1 + 2 + 3 + 4 + ... = -1/12???!!!

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oceanair

971 posts

oceanair

#1 h Sep 8, 2015, 12:21 AM • 4 Y

Y by LLJ, math101010, Adventure10, Mango247

https://www.youtube.com/watch?v=w-I6XTVZXww

uh... can someone tell me what is happening here?

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gamjawon

3496 posts

gamjawon

#2 h Sep 8, 2015, 12:30 AM • 9 Y

Y by _ComputerNerd12_, Iamteehee, abk2015, jonyj1005, alchemis80, Ridvik, math101010, Adventure10, Mango247

Redacted (Remove your downvotes?)
Hm it looks like the video is false (It's like proving $1=2$ )

The first series says
$1-1+1-1+1-1+1-1\cdot\cdot\cdot=\dfrac{1}{2}$

That is false. The sum does not converge.

Also by the arithmetic series test on the original expression, the series diverges.

This post has been edited 3 times. Last edited by gamjawon, Sep 8, 2015, 12:56 AM

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Eugenis

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#3 h Sep 8, 2015, 12:32 AM • 14 Y

Y by gamjawon, checkmatetang, _ComputerNerd12_, ninjasrule34, PiOfLife314, Dot22, Ridvik, MathSlayer4444, math101010, mathmaster2012, StarFrost7, oiler8, Adventure10, Mango247

Hello the series is divergent, therefore it does not converge to a specific value.

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_ComputerNerd12_

196 posts

_ComputerNerd12_

#4 h Sep 8, 2015, 12:44 AM • 2 Y

Y by Adventure10, Mango247

What's with all of the Numberphile posts all of a sudden?

Remember this when doing all weird math: Math = Logic + Arithmetic. Even when Arithmetic may seem correct, you also have to think about Logic.

ex: 0.99999... = 1

This post has been edited 1 time. Last edited by _ComputerNerd12_, Sep 8, 2015, 12:47 AM

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Cube1

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Cube1

#5 h Sep 8, 2015, 12:52 AM • 3 Y

Y by math101010, Adventure10, Mango247

First of all, this equality is true, counter-intuitively. Second, I found this equation in a physics book somewhere.

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Eugenis

2404 posts

Eugenis

#6 h Sep 8, 2015, 12:56 AM • 2 Y

Y by abk2015, Adventure10

_ComputerNerd12_ wrote:

What's with all of the Numberphile posts all of a sudden?

Remember this when doing all weird math: Math = Logic + Arithmetic. Even when Arithmetic may seem correct, you also have to think about Logic.

ex: 0.99999... = 1

$0.\overline{9}=1...$

This post has been edited 1 time. Last edited by Eugenis, Sep 8, 2015, 12:58 AM

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math129

700 posts

math129

#7 h Sep 8, 2015, 12:57 AM • 2 Y

Y by joshua123456, Adventure10

In the video to prove that $1-1+1-1+1...=\frac{1}{2}$ , it says that we can take the average of these sums of 0 and 1, and thus we get $1/2$ . You can't just do that, but I realize that there is an extended version of this proof (https://www.youtube.com/watch?v=PCu_BNNI5x4) , and there is one thing you can't do, and that is use algebra on a divergent series. In the video, it says that we can let the sum be $S$ , so then to prove that the answer is $1/2$ , it says to consider the sum $1-S$ , which you can't do because the series in a divergent series.

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_ComputerNerd12_

196 posts

_ComputerNerd12_

#8 h Sep 8, 2015, 12:59 AM • 1 Y

Y by Adventure10

Eugenis wrote:

Um $0.\overline{9}=1...$

However, by using the same logic, you can go up in tiny increments so that $0.\overline{9}=2$ by adding the tiniest number

Because 1 - 0. $\overline{9}$ = some minute number. You keep on adding that number, because apparently it doesn't matter in your proof that 1 = $0.\overline{9}$ and get that anything = anything

Logic...

This post has been edited 2 times. Last edited by _ComputerNerd12_, Sep 8, 2015, 1:07 AM
Reason: Haters...

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Iamteehee

2050 posts

Iamteehee

#9 h Sep 8, 2015, 1:01 AM • 8 Y

Y by Eugenis, gamjawon, thinkinavi, mathwizard888, TurtlePie, ShineBunny, StarFrost7, Adventure10

If you believe that, then you dont correctly understand it.

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gamjawon

3496 posts

gamjawon

#10 h Sep 8, 2015, 1:02 AM • 3 Y

Y by abk2015, Adventure10, Mango247

Proof of above

Apparently, Chrome does not render numbers on titles of hide tags.

This post has been edited 4 times. Last edited by gamjawon, Sep 8, 2015, 1:05 AM

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Maths4J

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#11 h Sep 8, 2015, 1:05 AM • 2 Y

Y by Adventure10, Mango247

Quote removed

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_ComputerNerd12_

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_ComputerNerd12_

#12 h Sep 8, 2015, 1:05 AM • 2 Y

Y by Adventure10, Mango247

gamjawon wrote:

Proof of above

Did you read my post? Math = Arithmetic + Logic. Arithmetic cant prove something by itself

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Maths4J

738 posts

Maths4J

#13 h Sep 8, 2015, 1:05 AM • 5 Y

Y by abk2015, math101010, StarFrost7, Adventure10, Mango247

And yes, 0.99999... is 1. That's all true.

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Iamteehee

2050 posts

Iamteehee

#14 h Sep 8, 2015, 1:07 AM • 5 Y

Y by gamjawon, ShineBunny, StarFrost7, Adventure10, Mango247

_ComputerNerd12_ wrote:

gamjawon wrote:

Proof of above

Did you read my post? Math = Arithmetic + Logic. Arithmetic cant prove something by itself

That's untrue. While logic is part of mathematics, you don't need logic to prove something. Arithmetic will suffice a lot of the time.
There also isn't any (correct) logic in your "proof" of $0.\overline{9}=2$

This post has been edited 1 time. Last edited by Iamteehee, Sep 8, 2015, 1:08 AM

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WhaleVomit

1452 posts

WhaleVomit

#15 h Sep 8, 2015, 1:13 AM • 4 Y

Y by gamjawon, ShineBunny, StarFrost7, Adventure10

Different proof:
$x=0.\overline{9}$ $10x=9.\overline{9}$ $9x=9$ $x=1$

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Iamteehee

2050 posts

Iamteehee

#16 h Sep 8, 2015, 1:16 AM • 2 Y

Y by Adventure10, Mango247

You might be wondering why we can say $x=0.9+0.09+0.009...$ but we can't say that $x=1-1+1-1..$
It is because the first x converges (approaches a certain value) and the second x diverges (does not approach a certain value) so we can't define $1-1+1-1...$ as x because this x does not exist.

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djmathman

7939 posts

djmathman

#17 h Sep 8, 2015, 1:38 AM • 20 Y

Y by laegolas, ThisIsASentence, abk2015, rkm0959, Eugenis, PiOfLife314, math101010, trumpeter, Mudkipswims42, checkmatetang, mathisawesome2169, bluecarneal, ShineBunny, spartan168, mathwizard888, Einsteinhead, StarFrost7, rafayaashary1, Adventure10, Mango247

^^ Correction: The sum $1-1+1-1+\cdots$ does not diverge - diverge means it would tend toward $\infty$ as the number of terms increased. Instead, it oscillates between $0$ and $1$ infinitely often, so the limit of the partial sums does not exist and thus we can't do the stuff we'd like to do with it.

As for this whole $1+2+3+\cdots=-\tfrac1{12}$ thing, the reason it exists is because of something called the Zeta function. For all $s>1$ , we can define the zeta function as $\zeta(s)=\sum_{n=1}^\infty\dfrac1{n^s}=\dfrac1{1^s}+\dfrac1{2^s}+\dfrac1{3^s}+\cdots.$ For example, $\zeta(2)=\tfrac{\pi^2}6$ . Of course, this is only defined for the positive $s$ ; how do we extend this definition to the negative and complex $s$ ? The existence of such an extension is based off the theory of analytic continuation, which basically says we can extend any analytic function defined along some domain into other domains. (It's kind of like how we can take the graph of $y=x$ only in the first quadrant and naturally extend it to include the negatives as well, only a lot more complex.) I'm not too familiar with the theory behind this (real/complex analysis isn't until sophomore or junior year or something) but a little internet research reveals that working magic on the zeta function allows us to extend its definition to negative real numbers through the formula $\zeta(s)=2^s\pi^{s-1}\sin\left(\dfrac{\pi s}2\right)\Gamma(1-s)\zeta(1-s).$ Here $\Gamma(\cdot)$ is the gamma function, which itself is an analytic continuation of the factorials. From this definition, we get $\zeta(-1)=\tfrac1{12}$ .

So although the normal zeta function doesn't make sense for negative numbers, we can extend the definition of this function through complex analysis to include something like $\zeta(-1)$ . This is what "allows" expressions like $1+2+\cdots$ to have "defined" values.

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FlakeLCR

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FlakeLCR

#18 h Sep 8, 2015, 1:54 AM • 7 Y

Y by MathSlayer4444, champion999, DrMath, math101010, hotstuffFTW, Adventure10, Mango247

gamjawon wrote:

Proof of above

I don't see why someone who doesn't accept the statement $0.\overline{9}=1$ would accept the statement $\dfrac{1}{9}=0.\overline{1}$ .

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math101010

1159 posts

math101010

#20 h Sep 9, 2015, 8:30 PM • 2 Y

Y by Adventure10, Mango247

I found this in the book Things to Make and Do in the Fourth Dimension by Matt Parker.

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TurtlePie

355 posts

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#21 h Sep 9, 2015, 10:31 PM • 2 Y

Y by Adventure10, Mango247

Proof: $S_1=1-1+1-1+1...$ $1-S_1=1-1+1-1+1-1...$ $1-S_1=S_1$ Therefore, $S_1 = \frac{1}{2}$ .

Also, as mentioned in the video, even though the sum of the natural numbers being equal to -1/12 is counter-intuitive, it is completely valid, and has been proven in many different ways.

This post has been edited 2 times. Last edited by TurtlePie, Sep 9, 2015, 10:33 PM

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Kezer

986 posts

Kezer

#22 h Sep 9, 2015, 10:44 PM • 3 Y

Y by Ridvik, Eugenis, Adventure10

djmathman wrote:

^^ Correction: The sum $1-1+1-1+\cdots$ does not diverge - diverge means it would tend toward $\infty$ as the number of terms increased. Instead, it oscillates between $0$ and $1$ infinitely often, so the limit of the partial sums does not exist and thus we can't do the stuff we'd like to do with it.

Actually, that depends on the definition of divergence. Most people use the term divergence as in the term/series/sequence doesn't converge, as in it either tends towards $\pm \infty$ or actually doesn't tend to any value.
I doubt everyone defines it the same way which is why probably why you replied like that; it's similar to the case whether $0$ is a natural number or not.
If I remember correctly, that one time I looked into the book we used in school (Germany; didn't really care about the trivial stuff done in school), convergence was even defined as the term tends to some value including $\infty$ .

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donot

1180 posts

donot

#23 h Sep 9, 2015, 10:47 PM • 2 Y

Y by Ridvik, Adventure10

TurtlePie wrote:

Proof: $S_1=1-1+1-1+1...$ $1-S_1=1-1+1-1+1-1...$ $1-S_1=S_1$ Therefore, $S_1 = \frac{1}{2}$ .

Also, as mentioned in the video, even though the sum of the natural numbers being equal to -1/12 is counter-intuitive, it is completely valid, and has been proven in many different ways.

Tell me why you can't write the sum of the series as $1+S=S$ , which obviously has no solution? Or why you can't multiply the terms in the sequence by $-1$ , and get that $-S=S$ , so $S=0$ ? By the way, the series is called Grandi's which may help to do research.

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Ridvik

129 posts

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#24 h Sep 9, 2015, 10:50 PM • 2 Y

Y by Adventure10, Mango247

i like yours i think the vid is false

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trumpeter

3332 posts

trumpeter

#25 h Sep 9, 2015, 10:54 PM • 10 Y

Y by Eugenis, thinkinavi, gamjawon, TurtlePie, math101010, ninjasrule34, mathwizard888, ShineBunny, StarFrost7, Adventure10

While we cannot say that $1+2+3+4+\ldots$ equals $-\frac{1}{12}$ , we can associate the series with the value $-\frac{1}{12}$ . As djmathman said, in an analytic continuation of the Zeta function, we essentially get that the sum acts like the number $-\frac{1}{12}$ .

About $1-1+1-1+\ldots$ , we also cannot say that it equals $\frac{1}{2}$ , but we can use something called Cesaro Summation to associate the series with $\frac{1}{2}$ .

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_ComputerNerd12_

196 posts

_ComputerNerd12_

#26 h Sep 14, 2015, 9:02 PM • 1 Y

Y by Adventure10

_ComputerNerd12_ wrote:

Eugenis wrote:

Um $0.\overline{9}=1...$

However, by using the same logic, you can go up in tiny increments so that $0.\overline{9}=2$ by adding the tiniest number

Because 1 - 0. $\overline{9}$ = some minute number. You keep on adding that number, because apparently it doesn't matter in your proof that 1 = $0.\overline{9}$ and get that anything = anything

Logic...

... did I get the record for the worst upvote:downvote ratio on a single post?

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_ComputerNerd12_

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_ComputerNerd12_

#27 h Sep 14, 2015, 9:06 PM • 2 Y

Y by Adventure10, Mango247

Xinyan wrote:

I suppose so. But I guess a large part of Greek mathematics was logic.

From this post

HAH!

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champion999

1530 posts

champion999

#29 h Oct 9, 2015, 2:20 AM • 5 Y

Y by ShineBunny, ninjasrule34, StarFrost7, Adventure10, Mango247

_ComputerNerd12_ wrote:

Eugenis wrote:

Um $0.\overline{9}=1...$

However, by using the same logic, you can go up in tiny increments so that $0.\overline{9}=2$ by adding the tiniest number

Because 1 - 0. $\overline{9}$ = some minute number. You keep on adding that number, because apparently it doesn't matter in your proof that 1 = $0.\overline{9}$ and get that anything = anything

Logic...

That "some minute number" is $0$ .

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to_chicken

2470 posts

to_chicken

#30 h Oct 9, 2015, 6:20 AM • 1 Y

Y by Adventure10

_ComputerNerd12_ wrote:

Eugenis wrote:

Um $0.\overline{9}=1...$

However, by using the same logic, you can go up in tiny increments so that $0.\overline{9}=2$ by adding the tiniest number

Because 1 - 0. $\overline{9}$ = some minute number. You keep on adding that number, because apparently it doesn't matter in your proof that 1 = $0.\overline{9}$ and get that anything = anything

Logic...

... did I get the record for the worst upvote:downvote ratio on a single post?

You're not even close to the record.

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tarzanjunior

849 posts

tarzanjunior

#31 h Oct 9, 2015, 7:32 AM • 3 Y

Y by RMO-prep, Adventure10, Mango247

The Arthemetic Mean of 0 and 1 is 1/2

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LLJ

241 posts

LLJ

#32 h Oct 9, 2015, 12:57 PM • 3 Y

Y by Maths4J, Adventure10, Mango247

I like watching these types of videos. Thank you for sharing.
video analysis

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Christopher03151

478 posts

Christopher03151

#33 h Oct 10, 2015, 7:35 PM • 1 Y

Y by Adventure10

We should get like a panel of AoPS smart people (teachers) and professors to tell us what the think of it...

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reaganchoi

5289 posts

reaganchoi

#34 h Oct 13, 2015, 12:28 AM • 2 Y

Y by Adventure10, Mango247

_ComputerNerd12_ wrote:

Eugenis wrote:

Um $0.\overline{9}=1...$

However, by using the same logic, you can go up in tiny increments so that $0.\overline{9}=2$ by adding the tiniest number

Because 1 - 0. $\overline{9}$ = some minute number. You keep on adding that number, because apparently it doesn't matter in your proof that 1 = $0.\overline{9}$ and get that anything = anything

Logic...

The "tiny portion" is actually 0.

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reaganchoi

5289 posts

reaganchoi

#35 h Oct 13, 2015, 12:30 AM • 2 Y

Y by Adventure10, Mango247

Also, if we do 1+2+3+... and subtract it from itself (but shifted one side across) like this:
(We are assuming that 1+2+3+4+5...=-1/12).
1+2+3+4+5.......=-1/12
. . 1+2+3+4.......=-1/12
Subtract to get
1+1+1+1+...=0
Subtract 1 from both sides
1+1+1+1+...=-1
Therefore, 0=-1.

This post has been edited 3 times. Last edited by reaganchoi, Oct 13, 2015, 12:31 AM

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math101010

1159 posts

math101010

#36 h Oct 13, 2015, 12:31 AM • 2 Y

Y by Adventure10, Mango247

what that doesn't make any sense

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StarFrost7

896 posts

StarFrost7

#37 h Oct 13, 2015, 12:34 AM • 2 Y

Y by Adventure10, Mango247

TurtlePie wrote:

Proof: $S_1=1-1+1-1+1...$ $1-S_1=1-1+1-1+1-1...$ $1-S_1=S_1$ Therefore, $S_1 = \frac{1}{2}$ .

Also, as mentioned in the video, even though the sum of the natural numbers being equal to -1/12 is counter-intuitive, it is completely valid, and has been proven in many different ways.

Try replacing $1 - 1$ for $1.$ Shouldn't this be also $\dfrac12$ according to your proof? Yet this yields $0 - S = S,$ a totally different value of $S.$

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ReaperLoser

36 posts

ReaperLoser

#38 h Oct 13, 2015, 12:36 AM • 2 Y

Y by Adventure10, Mango247

Infinite series cannot simply be summed like that.

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abean077

246 posts

abean077

#39 h Oct 13, 2015, 12:36 AM • 3 Y

Y by thinkinavi, Adventure10, Mango247

_ComputerNerd12_ wrote:

Because 1 - 0. $\overline{9}$ = some minute number.

OK, so say you have this number. Then it is 1/10 of itself, right? Because if it is any number it has to also fulfill the case where there is one more 9. But the only real number that is 1/10 of itself is 0.

Basically, if this minute number is nonzero it cannot exist. You can't have a number infinitely close to 0 but not 0 in the real numbers.

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StarFrost7

896 posts

StarFrost7

#40 h Oct 13, 2015, 12:41 AM • 2 Y

Y by Adventure10, Mango247

_ComputerNerd12_ wrote:

What's with all of the Numberphile posts all of a sudden?

Remember this when doing all weird math: Math = Logic + Arithmetic. Even when Arithmetic may seem correct, you also have to think about Logic.

ex: 0.99999... = 1

Sorry to digress, but remember the Hilbert Hotel and Cantor's proof that some infinities are greater than others? In both of these there is almost no logic whatsoever, but it's true.

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DarkDragon

455 posts

DarkDragon

#42 h Oct 13, 2015, 12:48 AM • 1 Y

Y by Adventure10

This is a very interesting video. To me physics is the most favorite part of mathematics and learning this wad very interesting. Thanks this helps me understand string theory much more better.

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reaganchoi

5289 posts

reaganchoi

#43 h Oct 13, 2015, 1:36 AM • 2 Y

Y by Adventure10, Mango247

math101010 wrote:

what that doesn't make any sense

What's wrong?

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