Editing Infinitesimal (section)

=== Hyperreals ===
{{Main|Hyperreal number}}
The most widespread technique for handling infinitesimals is the hyperreals, developed by [[Abraham Robinson]] in the 1960s. They fall into category 3 above, having been designed that way so all of classical analysis can be carried over from the reals. This property of being able to carry over all relations in a natural way is known as the [[transfer principle]], proved by [[Jerzy Łoś]] in 1955. For example, the transcendental function sin has a natural counterpart *sin that takes a hyperreal input and gives a hyperreal output, and similarly the set of natural numbers <math>\mathbb{N}</math> has a natural counterpart <math>^*\mathbb{N}</math>, which contains both finite and infinite integers. A proposition such as <math>\forall n \in \mathbb{N}, \sin n\pi=0</math> carries over to the hyperreals as <math>\forall n \in {}^*\mathbb{N}, {}^*\!\!\sin n\pi=0</math> .