This is not a proof, but an outline of an idea. I am not a mathematician.
First of all lets look at infinity.
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:
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:
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.
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.
