Is there something called next to 0 number?

Pffft such an easy question, the number is 1, see problem solved

>as any physics student knows

that doesn't mean the hyperreals aren't dense

are you a physics student?

explain the many uses of infinitesimals

>bonus points if can you do it without citing calculus, because differentials and 1-forms are not infinitesimals
>bonus BONUS points if you can do it without citing dual numbers

0+h

>bonus points if can you do it without citing calculus, because differentials and 1-forms are not infinitesimals

en.wikipedia.org/wiki/Differential_(infinitesimal)

>bonus points if you can explain something without actually being correct, therefore making me correct

epsilon

You can approximate the behavior of a physical system changing over time with great accuracy by considering infinitesimal components of the system behaving under infinitesimal time intervals and many of the times this will work despite being nonsense in a traditional mathematical framework.

i was trying to say that the ""actual"" infinitesimals aren't used in physics (except for a case where dual numbers are needed for some gauge theory wizardry), and that the constructions used in calculus are special cases of actual, mathematically precisely defined objects that we can talk about without referring to a philosopher

i'm pissed about this because "infinitesimal" has become a buzzword recently for some reason, and people who have no idea what it actually means are using it to refer to other things which they have no knowledge of,
because i am a math tutor, i hate to see such toxic misinformation being spread like this

and just to cover all bases, i do realize smooth infinitesimal analysis has been used in a topos-theoretical setting in some crazy branches of theoretical physics, though i'm not smart enough to figure out what actually goes on there

are you talking about linear approximation or something else

x ; x ---> 0

what about 0.5 thats even closer?

Don't listen to all of the brainlets in this thread OP. You certainly can find a number "next to" 0. Through use of the axiom of choice, one can well-order the set of all real numbers excluding 0 with a relation