Infinitesimal-Based Calculus

When Newton and Leibniz first developed calculus, they did so by using infinitesimals (really really small numbers). Infinitesimals were used until calculus was made more rigorous by Weierstass. The calculus that we are taught today is based on Weierstass’s ϵ-δ definition of the limit. However, people have been arguing that we go back to an infinitesimal-based calculus, not only for its historical roots, but because many proofs and concepts seem to be much cleaner when using infinitesimals. Using Keisler’s “Elementary Calculus: An Infinitesimal Approach,” our group set out to relearn calculus using infinitesimals. First we will define the hyperreal number line (an extension of the real line that contains the infinitesimals). Then we will walk through the familiar ideas and concepts of single variable calculus, such as limits, derivatives, and integrals, reformulated in terms of hyperreals.