Which is smaller. infinitesimal or 10-10000000000000000000000000000000000000000000000000000000 ?
Answers
Answer:
Explanation:
There are not smallest (biggest) infinitesimals in the surreal numbers.
Surreal numbers are built up incrementally with two sets of surreal numbers, the left (L) options and right (R) options of the number, such that there is no element of L greater than or equal to any element of R. [ | ] on day zero, when we have no numbers yet, both L and R are empty sets, and this is the number zero. Then we can construct [ | 0 ] and [ 0 | ] to make -1 and +1. Now this process continues to infinity, creating all dyadic rational numbers; the assignment of the left and right options creating either new forms of existing numbers or new “simplist” numbers. On day infinity (omega), we not only create the surreal numbers -omega and +omega, we also create numbers that aren’t dyadic rations, like 1/3.
These numbers can be used in arithmetic; addition, subtraction, multiplication, and division can be defined, as well as other operations. So one can create an infinitesimal, such as 1/omega (= epsilon).
The creation of additional surreal numbers continues; [omega | ] is omega+1 , and so on.
So there are many infinitesimals, as you can have larger ones than epsilon (e.g. 2 times epsilon) and smaller ones (e.g. epsilon divided by two). These are all greater than zero and less than any finite positive real number.
As surreal numbers are non-Archimedean, there is no limiting value of smallest or largest infinitesimal between 0 and the smallest finite positive real number.
If you remove the restriction that the left options must be surreal numbers (or empty) such that no element of L >= R, then you have the mathematical objects called games. I mention this, because the surreal numbers have a gap between zero and the infinitesimals, but there are games which have a value in this gap! These values are not surreal numbers, but you might consider them an infinitesimal smaller than any positive surreal number. For instance, [0|0] is a game whose value is star (*). An example of a game holding such a value is called UP, defined as [0 | *].
The story doesn’t end there, however, as these games can also be acted upon arithmetically, so UP * UP is a value less than UP and greater than zero, and UP * 2 is a value greater than UP and still less than any surreal number. There are infinitely many such games in the gap between zero and the smallest surreal number.
I hope my answer meets with readers’ approval; there are NO smallest (biggest) infinitesimal in our mathematics.
P.S. Mathematicians working with infinitesimals as defined by games and surreal numbers do not “treat them as numbers very freely.” Infinitesimal values are not like the dy/dx infinitesimals where an undergraduate may be told that one can just disregard squares or higher powers of infinitesimals as having no value to a real function, whose values at a limit are defined to be equal to the limiting value.