to align well above the Curie Temperature using AVEN cake fields to induce symmetry at room temperature?

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8 hours ago, Calligraphette_Coe said:

The Platypus Conjecture: a theory on how to cause magnetic dipoles to align well above the Curie Temperature using AVEN cake fields to induce symmetry at room temperature?

New career goal. :wub:

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On 22.06.2017 at 2:42 AM, Dodecahedron314 said:

I'd heard about this in the past, but I didn't know this was one of your old profs, Heart! Wow, science really is a small world sometimes.

It is... it is...

 

Hey! Hey! I'm switching to a math degree!

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19 minutes ago, Emery. said:

It is... it is...

 

Hey! Hey! I'm switching to a math degree!

 

Wait, actually? :D

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Just now, Heart said:

 

Wait, actually? :D

Yes :) Applied math.

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Just now, Emery. said:

Yes :) Applied math.

Welcome to the nerdier side of life :D We have cake.

 

8BKBTD+y60cY0AAAAASUVORK5CYII=

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Well, I was in physics and computer science before , so... I had a taste of a theoretical math too (a minor).

But thank you for the cake regardless :)

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