Note: This post can't be considered as a personal theory. These are the arguments which I figured out by reasoning and I just want to know it's validity.
Here I present multiplication and division operations of $0$ and $\infty$ by pure reasoning. $$\dfrac{x}{0}=\infty$$ (where $0$ is positive infinitesimal) $$AND$$ $$\dfrac{x}{0}=-\infty$$(where $0$ is negative infinitesimal)
If we divide $x$ by a small no., we get a big no. $$\dfrac{x}{small}=big$$
If we keep on dividing $x$ by a small no. and then by another small no. and so on, we get bigger and bigger no. $$\dfrac{x}{small\times small\times.....}=big\times big\times.....$$
Similarly if we divide $x$ by a negative small no. and then by another small no. and so on, we get bigger and bigger negative no. $$\dfrac{x}{-small\times small\times.....}=-big\times big\times.....$$
However while doing this, two facts should be noted:
(1) The small no. in the denominator of LHS will never reach $0$.
(2) The big no. in the RHS will never reach $\infty$ or $-\infty$.
Thus by reasoning, if we ever imagine that the small no. in the denominator of LHS is $0$ (which is impossible), we should also imagine that the positive or negative big no. in the RHS is $\infty$ or $-\infty$ respectively (which is also impossible). Is it a reasonable proof for "dividing a no. by zero, we get positive or negative infinity i.e. $\frac{x}{0}=\infty$ or $-\infty$"?
Now by algebraic manipulation,
$$0\times\infty=x$$(where $0$ is positive infinitesimal) $$AND$$ $$0\times-\infty=x$$(where $0$ is negative infinitesimal)
But is algebraic manipulation legal here?
Here, two facts should be noted:
(1) We can never add zero infinite times in positive or negative no. line
(2) We can never get $x=$ non-zero when we keep adding zero over and over.
Thus by reasoning, if we ever imagine that we added zero infinite times (which is impossible), we should also imagine that $x=$ non-zero (which is also impossible). Is it a reasonable proof for "infinity times zero is non-zero i.e. $0\times\infty=x$"?
If it is true, then by algebraic manipulation of $0$ and $\infty$, we are getting logically correct answers.
Now we observe $0\times\infty=x$ could be any no. and hence is undefined. If we want to make $x$ a well defined no., we should make no. lines of $0$ and $\infty$ as an extension of real no. line and give several values for $0$ and $\infty$ like:
Now if we choose particular values of $0$ and $\infty$, we get a well defined value of $x$. For example:
$$(0\times2)\times(\infty\times3)=(\infty^{-1}\times2)\times(\infty\times3)=2\times3\times\infty^{-1}\times\infty=6$$
Now can we consider this no. line of $0$ as the infinitesimal no. line of non-standard analysis?
In a similiar way, can we also define no. lines of infinitesimal of infinitesimal (second order differentials) and of infinite of infinite and so on?
Am I anywhere incorrect in my reasoning? Are my conclusions logically correct?