What is 1+2+3+4+5+...?

amirm

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Would you believe one correct answer to the sum of positive integers from 1 to infinity is... -1/12? Turns out it can be proven to be the case using multiple methods!

Here is one:


You can also use Euler's equations to prove the same. Rationalizing it in one's mind is an entirely different matter however :).
 

GaryProtein

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I do believe Peter is correct. I recall seeing this about 1-2 years ago along with a couple of other mathematical oddities.
 

amirm

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OK, I found the original thread and read through it: http://www.whatsbestforum.com/showthread.php?16365-The-sum-of-all-natural-numbers-1-12

Looks like the ultimate conclusion was not reached. Here is my read of it.

It is useful to be able to represent infinite sums as some value as to allow further analysis of physical world around us. The proof used to drive these answers must be correct at some level as the physics based on it works.

In some sense then, this sum may be a technique to solve a problem using much simpler math. Indeed there are much more complicated ways of proving some of these equations but why not use this simple concept to arrive at the same place quicker and faster?

Another similar concept is that of complex numbers. i = square root of -1. Such a concept is absurd as you can't take square root of -1. But we can use that to solve countless problems.

And remarkably just like the sum of natural integers being -1/12, we can use square root of -1 to solve real physical problems where this very value, actually means something in reality. In electronics, imaginary part of an equation represents the phase of an AC signal. In other words, it is not imaginary at all! By being able to encapsulate phase in the same number as its amplitude, we are to represent the entirety of an AC waveform and manipulate it using math of complex numbers.

The same scheme is used in string theory to come up with a sum of ways a string can resonate which is the original frequency and infinite multiples of that frequency. That sum instead of being infinite, can be considered to be 12. Multiple that by 2 and you get 24 for each dimension. Add the original X and Y plane and the total number of dimensions needed to express this infinite variety just becomes 26!

There: you have my plagiarized and embellished explanation of why this is a valid sum and so useful in real life. :)
 

ack

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Yes, and string theory depends on this result to eventually make a case and "justify" the purported 26 dimensions (not sure how you got 26 dimensions with your calculations, like what happened to the Z axis???); except that we have zero proof that string theory holds water, yet... and perhaps never, because things are not looking good for it. Let me send you a scathing PM about your analysis, and see if you want to make it public... LOL
 

GaryProtein

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I understand how they "proved" 1+2+3+4+5+6 . . . ? = -1/12,

BUT

that result is so counterintuitive that I can't help wonder if there is a logical fallacy in their proof.

It reminds me of the equation in 8th grade algebra where they "prove" 1 = 2 which has a logical fallacy in the proof.

The Fallacious Proof:
  • Step 1: Let a=b.
  • Step 2: Then
    ,
  • Step 3:
    ,
  • Step 4:
    ,
  • Step 5:
    ,
  • Step 6: and
    .
  • Step 7: This can be written as
    ,
  • Step 8: and cancelling the
    from both sides gives 1=2.


No matter how you look at it, the problem in the "proof" is that zero
behaves differently, and no account is taken of that. We're agreed
there, right?

You are suggesting that it is in the step I would simply call
"factoring" that something "goes wrong." But let's look at the whole
proof with an eye on the fact that a=b, so that we are aware of what
is zero:

written reality judgment
------- ------- --------
a=b a=a true
a^2=ab a^2=a*a true
a^2-b^2=ab-b^2 a^2-a^2=a*a-a^2 true (=0)
(a+b)(a-b)=b(a-b) (a+a)(a-a)=a(a-a) true (=0)
a+b=b a+a=a false (unless a=0!)
b+b=b a+a=a false
2b=b 2a=a false
2=1 2=1 false

Go through that carefully; you see that the factored step is the last
one that is true. It has not introduced the error, but it sets things
up for an error, by putting the equation into a form in which we are
multiplying by zero. It also introduces the 2, in a hidden form; so
you are right in saying that this is not a coincidence. It's sort of
like a burglar entering your house. When he walks up your street, he
has not yet broken in, but it is no coincidence that he is there. At
this point in the "proof," the burglar is on the scene, but isn't in
yet.

Now, you are right that at this step we have, formally,

2a * 0 = a * 0

and it already shows signs of the problem to come; but it is not wrong
yet, just as

2 * 0 = 1 * 0

is not wrong. It is only if you forget that 0 times anything is 0, and
that zero is special at this point, that there is a problem, and that
is exactly what is forgotten in the next step.

The problem, of course, is that what you call the "division property
of equality" has a condition:

if ab = ac, AND a is not zero, then b = c

If we omit that condition, then it is not true, because

0b = 0c = 0 for ALL b and c, regardless of whether b = c

The error in the proof is that this condition has not been checked,
so that a false "property" was applied.



 
Last edited:

YashN

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I understand how they "proved" 1+2+3+4+5+6 . . . ? = -1/12,

BUT

that result is so counterintuitive

Looks so much like a calculator/computer overflow error.

We may be living in a simulation :p
 

amirm

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Yes, and string theory depends on this result to eventually make a case and "justify" the purported 26 dimensions (not sure how you got 26 dimensions with your calculations, like what happened to the Z axis???);
What you saying? THat you want me to plagiarize and embellish more stuff??? OK here it goes.

The universe as we know it as four dimensions: the three you mention and time.

String theory says if we zoom into any particle we actually see a one-dimensional object called a string. In an X/Y space we can have two dimensions of them. Unlike our physical world that adds Z space and Time, strings according to the theory needs to have 24 more such dimensions for total of 26. And that if we back far enough, these dimensions fold back into four that we understand as our physical world.
 

amirm

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Seattle, WA
I understand how they "proved" 1+2+3+4+5+6 . . . ? = -1/12,

BUT

that result is so counterintuitive that I can't help wonder if there is a logical fallacy in their proof.

It reminds me of the equation in 8th grade algebra where they "prove" 1 = 2 which has a logical fallacy in the proof.

The Fallacious Proof:


No matter how you look at it, the problem in the "proof" is that zero
behaves differently, and no account is taken of that. We're agreed
there, right?

That is indeed the problem here. The substitutions seem correct superficially but checks as the rest of your post show, invalidate them from certain step down when replacement of one of the variables with zero does not work.

Not so with the topic at hand. There are no fallacies in it because mistakes like above are not made. And further, application in real physics validates the results.

What is going on is entirely different. One attempts to draw characteristics from sum of infinite numbers. The other is some derivation which makes mistakes half way through.
 

treitz3

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Common Core math says the answer is beer.

Tom
 

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GaryProtein

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Last edited:

RayDunzl

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Michio Kaku:

(explaining the mathematics of black holes)

"Here is the problem, right there: When R= 0 — the point at which physics itself breaks down. So 1/R = 1/0 = Infinity." (R = distance from the black hole)

"To a mathematician infinity is simply a number without limit. To a physicist it's a monstronsity! It means that first of all gravity is infinite at the center of a black hole.> Time stops. Space makes no sense. It means the collapse of everything we know about the physical universe. In the real world there is no such thing as infinity. Therefore there is a fundamental flaw in the formulation of Einstein's theory."

---

While trying to use quantum mathematics:

Dr. Michio Kaku returns to deliver the best line in the video:

"When I actually do this calculation... you get something which makes no sense whatsoever. Infinity. Total nonsense. In fact you get an infinite sequence of infinities. Infinitely worse than the divergences of Einstein's original theory."

"This is a nightmare beyond comprehension."

---

https://gist.github.com/zachharkey/6393108

From this video - http://www.dailymotion.com/video/x22yu61_bbc-horizon-who-s-afraid-of-a-big-black-hole_tech
 

GaryProtein

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GaryProtein

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YES!

This is a great explanation of the trick / logical fallacy the numberphile people used to "prove" their thesis.

In the real world, you cannot have infinity, division by zero or square roots of negative numbers. These things can only be used for mathematical abstractions. (or is it mathematical constructs? I don't know the preferred term)
 

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