What is an infinitesimal? What is an infinitesimal? @Mathematics

Question

anonymous · Answer

something that is practically immeasurably small

anonymous · Answer

and is it a number?

jamesj · Answer

It's a highly convenient, informal way of talking about and making manipulations with symbols in calculus.

e.g., if dy/dx = f(x)g(y), we can write it as f(x) dx = dy/g(y), even though that is meaningless in the theory of real numbers as "dx" and "dy" are not numbers.

anonymous · Answer

so we just use those to conveniently work with ideas, but not when we want to work our way out of some proof crisis?

jamesj · Answer

It turns out much, much later in higher mathematics you can formalize this idea and it works.  But using it here as I have or especially in physics/engineering is very convenient.

If you're writing formal proofs, you can't use them unless you have actually defined them.

anonymous · Answer

in physics, the teacher always uses such 'dx's

jamesj · Answer

that's fine.  It works and if you're getting stressed about it, just think of it as $ \Delta x $ and then take limits later when terms like dV/dx turn up.

anonymous · Answer

even limits can be more more stuffy using epsilon delta arguments , egad

anonymous · Answer

so in math you never have to say infinity . its taboo