Again inspired by a Numberphile video, I started to ponder the similarities between an "infinitesimal", often represented as , and the number .
Without diving too deeply into the meanings of infinitesimals (the video does this much better than I can), a basic definition of is this:
Where is any and all of the real numbers. In other words, is smaller than all other numbers, yet it is greater than .
Now please note that the following is more pondering and speculation than rigorous exposition. :)
What I realized is that if represents an "infinitely small" number, and if, as I recently speculated in another discussion, is really equal to , than perhaps we should just equate to .
As an equation, this could be stated as:
Another thing to note is that I am not proposing that , as it has been defined by mathematicians, is equal to . This is more of a reinterpretation of what an infinitesimal is. In this way, I have really just redefined an infinitesimal as actually representing . Just as infinity represents infinitely much, or everything, perhaps an infinitesimal represents infinitely little, or nothing.
This equation results in several satisfying results, but the result which most interested me was this: One of the axioms of infinitesimals is that there can be multiples of , so , , and so on. The parallel to this would be that there can be multiples of , so , , and so on.
Does anyone have any ideas/opinions on this? If so, I would love to hear them!