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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] scott com> 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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