A new general and unifying arithmetical concept: The Rational Mean, which allows to generate, among many other new algorithms, those celebrated Lucas?s, Bernoulli?s, Newton?s, Halley?s, Householder?s root-approximating methods which up to now were considered the exclusive achievement of Infinitesimal Calculus. No derivatives, no trial-and-error checkings, no geometry, no cartesian system, but just Simplest Arithmetic. These new high-order methods also embrace complex roots and the general algebraic equation, and from the solid evidence at hand, these arithmetical methods have no precedents in the mathematics literature. Indeed, it is really striking to realize that even ancient mathematicians had at hand the elementary tools for constructing these high-order algorithms. Actually, these new findings compel us to cogitate on the reasons mathematicians were forced to create the "elevated" infinitesimal system that we have inherited. The book also includes the new Generalized Continued Fractions for approximating the maximum and the minimum modulus root of the general algebraic equation, number Pi, number e, Golden Mean, and others with improved convergence rate.