My Math Forum 1+2+3+4... = -1/12, what the fresh hell is this?

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 July 1st, 2011, 07:33 PM #1 Member   Joined: Mar 2011 Posts: 49 Thanks: 1 1+2+3+4... = -1/12, what the fresh hell is this? Hey, I don't even know where to post this, maybe it could be moved to where it belongs. So when I look at very complex mathematical forumals, they look like gibberish to me, and I'm quite happy to find that they are gibberish, but this seems to quite directly conflict with my logic.. so I'm not asking to be explained this in a way that I understand, I would be more than happy if someone could translate this, which in my head right now is a fatal critical error contradiction, into just regular gibberish.. or something like that.. tell me it's something I must be misunderstanding.. or something lol http://en.wikipedia.org/wiki/1_%2B_2_%2 ... _%E2%80%A6 This is the wiki article about it.
 July 1st, 2011, 07:39 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,204 Thanks: 511 Math Focus: Calculus/ODEs Re: 1+2+3+4... = -1/12, what the fresh hell is this? $1+2+3+\cdots=-\frac{1}{12}\(\Re)$ where the notation $(\Re)$ indicates Ramanujan summation. http://en.wikipedia.org/wiki/Ramanujan_summation I moved this to Real Analysis & Topology.
 July 1st, 2011, 11:27 PM #3 Senior Member   Joined: Apr 2007 Posts: 2,140 Thanks: 0 Assuming no advanced mathematics knowledge, wouldn't 1 + 2 + 3 + ... diverge to infinity?
 July 2nd, 2011, 04:41 AM #4 Global Moderator     Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4 Re: 1+2+3+4... = -1/12, what the fresh hell is this? Or it converges to some p-adic number, perhaps
 July 8th, 2011, 01:07 AM #5 Member   Joined: Mar 2011 Posts: 49 Thanks: 1 Re: 1+2+3+4... = -1/12, what the fresh hell is this? Okay, I think I'll leave this as it is for now, I was showed it by someone who just wanted to annoy me xD Thanks from manus
 December 15th, 2015, 08:36 AM #6 Senior Member   Joined: Dec 2015 From: holland Posts: 163 Thanks: 37 Math Focus: tetration S1 = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … = 1/2 this is not true 1-1+1-1....=0 or 1-1+1-1....=1 the outcomes can not be mediated. On the other hand true is .....99999.0 = -1 so 9+90+900+..... = -1 and 1+10+100.....=-1/9
December 15th, 2015, 09:00 AM   #7
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Quote:
 Originally Posted by manus S1 = 1 − 1 + 1 − 1 + 1 − 1 + 1 − 1 + … = 1/2 this is not true 1-1+1-1....=0 or 1-1+1-1....=1 the outcomes can not be mediated.
It very much depends on what you mean by your notation. The sequence of finite sums does not converge, but there is a meaningful sense in which we can attach the value $-{1 \over 12}$

Quote:
 Originally Posted by manus On the other hand true is .....99999.0 = -1
What does this mean?

 December 15th, 2015, 10:48 AM #8 Senior Member   Joined: Dec 2015 From: holland Posts: 163 Thanks: 37 Math Focus: tetration Informally, non-terminating decimals are easily understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating decimal. If two decimal expansions differ only after the 10th decimal place, they are quite close to one another; and if they differ only after the 20th decimal place, they are even closer. 10-adic numbers use a similar non-terminating expansion, but with a different concept of "closeness". Whereas two decimal expansions are close to one another if their difference is a large negative power of 10, two 10-adic expansions are close if their difference is a large positive power of 10. Thus 3333 and 4333, which differ by 10^3, are close in the 10-adic world, and 33333333 and 43333333 are even closer, differing by 10 ^7. More precisely, a positive rational number r can be expressed as 10^e·(p/q), where p and q are positive integers and q is relatively prime to p and to 10. For each r ≠ 0 there exists the maximal e such that this representation is possible. Let the 10-adic absolute value of r to be |r|= 10^-e |0| = 0. Closeness in any number system is defined by a metric. Using the 10-adic metric the distance between numbers x and y is given by |x − y|. An interesting consequence of the 10-adic metric (or of a p-adic metric) is that there is no longer a need for the negative sign. As an example, by examining the following sequence we can see how unsigned 10-adics can get progressively closer and closer to the number −1: 9=-1+10 so |9-(-1)|= 0.1 99=-1+10^2, so |99-(-1)|=0.01 999=-1+10^3 so |999-(-1)|= 0.001 9999=-1+10^4 so |9999-(-1)|=0.0001 and taking this sequence to its limit, we can say that the 10-adic expansion of −1 is ...... 9999=-1 Last edited by manus; December 15th, 2015 at 11:10 AM.
 December 15th, 2015, 11:13 AM #9 Senior Member   Joined: Dec 2015 From: holland Posts: 163 Thanks: 37 Math Focus: tetration 5-adic: -1 = .......444 i = .......12013233 -i = ......32431212
 December 15th, 2015, 01:48 PM #10 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,405 Thanks: 2477 Math Focus: Mainly analysis and algebra That's a completely different number system, so it's rather muddying the waters here.

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