Quantification of the alkalinity of natural water by applying Bernoulli numbers and Simpson's approximate integral

Abstract

The calculation of Bernoulli numbers is because there are numerous mathematical expressions and environmental engineering phenomena in which they can be applied. Although it is true that, with the aid of the computer and the use of programs, there are a great number of them; in the current references they are published in a reduced number and, therefore, when using them, unsatisfactory approximations are made. The purpose was to calculate the first 20 Bernoulli numbers, apply them to mathematics (∫ ln(cosx)dx) and environmental engineering, quantifying the alkalinity of natural water in the form of calcium carbonate (CaCO3)(mg/l ), measuring carbon dioxide (CO2). There is a great variety of relations containing Bernoulli numbers presented in several practical applications of mathematics. The best relationships used here are binomial developments. Employing the appropriate equations, the Bernoulli numbers were calculated mechanically, arriving to determine the target number of numbers. Finally, the Bernoulli numbers contribute significantly to mathematical calculations such as calculating integrals of the type ln(cosx), as well as to applications to environmental engineering, evaluating the alkalinity of water as calcium carbonate (CaCO3).