Thread 16758736

Are infinitesimals considered finite or non-finite?

>>16758736 (OP)
I think it's considered non-finite.

>>16758736 (OP)
Finite, in the Archimedean sense (less than 1 + 1 + ... + 1 for some such finite sum)

>>16758736 (OP)
they violate the archimedean axiom, so they are infinite

>>16758736 (OP)
dont exist and neither do infinities

>>16758736 (OP)
They are exactly equal to zero.

>>16758736 (OP)
Rational numbers are finite in some integer base. Irrational numbers are also always finite in some transfinite ordinal base, whether because they repeat in that base or because they simply end finitesimally without an infinitesimal part, depending on how you want to represent them. So all infinitesimals are both rational and finite.

>>16758904
They violate it by skipping a stone on a lake and pretending each bounce on the lake is a new infinite set. Nothing really changes other than the order of what you're counting. The lake is still there, the stone is still there, you're still there, the camera capturing it all is still there. Only the order of the numbers you choose to highlight are different. Infinitesimals are, at best, a permutation of the regular number line.

>>16758736 (OP)
It depends on the context. In mathematics 'finite' is an informal term. Infinitesimals are less than infinite, and thus could be described as finite. On the other hand something can be described as small but finite, in the sense of not infinitesimal nor zero. The word is used in the latter sense in mechanics, as a distinction between finite and infinitesimal strain theories.

>>16759917
>rational
That's nonsense, and easy to contradict. A real multiplied by an infinitesimal yields an infinitesimal, yet a rational multiplied by irrational yields an irrational

>>16759934
>That's nonsense, and easy to contradict. A real multiplied by an infinitesimal yields an infinitesimal, yet a rational multiplied by irrational yields an irrational
I don't follow how that's a contradiction, spell it out a bit.

>>16759929
not even close fucko, read
https://en.wikipedia.org/wiki/Archimedean_property

>>16759952
Tell me what part of that page isn't even close.

>>16759951
An irrational real multiplied by an infinitesimal yields an irrational infinitesimal

>>16759967
How do you write it?

>>16759967
>>16759973
Like what's the first thing you can say about that "irrational infinitesimal" that's different from the irrational real itself?

>>16759980
It is smaller

>>16759998
So's the next digit of 0.3... which is finite in base 3.

>>16760002
Well in this case it is a lot smaller, as we obtained the infinitesimal by multiplying the real with another (arbitrary) infinitesimal. One could say infinitely smaller, but that might be confusing. Infinitesimals are smaller than the smallest real, much like infinities are larger than the largest real. You can keep adding zeros to ...0001 forever, much like you can add zeros to 1000..., but both will stay real. Infinitesimals and infinities go beyond those, extending the real number line in different, yet intuitively similar ways

>>16760006
>we obtained the infinitesimal by multiplying the real with another (arbitrary) infinitesimal
Right, but what can you say about the infinitesimal part of the product?

>>16760009
It is infinitesimal, ie very small. Whether it's irrational or not does not matter, as the other factor is irrational

>>16760006
>>16760009
Like you can write pi as 3.14... even if you can't finish it and it doesn't repeat. I don't think you can write anything at all about pi times an arbitrary infinitesimal.

>>16760014
>>16760016
But you can write something about pi times a rational infinitesimal, because you can properly decimate the digits of pi by the base of that infinitesimal. And what you can write is rational in the base of that infinitesimal.

>>16760016
You can multiply π with a small number, and the product will be also irrational. Basic arithmetic rules don't change when working with infinitesimals (or hyperreals in general)

>>16760018
The product is irrational but the irrationality is completely stuck in the real part. You can't say anything about the infinitesimal part. You'd have to write 3.14...;... which is the same as writing 3.14....

>>16760024
There is no real part in the result, the result is infinitesimal, it is not some sort of composite. Much like multiplying multiplying a natural number with a real will result in a real (which might also be natural, but that's beside the point)

>>16760049
Take any infinitesimal h = 0...;...275634... where the digits don't repeat.
Then 3.14...;... times h = ?

>>16760049
I'm starting to typo, I think I need some sleep

>>16760054
Peace out, sleep well.

>>16760052
If you want to think it like that, you can treat the as a product of a real and 10^n, where n is large. But don't fool your self into thinking that the number itself is not infinitesimal. It is no more composite than 6.283... is a composite of π and 2

>>16760056
cheers