I wish to apologise to reader about not writing in Arabic as I have no means to do so.
I was interested in the article, which contained abstract arguments concerning complex numbers and infinity, so I sought to put my thoughts in writing.
To say that infinity, : = x is to introduce 2 different arguments both of which prove that,
x + x ! 2x , 3x, or any other than :
1) If infinity = x, then x is undefined and it follows that it is not subject to mathematical assumptions and operations, and although not a singularity it exhibits some of its properties.
Therefore to say, x + x = 2x,
is like saying, x + y = 2x, 2y or 2xy. All of which would violates basic mathematics.
The argument should be expressed as,
x + y = :
This holds true with either or both variables are :. If either or both variables change then the problem becomes one of dynamics, i.e. changes with time and the variables would require redefinition with each new value. A solution can be found, but life is too short!
2) If infinity = x, and x belongs to a defined set or a known group, then to say,
x + x = 2x,
is a gross misrepresentation of the initial assumption that infinity includes all constants beyond a specific set deemed practical to justify an argument. Here, the addition of any constant to x can only lead to redefinition of x as being a larger infinity.
It therefore follows that x + x = x
Which agrees with true logic that
1 + 1 = 1, albeit : is always < 1
Therefore 1 + 1 [ 1,
And, x + x ! 2x
X, but not an infinite one.