 |
|
|
|
|
|
| |
| |
|
|
|
|
| |
| |
|
|
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
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
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
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
>> 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
>
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
> 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!
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
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...
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
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.)
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
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
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
>> 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).
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
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
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
| |
|
|
>> -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.
Post a reply to this message
|
|
| |
| |
|
|
|
|
| |
