The sum of all natural numbers = -1/12

ack · Dec 16, 2014

I am tired of reading defensive posts when everyone else seems to like the "other" competing products; or about the Joe Newcomer with SO and SO out-of-this-world product that now beats every other established line out there; or why a poster's own system is the best he's ever heard and cannot imagine better...

So, how about some math magic... and this is real, as it shows up in physics (e.g. in understanding the Casimir force), and as you will see, is also used in String Theory... The sum of all natural numbers equals not infinity (infinity is not a number), but -1/12. This is real science, not some ... totally translucent cable

Not convinced? Here's and Euler's proof http://math.ucr.edu/home/baez/qg-winter2004/zeta.pdf and the longer version of the video

No one has been able to find a flaw yet...

FrantzM · Dec 16, 2014

Counterintuitive is the word that conmes to mind.Regardless of the"proof" I cannot fathom how adding positive numbers leads to a negative result, thus a number smaller than the any of the numbers of the series.

That result would reject that if a and b are positive numbers a +b >a and a+b >b

marty · Dec 16, 2014

Interesting but it seems to me the flaw is obvious. Stating that the answer to S1 is determined by whether you stop at an even or odd number is a false premise. By definition, you are not asked or requested to interpret S1 by assuming you can stop at an odd or even number and get an answer in attempting to solve for "S1". Therefore, the answer to S1 is not 1/2 as it is not the average of anything, certainly not the 2 equations he would like you to believe is the same thing as S1 itself. Jeez, is this how they passed Obamacare? No wonder.

ack · Dec 16, 2014

marty said:
Interesting but it seems to me the flaw is obvious. Stating that the answer to S1 is determined by whether you stop at an even or odd number is a false premise. By definition, you are not asked or requested to interpret S1 by assuming you can stop at an odd or even number and get an answer in attempting to solve for "S1". Therefore, the answer to S1 is not 1/2 as it is not the average of anything, certainly not the 2 equations he would like you to believe is the same thing as S1 itself. Jeez, is this how they passed Obamacare? No wonder.

it's not so easy; as he says, what he claims, can be proven. S1 is a divergent series called Grandi's Series - start with http://en.wikipedia.org/wiki/Grandi's_series if you care to read more on it (the series' Cesaro summation is shown here http://en.wikipedia.org/wiki/Cesàro_summation). The first video is the layman's proof, the second uses Euler's and Reimann's approaches. You can't beat this (the Sum of all natural numbers)... if you do, you will be able to disprove String Theory (many have tired many ways, no one yet can), and real physical phenomena.

es347 · Dec 16, 2014

..I am so thankful I'm at a point in my life where I can look back on my field theory classes...with total disdain I might add, and when I come upon a post like this can simply hit the back arrow multiple times and say been there, done that, got the T-shirt. Granted I never tried to disprove String Theory but I certainly paid my dues in those fields classes....shudder..

Mike · Dec 16, 2014

Playing with divergent series could be problematic...I thought that John Baez was a category theorist, not a physicist.

GaryProtein · Dec 16, 2014

Logical fallacies abound with 1+2+3+4+5+6....=-1/12
It is a false proof. There are things you cannot do with zero and infinity.

See these other examples:

One Equals Zero!

The following is a "proof" that one equals zero.
Consider two non-zero numbers x and y such that
x = y.
Then x[SUP]2[/SUP] = xy.
Subtract the same thing from both sides:
x[SUP]2[/SUP] - y[SUP]2[/SUP] = xy - y[SUP]2[/SUP].
Dividing by (x-y), obtain
x + y = y.
Since x = y, we see that
2 y = y.
Thus 2 = 1, since we started with y nonzero.
Subtracting 1 from both sides,
1 = 0.What's wrong with this "proof"?
Presentation Suggestions:
This Fun Fact is a reminder for students to always check when they are dividing by unknown variables for cases where the denominator might be zero.
The Math Behind the Fact:
The problem with this "proof" is that if x=y, then x-y=0. Notice that halfway through our "proof" we divided by (x-y).
For a more subtle "proof" of this kind, see One Equals Zero: Integral Form.

==========================================================

try this:

There are lots of "proofs" that claim to prove something that is obviously not true, like 1 + 1 = 1 or 2 = 1.

All of these "proofs" contain some error that most people aren't likely to notice. The most common trick is to divide an equation by zero, which is not allowed (in fact, you cannot ever divide by zero.) If a "proof" divides by zero, it can "prove" anything it wants to, including false statements. It's important to recognize that while these "proofs" may be funny and cute, they always contain some error, and are therefore not real proofs.

Here's one from the Dr. Math archives:

"Proof" that 1 + 1 = 1a = 1
b = 1
a = b
a[SUP]2[/SUP] = b[SUP]2[/SUP]
a[SUP]2[/SUP] - b[SUP]2[/SUP] = 0
(a-b)(a+b) = 0
(a-b)(a+b)/(a-b) = 0/(a-b)
1(a+b) = 0
(a+b) = 0
1 + 1 = 0
2 = 0
1 = 0
1 + 1 = 1
You'll want to read the detailed discussion in the archives. This false proof relies on dividing both sides of an equation by zero (disguised as a-b), whereas you can divide both sides of an equation by the same thing only as long as you are NOT dividing by zero.

And another:

"Proof": 2 = 1a = b
a[SUP]2[/SUP] = ab
a[SUP]2[/SUP] - b[SUP]2[/SUP] = ab-b[SUP]2[/SUP]
(a-b)(a+b) = b(a-b)
a+b = b
b+b = b
2b = b
2 = 1
What is wrong with this proof? Well, the fact that we got 2 = 1 is proof that the method of getting the solution was not mathematically sound. (Read a detailed discussion and find the fallacy in our archives.)

Here's a similar example from Eric Weisstein's World of Mathematics.

Here's a proof that doesn't use division by zero:

"Proof": 2 = 1-2 = -2
4 - 6 = 1 - 3
4 - 6 + 9/4 = 1 - 3 + 9/4
(2 - 3/2)[SUP]2[/SUP] = (1 - 3/2)[SUP]2[/SUP]
2 - 3/2 = 1 - 3/2
2 = 1
What's wrong with this? Taking square roots requires the use of the double plus-or-minus sign (or absolute values). In this case, the plus sign gives an extraneous result, and the minus sign is the one that gives the right conclusion.

Also see this about zero and infinity:

https://www.youtube.com/watch?v=BRRolKTlF6Q

FrantzM · Dec 17, 2014

GaryProtein said:
Logical fallacies abound with 1+2+3+4+5+6....=-1/12
It is a false proof. There are things you cannot do with zero and infinity.

See these other examples:

One Equals Zero!

The following is a "proof" that one equals zero.
Consider two non-zero numbers x and y such that
x = y.
Then x[SUP]2[/SUP] = xy.
Subtract the same thing from both sides:
x[SUP]2[/SUP] - y[SUP]2[/SUP] = xy - y[SUP]2[/SUP].
Dividing by (x-y), obtain
x + y = y.
Since x = y, we see that
2 y = y.
Thus 2 = 1, since we started with y nonzero.
Subtracting 1 from both sides,
1 = 0.What's wrong with this "proof"?
Presentation Suggestions:
This Fun Fact is a reminder for students to always check when they are dividing by unknown variables for cases where the denominator might be zero.
The Math Behind the Fact:
The problem with this "proof" is that if x=y, then x-y=0. Notice that halfway through our "proof" we divided by (x-y).
For a more subtle "proof" of this kind, see One Equals Zero: Integral Form.

==========================================================

try this:

There are lots of "proofs" that claim to prove something that is obviously not true, like 1 + 1 = 1 or 2 = 1.

All of these "proofs" contain some error that most people aren't likely to notice. The most common trick is to divide an equation by zero, which is not allowed (in fact, you cannot ever divide by zero.) If a "proof" divides by zero, it can "prove" anything it wants to, including false statements. It's important to recognize that while these "proofs" may be funny and cute, they always contain some error, and are therefore not real proofs.

Here's one from the Dr. Math archives:

"Proof" that 1 + 1 = 1a = 1
b = 1
a = b
a[SUP]2[/SUP] = b[SUP]2[/SUP]
a[SUP]2[/SUP] - b[SUP]2[/SUP] = 0
(a-b)(a+b) = 0
(a-b)(a+b)/(a-b) = 0/(a-b)
1(a+b) = 0
(a+b) = 0
1 + 1 = 0
2 = 0
1 = 0
1 + 1 = 1
You'll want to read the detailed discussion in the archives. This false proof relies on dividing both sides of an equation by zero (disguised as a-b), whereas you can divide both sides of an equation by the same thing only as long as you are NOT dividing by zero.

And another:

"Proof": 2 = 1a = b
a[SUP]2[/SUP] = ab
a[SUP]2[/SUP] - b[SUP]2[/SUP] = ab-b[SUP]2[/SUP]
(a-b)(a+b) = b(a-b)
a+b = b
b+b = b
2b = b
2 = 1
What is wrong with this proof? Well, the fact that we got 2 = 1 is proof that the method of getting the solution was not mathematically sound. (Read a detailed discussion and find the fallacy in our archives.)

Here's a similar example from Eric Weisstein's World of Mathematics.

Here's a proof that doesn't use division by zero:

"Proof": 2 = 1-2 = -2
4 - 6 = 1 - 3
4 - 6 + 9/4 = 1 - 3 + 9/4
(2 - 3/2)[SUP]2[/SUP] = (1 - 3/2)[SUP]2[/SUP]
2 - 3/2 = 1 - 3/2
2 = 1
What's wrong with this? Taking square roots requires the use of the double plus-or-minus sign (or absolute values). In this case, the plus sign gives an extraneous result, and the minus sign is the one that gives the right conclusion.

Also see this about zero and infinity:

https://www.youtube.com/watch?v=BRRolKTlF6Q

Thanks Gary!

ack · Dec 17, 2014

There are no tricks and false operations here... but as always, the devil is in the details. If you carefully read the Grandi Series wikipedia article I quoted, you will notice the following:

The above manipulations do not consider what the sum of a series actually means. Still, to the extent that it is important to be able to bracket series at will, and that it is more important to be able to perform arithmetic with them, one can arrive at two conclusions:

The series 1 ? 1 + 1 ? 1 + … has no sum.[SUP][1][/SUP][SUP][2][/SUP]

...but its sum should be 1/2.[SUP][2][/SUP]

In fact, both of these statements can be made precise and formally proven, but only using well-defined mathematical concepts that arose in the 19th century. After the late 17th-century introduction of calculus in Europe, but before the advent of modern rigor, the tension between these answers fueled what has been characterized as an "endless" and "violent" dispute between mathematicians.[SUP][3][/SUP][SUP][4][/SUP]

The highlighted is key. Therefore, the first video is not lying to you, but they don't tell you the entire truth either. If you rely on one proof of the Grandi Series' value, you prove the result; if you rely on the other, well, then there is no proof of the result = -1/12.

This is one of those problems where the final result depends on your approach; you can prove it one way, but you cannot entirely disprove it. Or can you... The most annoying thing to me is the result's applicability in physics. Finally, Euler's approach avoids the Grandi Series, so what's the issue there...

PS: Thankfully, apparently we are not the only ones violently sparring over proofs

BobM · Dec 17, 2014

So, in a nutshell, you can be fooled/tricked when using abstract x's and y's instead of real numbers. Or the corallary, accountants (and Physicists/Mathematicians) can sometimes make the numbers do whatever they want them to do, and unless you have the training and are paying attention, you can be duped.

Wanna buy a timeshare?

ack · Dec 17, 2014

.. or as my wife would say, just buy the damn thing; nope, need to understand what it does and how it works first

PeterA · Dec 17, 2014

FrantzM said:
Counterintuitive is the word that conmes to mind.Regardless of the"proof" I cannot fathom how adding positive numbers leads to a negative result, thus a number smaller than the any of the numbers of the series.

That result would reject that if a and b are positive numbers a +b >a and a+b >b

Thanks Ack for introducing a thread that is not arguing about what happens inside a Stillpoint or a Transparent network box.

Here is my layman's analysis. I'm with Frantz that this is very counterintuitive. But more than that, I would think that the sum of all natural numbers is unknowable, that is, it is uncountable. One would never get to the end of the sum, just as we can never know the last digit in pi.

Also, how can the sum of any natural numbers, let alone, the sum of all natural numbers be something other than a natural number?

If one represents each natural number with a corresponding number of peas, m&m's or anything, the act of counting them would never end and the pile or number of peas created by adding them together would never get smaller, let alone be a negative fraction of a number. That is why the sum is unknowable.

How can the fellow use negative numbers in an equation used to describe natural numbers? If the fellow's two equations for S1 and S2 actually represent what is happening, he would be forced to cut one pea in half for his S1 solution because he presumes an "average" value to make it work and stop the counting. To get to that point, he wants us to accept that we must stop counting the peas. That seems to me to be an unreasonable request and a false presumption.

MadFloyd · Dec 17, 2014

Why doesn't anyone ever discuss audio on audio forums anymore. It's always politics, TV shows, coffee machines or whatever. Now String Theory. Jeez.

Anyone know any REAL audio forums?

Steve Williams · Dec 17, 2014

MadFloyd said:
Why doesn't anyone ever discuss audio on audio forums anymore. It's always politics, TV shows, coffee machines or whatever. Now String Theory. Jeez.

Anyone know any REAL audio forums?

Floyd....and I thought you and PeterA and ack were all friends in the same burbs

microstrip · Dec 17, 2014

MadFloyd said:
(...) Anyone know any REAL audio forums?

I do my best, but no one seems interested anymore in the scientific reason why silver and copper cables sound different ...

PeterA · Dec 17, 2014

Steve Williams said:
Floyd....and I thought you and PeterA and ack were all friends in the same burbs

Indeed we are, except for the times we discuss MIT vs Transparent (Madfloyd seems conflicted on this) or Spectral vs Pass. On Magico (well not S vs Q), most things analog and the BSO, we all agree, I think. And we never discuss "accurate" vs "musical". Just kidding.

ack · Dec 17, 2014

I love this thread... Here's another fun video on Grandi's Series, that includes both another proof that the series converge on 1/2 (math is NOT logic), and a fun experiment showing that the series cannot really have a sum at all... essentially depicting in layman's terms what the wikipedia article says regarding the Series... while still looking to poke holes in Euler's/Reimann's approach.

PeterA · Dec 17, 2014

1-1+1-1...is not a sum of all natural numbers. 1+2+3+4... is a sum of all natural numbers. What am I missing, Ack?

ack · Dec 17, 2014

PeterA said:
1-1+1-1...is not a sum of all natural numbers. 1+2+3+4... is a sum of all natural numbers. What am I missing, Ack?

Read the thread carefully; the sticky point is 'S1' in the original video (this 1-1+1-1+... sequence, aka Gradi's Sequence), based on which he derives the rest of the proof about natural numbers in that video. I made it quite clear I thought that this latest video is about this sequence.

PeterA · Dec 17, 2014

ack said:
Read the thread carefully; the sticky point is 'S1' in the original video (this 1-1+1-1+... sequence, aka Gradi's Sequence), based on which he derives the rest of the proof about natural numbers in that video. I made it quite clear I thought that this latest video is about this sequence.

Yes, I'm talking about that same Gradi's Sequence. I don't think it can be used in the proof of the original video precisely because it is not a sum of all natural numbers. The two premise formulae, S1 and S2, upon which the "proof" in the original video is based, IMO, is the problem. The conclusion, or proof, is based on a faulty premise. Arbitrarily deciding to average the two results of the sequence to get 1/2 is flawed.

I also think S2 is flawed because there is no requirement that all natural numbers have to be added in sequence. If you took 2+3+127865+99+872...and never repeated a number and then applied his trick of shifting the numbers over with a different randomly ordered sequence, the result would be totally different and not result in the 1/4 solution.

I'm left with the obvious analogy of adding a whole bunch of peas and never getting to a final sum, much less one that is a negative fraction and not itself a natural number.

The sum of all natural numbers = -1/12

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