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Maybe I'm a bit late to the party here, probably because I'm an Engineer
rather than a Mathematician, but this seemed a pretty crazy "proof" of
what you get if you sum all the natural numbers up:
s= 1+2+3+4+5+6+...
4s= 4+8+12+16+...
(s-4s) = 1+2+3+4+5+ 6+...
-4 -8 -12-...
-3s = 1-2+3-4+5-6+...
-3s-3s = 1-2+3-4+5-6+...
+1-2+3-4+5-6+...
-6s = 1-1+1-1+1-1+1-...
1-(-6s)= 1-(1-1+1-1+1-1+1-...)
= 1-1+1-1+1-1+1-...
= -6s
1+6s = -6s
12s = -1
s = -1/12
Crazy huh?
https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
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Le 27/07/2015 11:19, scott a écrit :
> Maybe I'm a bit late to the party here, probably because I'm an Engineer
> rather than a Mathematician, but this seemed a pretty crazy "proof" of
> what you get if you sum all the natural numbers up:
>
> s= 1+2+3+4+5+6+...
>
Nah, s does *NOT* converge, insisting that s exists get you what you
deserve: bullshit (unless you are interested in classification of
divergent series).
Same as asking the maximal value of a Dirac function... lovely object of
theory, no practical existence.
> https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
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>> s= 1+2+3+4+5+6+...
>>
>
> Nah, s does *NOT* converge, insisting that s exists get you what you
> deserve: bullshit (unless you are interested in classification of
> divergent series).
It does seem absurd, that the result comes out negative and less than
even the smallest term in the sequence. Saying that though the steps
seem logical enough (from a practical point of view rather than a
mathematical point of view) to come to the answer of -1/12.
> Same as asking the maximal value of a Dirac function... lovely object of
> theory, no practical existence.
I thought of that as a curve with area underneath equal to unity with no
width (so height has to be infinite).
But according to the wikipedia page below the -1/12 thing does have some
practical uses? I couldn't find any actual information about those
practical uses though.
>> https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
>
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> But according to the wikipedia page below the -1/12 thing does have some
> practical uses? I couldn't find any actual information about those
> practical uses though.
>
>>> https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
>>
Mathematicians are crazy! Practical uses to them means something
different. What? I don't know, I'm not a Mathematician. But I do love to
play with numbers!
This part
(s-4s) = 1+2+3+4+5+ 6+...
-4 -8 -12-...
-3s = 1-2+3-4+5-6+...
Makes me think: Infinity strikes again!
Even though 4s and s can be put in one to one correspondence to Infinity
In real life you have s=n(n+1)/2 and you choose the n.
So using the grouping as above there would always be some of the 4s left
over.
Infinity is tricky! That's my two cents.
Have Fun!
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On 27/07/2015 01:56 PM, scott wrote:
>>> s= 1+2+3+4+5+6+...
>>>
>>
>> Nah, s does *NOT* converge, insisting that s exists get you what you
>> deserve: bullshit (unless you are interested in classification of
>> divergent series).
>
> It does seem absurd, that the result comes out negative and less than
> even the smallest term in the sequence. Saying that though the steps
> seem logical enough (from a practical point of view rather than a
> mathematical point of view) to come to the answer of -1/12.
It seems the idea is to replace Sum[n] with Sum[n^-s], which is the
definition of the Riemann zeta function. The new series doesn't converge
for the value of interest, but by analytic continuation you can figure
out a suitable value that makes it "fit in with" the other values.
It's a little like... what is b^0.5? How do you multiply something by
itself half a time? That doesn't even make *sense*! But if you
extrapolate from the values that *do* make sense... you come to a simple
and even rather useful result.
> But according to the wikipedia page below the -1/12 thing does have some
> practical uses? I couldn't find any actual information about those
> practical uses though.
Well, as "practical" as the Riemann zeta function I guess...
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On 27/07/2015 10:19 AM, scott wrote:
> s= 1+2+3+4+5+6+...
>
> s = -1/12
>
> Crazy huh?
If you think that's mad, watch this:
10 + 4 = 2
2 / 5 = 10
Wait, whaaat?!
Well now, let's try that again. If the time is currently 10 PM, then
what time will it be in 4 hours' time? Hint: not 14 PM.
In "normal" arithmetic, claiming that 10 + 4 = 2 is just flat wrong. But
change your definitions (say, to agree that after counting past 12 we go
back to 1 again), and suddenly this makes a whole lot of sense, and is
"useful" in that billions of people do this exact type of calculation
all over the world every single day. It doesn't get much more
"practical" than that.
To convince yourself that 2 / 5 = 10, start at 12 o'clock, and keep
adding on 5 hours until you land on 2 o'clock. I promise you, it takes
10 steps to do this. Hence, 5 * 10 = 2, and therefore surely 2 / 5 = 10.
This latter type of shenanigans is mostly used in cryptography and
number theory, but does also pop up in places like error-correcting
codes. (If you've ever tried to scan a bar code or play a CD, you care
about error-correcting codes.)
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On Mon, 27 Jul 2015 18:47:22 +0100, Orchid Win7 v1 wrote:
> It's a little like... what is b^0.5? How do you multiply something by
> itself half a time?
Isn't that called a square root?
Jim
--
"I learned long ago, never to wrestle with a pig. You get dirty, and
besides, the pig likes it." - George Bernard Shaw
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>> But according to the wikipedia page below the -1/12 thing does have some
>> practical uses? I couldn't find any actual information about those
>> practical uses though.
>
> Well, as "practical" as the Riemann zeta function I guess...
Yes I suppose "complex analysis, quantum field theory, and string
theory" are all quite theoretical non-practical things (from an
Engineering point of view). I was hoping it would be something like
complex numbers, that do actually have real world proper practical uses
(like analysing AC circuits or mechanical vibrations).
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scott <sco### [at] scott com> wrote:
> Maybe I'm a bit late to the party here, probably because I'm an Engineer
> rather than a Mathematician, but this seemed a pretty crazy "proof" of
> what you get if you sum all the natural numbers up:
> s= 1+2+3+4+5+6+...
> 4s= 4+8+12+16+...
> (s-4s) = 1+2+3+4+5+ 6+...
> -4 -8 -12-...
> -3s = 1-2+3-4+5-6+...
> -3s-3s = 1-2+3-4+5-6+...
> +1-2+3-4+5-6+...
> -6s = 1-1+1-1+1-1+1-...
Which is equal to:
-6s = (1-1)+(1-1)+(1-1)+...
= 0+0+0+0+... = 0
s = 0/-6 = 0
Therefore:
1+2+3+4+5+6+... = 0
Crazy, huh?
--
- Warp
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>> -6s = 1-1+1-1+1-1+1-...
>
> Which is equal to:
>
> -6s = (1-1)+(1-1)+(1-1)+...
> = 0+0+0+0+... = 0
I'm no mathematician, but to do that you must make the assumption that
there are an even number of terms in the infinite sum (ie every +1 has a
-1 to pair with it). You could have assumed an odd number of terms and
got a sum of 1 instead.
Writing the sum equals 1 minus the sum seems to avoid the need to make
such an assumption.
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scott <sco### [at] scottcom> wrote:
> >> -6s = 1-1+1-1+1-1+1-...
> >
> > Which is equal to:
> >
> > -6s = (1-1)+(1-1)+(1-1)+...
> > = 0+0+0+0+... = 0
> I'm no mathematician, but to do that you must make the assumption that
> there are an even number of terms in the infinite sum (ie every +1 has a
> -1 to pair with it). You could have assumed an odd number of terms and
> got a sum of 1 instead.
> Writing the sum equals 1 minus the sum seems to avoid the need to make
> such an assumption.
An infinite sum can't have an "odd" or an "even" number of terms.
But you bring a good point. If you pair the elements differently, you get:
-6s = 1+(-1+1)+(-1+1)+(-1+1)+...
= 1+0+0+0+0+... = 1
Therefore s = -1/6.
In fact, you can get basically any integer value you want for -6s when
you group the elements appropriately.
This goes to show that when you are dealing with infinities, you can
"prove" anything you want.
I think the original "proof" is bogus.
The "proof" using Riemann's zeta function is also bogus in a sense.
Riemann's zeta function is the infinite sum of 1/n^s, but only for
values of s so that Real(s) > 1. (The sum would give 1+2+3+4+... when
s = -1, but the zeta function is not 1/n^s for values of s < 1. It's
something a lot more complicated.)
--
- Warp
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>> I'm no mathematician, but to do that you must make the assumption that
>> there are an even number of terms in the infinite sum (ie every +1 has a
>> -1 to pair with it). You could have assumed an odd number of terms and
>> got a sum of 1 instead.
>
>> Writing the sum equals 1 minus the sum seems to avoid the need to make
>> such an assumption.
>
> An infinite sum can't have an "odd" or an "even" number of terms.
Yes that was my point, "grouping" in any way is invalid because you must
make assumptions about the total number of terms, which you can't for an
infinite list.
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scott <sco### [at] scottcom> wrote:
> >> I'm no mathematician, but to do that you must make the assumption that
> >> there are an even number of terms in the infinite sum (ie every +1 has a
> >> -1 to pair with it). You could have assumed an odd number of terms and
> >> got a sum of 1 instead.
> >
> >> Writing the sum equals 1 minus the sum seems to avoid the need to make
> >> such an assumption.
> >
> > An infinite sum can't have an "odd" or an "even" number of terms.
> Yes that was my point, "grouping" in any way is invalid because you must
> make assumptions about the total number of terms, which you can't for an
> infinite list.
I don't think that's how it works. (If it were, then that original "proof"
would be invalid from the get-go, because it's grouping elements and
summing those groups.)
--
- Warp
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>> Yes that was my point, "grouping" in any way is invalid because you must
>> make assumptions about the total number of terms, which you can't for an
>> infinite list.
>
> I don't think that's how it works. (If it were, then that original "proof"
> would be invalid from the get-go, because it's grouping elements and
> summing those groups.)
There's no grouping like you did in the original proof. Which part of
the original proof assumes the length of the summation is anything other
than infinite?
Welcome back BTW :-)
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Am 27.07.2015 um 19:54 schrieb Orchid Win7 v1:
> This latter type of shenanigans is mostly used in cryptography and
> number theory, but does also pop up in places like error-correcting
> codes. (If you've ever tried to scan a bar code or play a CD, you care
> about error-correcting codes.)
Nobody cares about error-correcting codes when playing an audio CD.
Unlike DVD or even data CDs (aka CD-ROMs), Sony's audio CD format
doesn't waste any data capacity on bit error recovery.
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Am 27.07.2015 um 11:19 schrieb scott:
> https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
You may also like the proof that all triangles are equilateral:
https://youtu.be/Yajonhixy4g
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On 31/07/2015 11:22 PM, clipka wrote:
> Am 27.07.2015 um 19:54 schrieb Orchid Win7 v1:
>
>> This latter type of shenanigans is mostly used in cryptography and
>> number theory, but does also pop up in places like error-correcting
>> codes. (If you've ever tried to scan a bar code or play a CD, you care
>> about error-correcting codes.)
>
> Nobody cares about error-correcting codes when playing an audio CD.
> Unlike DVD or even data CDs (aka CD-ROMs), Sony's audio CD format
> doesn't waste any data capacity on bit error recovery.
In fact, the audio CD format uses cross-interleaved Reed-Solomon codes
for error recovery.
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Am 01.08.2015 um 11:24 schrieb Orchid Win7 v1:
> On 31/07/2015 11:22 PM, clipka wrote:
>> Am 27.07.2015 um 19:54 schrieb Orchid Win7 v1:
>>
>>> This latter type of shenanigans is mostly used in cryptography and
>>> number theory, but does also pop up in places like error-correcting
>>> codes. (If you've ever tried to scan a bar code or play a CD, you care
>>> about error-correcting codes.)
>>
>> Nobody cares about error-correcting codes when playing an audio CD.
>> Unlike DVD or even data CDs (aka CD-ROMs), Sony's audio CD format
>> doesn't waste any data capacity on bit error recovery.
>
> In fact, the audio CD format uses cross-interleaved Reed-Solomon codes
> for error recovery.
Damn, I hate to stand corrected.
But there was /something/ with regards to error recovery that CD-ROMs
have but CD-DAs don't.
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On 8/1/2015 4:36 PM, clipka wrote:
>> In fact, the audio CD format uses cross-interleaved Reed-Solomon codes
>> for error recovery.
>
> Damn, I hate to stand corrected.
Write this Date in our calenders.
On the first of August two thousand and fourteen. Clipka admitted he was
wrong. :-P
--
Regards
Stephen
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On 01/08/2015 04:36 PM, clipka wrote:
> Am 01.08.2015 um 11:24 schrieb Orchid Win7 v1:
>> In fact, the audio CD format uses cross-interleaved Reed-Solomon codes
>> for error recovery.
>
> Damn, I hate to stand corrected.
>
> But there was /something/ with regards to error recovery that CD-ROMs
> have but CD-DAs don't.
Yes, I have that vague recollection as well. I should think ISO-9660
probably has block-level checksums or similar, to allow corruption to be
detected. For audio CDs, the player is supposed to just fill in any
unreadable chunks with silence.
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Am 01.08.2015 um 17:49 schrieb Stephen:
> On 8/1/2015 4:36 PM, clipka wrote:
>>> In fact, the audio CD format uses cross-interleaved Reed-Solomon codes
>>> for error recovery.
>>
>> Damn, I hate to stand corrected.
>
> Write this Date in our calenders.
> On the first of August two thousand and fourteen. Clipka admitted he was
> wrong. :-P
No I didn't. Not back /then/. :-P
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On 01/08/2015 08:25 PM, clipka wrote:
> Am 01.08.2015 um 17:49 schrieb Stephen:
>> Write this Date in our calenders.
>> On the first of August two thousand and fourteen. Clipka admitted he was
>> wrong. :-P
>
> No I didn't. Not back /then/. :-P
Haha, 0wn3d!
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On 8/1/2015 8:25 PM, clipka wrote:
> Am 01.08.2015 um 17:49 schrieb Stephen:
>> On 8/1/2015 4:36 PM, clipka wrote:
>>>> In fact, the audio CD format uses cross-interleaved Reed-Solomon codes
>>>> for error recovery.
>>>
>>> Damn, I hate to stand corrected.
>>
>> Write this Date in our calenders.
>> On the first of August two thousand and fourteen. Clipka admitted he was
>> wrong. :-P
>
> No I didn't. Not back /then/. :-P
>
Bugrit!
--
Regards
Stephen
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On 8/1/2015 8:47 PM, Orchid Win7 v1 wrote:
> On 01/08/2015 08:25 PM, clipka wrote:
>> Am 01.08.2015 um 17:49 schrieb Stephen:
>>> Write this Date in our calenders.
>>> On the first of August two thousand and fourteen. Clipka admitted he was
>>> wrong. :-P
>>
>> No I didn't. Not back /then/. :-P
>
> Haha, 0wn3d!
--
Regards
Stephen
Post a reply to this message
Attachments:
Download 'something 1.wav.dat' (217 KB)
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On 8/1/2015 9:07 PM, Stephen wrote:
> ""
A prize if anyone recognises the voice of the actor.
--
Regards
Stephen
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scott <sco### [at] scottcom> wrote:
> There's no grouping like you did in the original proof. Which part of
> the original proof assumes the length of the summation is anything other
> than infinite?
This part of the proof is grouping elements in pairs and summing them up:
(s-4s) = 1+2+3+4+5+ 6+...
-4 -8 -12-...
-3s = 1-2+3-4+5-6+...
> Welcome back BTW :-)
I had some problems with the dreaded "can't get fully qualified domain
name" error (which was incidentally solved by ticking one checkbox in
an obscure system setting. But damned it was hard to figure that out.)
--
- Warp
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Warp <war### [at] tagpovrayorg> wrote:
> scott <sco### [at] scottcom> wrote:
> > There's no grouping like you did in the original proof. Which part of
> > the original proof assumes the length of the summation is anything other
> > than infinite?
>
> This part of the proof is grouping elements in pairs and summing them up:
>
> (s-4s) = 1+2+3+4+5+ 6+...
> -4 -8 -12-...
> -3s = 1-2+3-4+5-6+...
>
> > Welcome back BTW :-)
>
> I had some problems with the dreaded "can't get fully qualified domain
> name" error (which was incidentally solved by ticking one checkbox in
> an obscure system setting. But damned it was hard to figure that out.)
>
> --
> - Warp
As my knowledge of mathematics only goes up to differential equations (and is
more than somewhat spotty around the edges) I have no idea what zeta functions
are, so I had to actually do some research on why this astoundingly bad
wikipedia article is also astoundingly wrong. Or as they say on TV Tropes
(Warning: Timesink) Not Even Wrong.
https://plus.maths.org/content/infinity-or-just-112
Firstly, the article title is confusing as hell. Secondly, it opens by stating
that the sum of the natural numbers is equal to a value lower than the smallest
term in that sequence. It then purports to offer proof of this concept which is
akin to taking a true statement and tacking on another true statement which
leads to a logical result that is mathematically correct, given the whole, but
which has nothing at all to do with the original premise of the foundational
statement.
So it turns out that the result is mathematically valid, and relates to the
Casimir Effect. The problem I have with it is the astounding level of
intellectual irresponsibility in conflating the sum of Natural Numbers with a
result that clearly requires advanced analytical mathematics that few of the
people reading it will know about. Was this done intentionally in order to make
people research the topic? If so, then it's still quite unethical, since not
everybody will research it properly, and those with only a vague understanding
of the first principles will be confused by it. It's bad enough that the
youtube video presented it the way it did, but the fact that it's now on
wikipedia where the first line of the article can be taken as fact without
considering the rest of the article, and the underlying math and physics
principles, is inexcusable.
Regards,
A.D.B.
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>> There's no grouping like you did in the original proof. Which part of
>> the original proof assumes the length of the summation is anything other
>> than infinite?
>
> This part of the proof is grouping elements in pairs and summing them up:
>
> (s-4s) = 1+2+3+4+5+ 6+...
> -4 -8 -12-...
> -3s = 1-2+3-4+5-6+...
This one is different, as there is no assumption that the "..."
(infinite list) has any other properties other than "it goes on
forever". So long as both parts of the sum "go on forever" then there
will always a pair for each item.
However if you try and group elements like:
s = 1-1+1-1+1-1+1-...
s = (1-1)+(1-1)+(1-1)+(1-1)+...
s = 0+0+0+...
s = 0
Then the "..." in the 3rd (and perhaps 2nd) line makes the assumption
that there are an even number of terms, that the series ends in a "-1".
That (I think) is a wrong assumption, an infinite list/sum doesn't have
any concept of a "last" item.
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scott <sco### [at] scottcom> wrote:
> However if you try and group elements like:
> s = 1-1+1-1+1-1+1-...
> s = (1-1)+(1-1)+(1-1)+(1-1)+...
> s = 0+0+0+...
> s = 0
> Then the "..." in the 3rd (and perhaps 2nd) line makes the assumption
> that there are an even number of terms
No, it doesn't. It simply makes the assumption that you can choose
each odd-placed and even-placed number in the series (which is true)
and sum them together (which is also true). This can be done forever.
--
- Warp
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>> s = 1-1+1-1+1-1+1-...
>> s = (1-1)+(1-1)+(1-1)+(1-1)+...
>> s = 0+0+0+...
>> s = 0
>
>> Then the "..." in the 3rd (and perhaps 2nd) line makes the assumption
>> that there are an even number of terms
>
> No, it doesn't. It simply makes the assumption that you can choose
> each odd-placed and even-placed number in the series (which is true)
> and sum them together (which is also true). This can be done forever.
Yes, I just realised that you could also write s as:
s =+1+1+1+1+1+...
-1-1-1-1-1-...
= 0+0+0+0+0+...
Which makes no such assumptions.
But then you could probably just as validly (which might not be valid at
all) write s as:
s = +1+1+1+1+1+1+...
-1-1-1-1-...
= 1+1+0+0+0+0+...
So essentially you could "prove" any value you like for s.
Funnily enough if you use the standard formula for the infinite sum of
geometric progressions, you also get 1/2.
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>> https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
>
> You may also like the proof that all triangles are equilateral:
>
> https://youtu.be/Yajonhixy4g
Doesn't that come under a different category of just being a trick/hoax
though (a bit like all the 1=2 type "proofs")? As opposed to this
assuming 1+2+3+...=-1/12 thing is actually useful in other areas of
maths and science.
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Le 2015-07-27 05:19, scott a écrit :
> Maybe I'm a bit late to the party here, probably because I'm an Engineer
> rather than a Mathematician, but this seemed a pretty crazy "proof" of
> what you get if you sum all the natural numbers up:
>
> s= 1+2+3+4+5+6+...
>
> 4s= 4+8+12+16+...
>
> (s-4s) = 1+2+3+4+5+ 6+...
> -4 -8 -12-...
> -3s = 1-2+3-4+5-6+...
^
TYPO +2. Not -2.
Likewise for +6, +10, +14...
So:
-3s = 1+2+3+(4-4)+5+6+7+(8-8)+9+10+11+...
-3s = 1+2+3+(0)+5+6+7+(0)+9+10+11+...
> -3s-3s = 1-2+3-4+5-6+...
> +1-2+3-4+5-6+...
> -6s = 1-1+1-1+1-1+1-...
No.
-6s = 2+4+6+10+12+14+18+20+22...
Then the rest is wrong.
>
> 1-(-6s)= 1-(1-1+1-1+1-1+1-...)
> = 1-1+1-1+1-1+1-...
> = -6s
> 1+6s = -6s
> 12s = -1
>
> s = -1/12
>
> Crazy huh?
>
> https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
--
/*Francois Labreque*/#local a=x+y;#local b=x+a;#local c=a+b;#macro P(F//
/* flabreque */L)polygon{5,F,F+z,L+z,L,F pigment{rgb 9}}#end union
/* @ */{P(0,a)P(a,b)P(b,c)P(2*a,2*b)P(2*b,b+c)P(b+c,<2,3>)
/* gmail.com */}camera{orthographic location<6,1.25,-6>look_at a }
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scott <sco### [at] scottcom> wrote:
> >> https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF
> >
> > You may also like the proof that all triangles are equilateral:
> >
> > https://youtu.be/Yajonhixy4g
>
> Doesn't that come under a different category of just being a trick/hoax
> though (a bit like all the 1=2 type "proofs")? As opposed to this
> assuming 1+2+3+...=-1/12 thing is actually useful in other areas of
> maths and science.
No professor I've ever met would accept this statement as true without the
intermediate theorems which would show (if I'm understanding correctly) that the
intent is to subtract infinity from the sum of all natural numbers.
As written, this is a false premise.
The people that made the video -knew- that they were oversimplifying the
premise. They did this to create a mystery where there was no mystery in order
to "Engage the wider public". The idea was that people would research the
topics more fully in order to understand how this could be, but the problem is
that this whole topic is useless unless you're working with quantum mechanics,
in which case you have a great deal more knowledge about mathematics, and this
becomes a carny trick.
There are plenty of interesting areas of mathematics that can be showcased to do
what they were attempting to do without resorting to mathematical slight of
hand.
Regards,
A.D.B.
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Le 04/08/2015 22:46, Francois Labreque a écrit :
>>
>> (s-4s) = 1+2+3+4+5+ 6+...
>> -4 -8 -12-...
>> -3s = 1-2+3-4+5-6+...
> ^
> TYPO +2. Not -2.
> Likewise for +6, +10, +14...
>
>
> So:
>
> -3s = 1+2+3+(4-4)+5+6+7+(8-8)+9+10+11+...
> -3s = 1+2+3+(0)+5+6+7+(0)+9+10+11+...
You are on something.
It was not a typo per itself, but the intent to make the -4s part more
dense than the s part (so as to remove the 4s every 2 terms of s,
instead of nullifying every 4 terms).
Of course, such intent is dishonest when dealing with infinite number of
terms. Is ((s -2s) -2s ) more honest ?
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> The people that made the video -knew- that they were oversimplifying the
> premise. They did this to create a mystery where there was no mystery in order
> to "Engage the wider public". The idea was that people would research the
> topics more fully in order to understand how this could be, but the problem is
> that this whole topic is useless unless you're working with quantum mechanics,
> in which case you have a great deal more knowledge about mathematics, and this
> becomes a carny trick.
It worked though - I only actually found the video after a friend at
work sent me the "proof" and started researching further. Learning a bit
more maths is never a bad thing :-)
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scott <sco### [at] scottcom> wrote:
> Doesn't that come under a different category of just being a trick/hoax
> though (a bit like all the 1=2 type "proofs")? As opposed to this
> assuming 1+2+3+...=-1/12 thing is actually useful in other areas of
> maths and science.
I'm still not sure how valid it is to use "1+2+3+..." here, even when
talking about physics.
The so-called "Euler zeta function" is the infinite sum, where n goes
from 1 to infinity, of 1/n^s, where 's' is a real number. This infinite
sum converges to a finite value for any value of s > 1. For any value
of s <= 1 the sum converges to infinity (and thus is undefined).
Bernhard Riemann had an epiphany about said function when he was
studying it (something about it being related to the density of
prime numbers), and he extended it for all complex values of s.
The infinite sum still converges to a finite value when the real
part of s is > 1 (the imaginary part can be anything), and to infinity
when the real part is <= 1 (and thus is undefined.)
There is a way, however, to extend such functions to cover the entire
complex plane in such a manner that the result is still the same for
all Real(s)>1, but defined for all the remaining complex values of s
as well (except for the single singularity at s=1+0i, which remains
undefined).
This so-called analytical continuation of the Euler zeta function is
the so-called Riemann zeta function. Said function gives the exact
same values as the former for all Real(s)>1. However, the function
is rather different from the much simpler Euler zeta function. It's
not the same function.
It turns out that the Riemann zeta function gives a value of -1/12
when s = -1. If you were to plug s = -1 into the Euler zeta function,
you would get the infinite sum 1+2+3+4+5... (try it to see.)
However, the Euler zeta function is *not* the Riemann zeta function.
They give different results for all Real(s) <= 1. When you plug s=-1
into the Riemann zeta function, you are *not* getting 1+2+3+4+5+...
You are getting something completely different (something that results
in -1/12).
Why they are somehow considered "equal", I don't understand. (I'm not
a mathematician.)
--
- Warp
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