Infinitesimals do not exist in the standard real number system. 'tiny' seems to be more related to the smallest positive representable (normal or subnormal) IEEE 754 float/double type of value which is a real number.
Infinitesimals tautologically exist in any finite representation of numbers. For floats, it's the smallest representable positive number. For integers of any type, it's 1.
As for how common they are, we learn about them in any introductory calculus course when defining derivatives. You come across the idea whenever discussing limits, if somewhat obliquely.
If I learned about it in high school math, and again in "real" math courses at my university, I'd say it's pretty standard.
I've never heard of that, and your definition of 1 as infinitesimal is incompatible withbits properties (infinitesimal + infinitesimal + infinitesimal is greater than a non-infinitesimal 2?!) and I don't see a mention on Wikipedia, and it goes against the plain read it of "in-finite-simal".
Also, you seem to be conflating "common" with "standard". "standard" is a mathematical term. Infinitesimal are handwavy in standard analysis (epsilon-delta are the rigorous alternative), but exist rigorously in nonstandard analysis.
I guess I'm using the wrong term, then. I often find it useful to have a concept of "smallest representable positive number," specifically for handling edge cases such as the one I gave up-thread. I see how that doesn't map to infinitesimal as defined in the shared link.
There are other instances where I've had a need for such a smallest positive number, where logic is simplified as opposed to checking for 0 in a special way. Whether there's an agreed upon term for that, I know where I've found value in programming tasks.
When I need such a thing, it is almost invariably in comparisons, so I am not doing arithmetic with multiple instances of that smallest representable positive number.
