The Most Counter-Intuitive Result In Mathematics

[the_lock_man]

Now, I'd have thought that if you add all the integer numbers from 1 to infinity, then the answer would infinity.
 
But no, apparently, the answer is -1/12 (minus one-twelfth).
 
Bizarre, but here is the explanation.
 
https/www.youtube.com/watch?v=w-I6XTVZXww

 

[fluttermoth]

I've watched that twice now, and I still don't really understand it!  
 
I love maths, but my brain doesn't. It insists that it's too hard and tries to shut down 

 

[daizeUK]

I'm glad I had a beer in my hand before trying to understand this.

 

[malfunction]

Made my brain sore!

 

[Ch4rlie]

Was that maths? 

 
it was just...............it was just...............

 

[Freedom]

Love it! 

  I was a Mathematics major in college, spent ages 'playing' in 13 dimensional space.  11th dimension was alright, but 13 was awesome!

 

[Donya]

The reason this looks bizarre is that their "sum" is not how it would be treated in other mathematical fields, and they are not making that clear. What they are doing is a valid operation, but it's a different meaning to the big sigma symbol than would be the case for other math-related fields, since it's really a limit or average of partial summations and not just a simply tally. It looks less bizarre when done more formally: http/en.wikipedia.org/wiki/Ces%C3%A0ro_summation (first example is the +/-1 series from the video).

 

[TwoTankAmin]

This is my idea of math- 7x13 = 28
 
[media]https/www.youtube.com/watch?v=XnICFjDn97o[/media]

 

[malfunction]

The reason this looks bizarre is that their "sum" is not how it would be treated in other mathematical fields, and they are not making that clear. What they are doing is a valid operation, but it's a different meaning to the big sigma symbol than would be the case for other math-related fields, since it's really a limit or average of partial summations and not just a simply tally. It looks less bizarre when done more formally: http/en.wikipedia.org/wiki/Cesàro_summation (first example is the +/-1 series from the video).

Am pretty sure that just broke my brain. What I can't get my head around is that the sum of all numbers to infinity is smaller than the sum of all numbers to any large number (which by definition is smaller than infinity). Huh??

This is my idea of math- 7x13 = 28
 
[media]https/www.youtube.com/watch?v=XnICFjDn97o[/media]

That gent clearly has a promising career in our government!

 

[Donya]

 

The reason this looks bizarre is that their "sum" is not how it would be treated in other mathematical fields, and they are not making that clear. What they are doing is a valid operation, but it's a different meaning to the big sigma symbol than would be the case for other math-related fields, since it's really a limit or average of partial summations and not just a simply tally. It looks less bizarre when done more formally: http/en.wikipedia.org/wiki/Cesàro_summation (first example is the +/-1 series from the video).

Am pretty sure that just broke my brain. What I can't get my head around is that the sum of all numbers to infinity is smaller than the sum of all numbers to any large number (which by definition is smaller than infinity). Huh?

?
 

 
The problem is that the guys in the physics video are being informal/sloppy in their notation, and anyone who isn't already familiar with those particular notation abuses will be confused as a result (which is probably intentional - otherwise it wouldn't look half as clever). The standard definition of a sum of a set of numbers is just adding them together - but that's not what they're doing. Their "sum" is really more like averaging when they explain what they're actually calculating. Although this is an extreme analogy, it's a bit like if I said 1+1=3 by redefining the "a+b" operation to mean a+b+1. If you watch/listen carefully when they're going through the first example, they do verbally slip in the fact that an averaging-like thing is involved rather than a standard sum, but that critical part gets shoved under the carpet straight away. When it's not shoved under the carpet, as in the wiki page, you get a much uglier looking statement involving limits. 

 

[malfunction]

Ok, I get that they're not simply adding the numbers together in the conventional sense - they're taking averages instead. But why does the average add to -1/2? Surely the result of the procedure will vary depending on how long they continue adding averages for? Why do they stop at that particular point? Why is the point they stop at better for calculating the sum of all numbers to infinity than any other point?

 

[snazy]

Their "sum" is really more like averaging when they explain what they're actually calculating.If you watch/listen carefully when they're going through the first example, they do verbally slip in the fact that an averaging-like thing is involved rather than a standard sum, but that critical part gets shoved under the carpet straight away.

No, it's not averaging. You can prove that S=1-1+1-1+1.......=1/2 by adding
S=1-1+1-1+1-1...
+
S= 1-1+1-1+1....
=
2S=1+0+0+0+0+0......=1 therefore 2S=1 therefore S=1/2, so S=S=1-1+1-1+1-1.....=1/2

The forum can't display it correctly, but when adding the two Ss together, just move the 2nd sum a bit further to the right so the second 1 in the first sum matches the first number in the second sum, this way top and bottom add up to 1+0+0+0, etc..

 

[daizeUK]

Yep I think they explain this in their link out video, it made a lot more sense to me after I watched that.
 
The main thing I take from all this is that there is something absolutely fundamental about the universe that we just don't get!

 

[Donya]

Snazy, I said "more like" averaging, not literally that everything is averaged. The guy in the video even says as much. The formalization I linked to involves averaging of partial sums, which is shown on the page I linked.

 

[snazy]

I was referring to the video. There's no averaging involved from what I can tell. They just didn't go into detail why the first sum equals 1/2, which I gave of an example above.