Finite Infinity is actually a
misleading term. I am simply referring to a
theory I have formed to do with the nature of
infinity in a
mathematical sense. Keep in mind that for this concept to work, you have to be a subscriber to the theory that
a number divided by zero is equal to
infinity. I agree with this on the very
straightforward concept that
zero will go into any number an
infinite number of times, and that
infinity will divide into any number
zero times.
Also, accept the notation ^- to refer to a repeating symbol (a line) above the preceding number, as in repeating decimals. So 0.3 recurring would be written as 0.3^- and 0.54 recurring would be 0.5^-4^-, and 0.23 with only the 3 recurring would be 0.23^-. I hope that makes sense. Also, I will use the variable 'I' to refer to Infinity, for lack of the infinity symbol. Do not confuse this with 'i', the imaginary number symbol, as for the purposes of this demonstration imaginiray numbers are not required. So here it goes:
First, some basic theory: if x/y = z then x = zy and y = x/z. Let us apply this to the zero-division rule:
x/0 = I
Rearrange:
I*0 = x
Again, with x as 5:
5/0 = I
I*0 = 5
This rearrangement of equations brings up the point that infinity times 0 is equal to a number. However, as x can have any value, we need to distinguish between multiple infinities. This is a concept most consider ludicrous, including me, however I feel it is something interesting to toy with.
Allright, let us write our infinites with a new use of the repeating concept. If we can have a repeating decimal, why not a repeating numeral? So, the particular infinity that comes from 5 divided by 0 would be 50^- (here is the first instance of my repeating notation, so be clear that this refers to 5000000... into infinity). The particular infinity that comes from 12 divided by 0 would be 120^-, and so forth.
Now we must break another fundamental rule of mathematics, which is that any number multiplied by zero must equal zero. As the above equations demonstrate, this rule is defied when the number in question is infinity. So, 50^- * 0 would be 5, 120^- * 0 would be 12, and so forth. This implies that zero is not actually zero. Again, this can be hard to swallow, but let us assume that it might be possible to have a repeating decimal where the particular digit that repeats is not the last one! This does not immediately make sense, but look at:
0 = 0.0^-1.
This is, at first, apparently quite insane. It refers to a decimal composed of an infinite string of zeroes, with a one at the end! How can their be an end to infinity? Their can't. Unless of course we can reduce infinity! Hence, an infinitely large number (say 50^-) is multiplied by an infinitely small one, (in this case Zero, but the special zero that is 0.0^-1), you could, in theory, end up with a finite number!
The main problem with this concept is that it defies many mathematical principles, but if you look at it logically, it sort of makes sense. For demonstrative purposes, let us apply it to a simple infinity-based operation:
20^- * 30^- = 60^-
This works, because we can look at it as the 0 denominated fractions like this:
(2/0) * (3/0) = (6/0)
And, following basic fractional calculation rules, it works. The same applies for division, addition and subtraction.
There are further issues brought up here. For example, if zero is not actually zero, then what is 5 * 0. Is it 0, or is it 0.0^5? I would say the latter, but then we raise the question of whether or not 50^- * 0.0^-1 is the same as 50^- * 0.0^-5. Some would suggest that the former is equal to 5, while the latter is equal to 25, but then you must wonder if 25/0 can be equal to either 250^- or 50^-. There is an infinite number of problems raised, but it is fun to think about, even if it (as is most likely the case) has no mathematical merit.