New high-order arithmetical root-approximating methods. Approximations at any desired convergence rate. Similar to and even better than those from infinitesimal calculus. 
These new straightforward arithmetical methods do not require using derivatives, geometry, or Trial-&-Error tests and also embrace the well-known methods from Daniel Bernoulli, Newton, Householder, and Halley, among others, just by using the Simplest Arithmetic.

The new high-order arithmetical methods are based on the generalization of an operation called Mediant in Number Theory, which has been rarely used in the literature and is commonly treated as a curiosity (Farey fractions, Brocot sequence, Ford's circles, Lattices, and some others).

https://www.youtube.com/watch?v=VKy0UPOf1Ew