Mathematics, Department of


Date of this Version



arXiv:2311.13551v1 [math.HO] 20 Nov 2023

Published (2023) Notices of the American Mathematical Society, 70 (11), pp. 1772-1779.


It is likely a fair assumption that you, the reader, are not only familiar with but even quite adept at differentiating by x. What about differentiating by 13? That certainly didn’t come up in my calculus class! From a calculus perspective, this is ridiculous: are we supposed to take a limit as 13 changes? One notion of differentiating by 13, or any other prime number, is the notion of p-derivation discovered independently by Joyal [Joy85] and Buium [Bui96]. p-derivations have been put to use in a range of applications in algebra, number theory, and arithmetic geometry. Despite the wide range of sophisticated applications, and the fundamentally counterintuitive nature of the idea of differentiating by a number, p-derivations are elementary to define and inviting for exploration. In this article, we will introduce p-derivations and give a few basic ways in which they really do act like derivatives by numbers; our hope is that you will be inspired and consider adding p-derivations to your own toolkit!