A somewhat intuitive reason for -1/12

This is not a proof, but an outline of an idea. I am not a mathematician.

First of all lets look at infinity.

inf/inf is undefined,​

but x/x is 1 regardless of what x is, even if x = inf. This could be explained by inf being a range rather than a number,
and when we say "x = inf" we mean x is a single specific number from that range.

Hinge 1: This idea hinges on viewing infinity as a range.

Now for the sum of all the natural numbers. Imagine that you have a line K that starts at ( 0, 0 ) and extends towards ( s, 0 ) where s is the sum "so far". The line K can be seen as a connection of line segments, each of length k where k is a natural number. The length of the 3rd segment k3 is 3. When we add the nth segment to K we extend it with the line segment kn.

Lets look at the length of kn in terms of the Pythagoras theorem:

kn = sqrt( n^2 + d^2 )​

where d is the distance along the y axis of the two connection points at kn and k(n+1). The distance d would be zero, right? Lets change the last formula to calculate it:

d = sqrt( kn^2 - n^2 )​

I believe that when n=inf, n+1 and n can not be assumed to be ordered sequentially.

The numbers kn and d can now become imaginary because n can be larger that kn. I realize that n>kn doesn't make sense with finite numbers, but when dealing with infinity in this particular context it could make sense.

Hinge 2: This idea hinges on n>kn sometimes being true in this context.

To refer to the inf/inf vs x/x example, n>kn could come up if "traversing" within the infinity range in a particular way, because we are dealing with different infinite numbers, basically doing x=inf over and over getting different infinite values for x. Infinity could be seen as an "unordered range" due to inf+1 being "weird". For example a/b would be undefined if a=inf and b=inf because a and b are different infinite numbers. If, however, we said a=b=inf they would be equal and a/b would be equal to 1.

The point is: Because of the sqrt of potentially negative numbers we get imaginary numbers and our line segments can now have a direction other than parallel to the x-axis!

The line K could go all the way around the first quadrant and while parallel to the y-axis, cross the x-axis at -1/12.

It could be easier to reason about this with the sum of the squares of natural numbers, which would circumvent the first quadrant and cross the x-axis at y=0.

 

This is good! Well done. Yet....

x/x=x....in all cases is NOT true if x=0

That said there is much for me to unpack here. Please give me time to think before I reply further. Also I have some questions for you. But would like to think a minute before I ask. I also am not a mathematician.

Lastly....almost any other math website would receive you better. This one is infested with philosophers like me. If a mathematician cannot stomp me out then they leave for another site....sigh.

 

Ok...I think/hope that I am ready for you. Please bear with me as my questions to you will reveal my ignorance in this subject.

1. I take serious issues with "imaginary" numbers. They ARE real. It is only that they are abstract. An electron "without" computers is purely abstract. Yet we know that it is real. This is largely "beside" your point. Yet I feel it is not so. I will let this go until you agree to discuss its relativity to this subject.

2. I think inf/inf is defineable. Just as I believe 0/0 is defineable. The point here is relative to your beautiful thought that infinite is a range rather than a number. Yet...you state inf/inf = undefined. Then you state if x = inf then x/x is still 1. This is contradiction as you point out. Then you continue based on that contradiction.

3. What is the significance of having directions other than parallel to the y axis? We have these all the time in other scenarios? My ignorance is screaming here. Yet the fact we do have them in other scenarios and not in this one speaks beautifully to your point.

4. (-1/12) to me or the layman is just a number. Your title might also have been "intuitive reason for 2". Intuitive for what? If the line crosses there why? Point? Meaning? Here I admit meaning truth and beauty are not mutually exclusive.

5. This is relative to my own paper, which I am not asking you to read. But your idea does affect mine...so. If infinite is a range and not a number...if division by zero and by infinite are defined. Then we can show that zero leads to infinite, and infinite leads to zero. This then is a continuum theory!

6. If then infinite is a range, then new notation is needed. You need to show with this new notation that x is the same value yet different from (x not in the range of infinite). You say x=(a number from the range of infinite). Yet x "on its own" or not in the range is the same yet different then x in the range. I imagine absolute value here. The absolute value of -1=1. The same value but different.

I hope I have done you justice. Your greatest thought for me is that infinite is a range not a number. Perhaps this is already so and the idea is not yours. I don't care it is jaw dropping and I wonder how I have not considered it before. I don't think I will ever consider infinite as anything else after this .

Again...well done...and good luck in all you do. You are loved by someone somewhere. That is more important than this!

 

Lol....see what I mean? What exactly did I say to run you off? I know that you read this post....but no reply? You worked hard, yet you do not answer? I know what its like to be on the other side. The only reason you do not reply is embarrassment. You are embarrassed that I replied. Someone who is a dunning-kruger. I am proud of this. Because the only other way to learn...is from "them". You will not care, but you cannot learn from "them" without picking up their ego, narcissim , and arrogance
Good luck in your attempt to fit in!