In this article, we will show that the Pythagorean approximations of Formula coincide with those achieved in the 16th century by means of continued fractions. Assuming this fact and the known relation that connects the Fibonacci sequence with the golden section, we shall establish a procedure to obtain sequences of rational numbers converging to different algebraic irrationals. We will see how approximations to some irrational numbers, using known facts from the history of mathematics, may perhaps help to acquire a better comprehension of the real numbers and their properties at further mathematics level.
Javier Peralta from Madrid, Spain wrote an article that was recently published online in Teaching Mathematics and its Applications. The article is entitled Pythagorean approximations and continued fractions, and it relates to the Fibonacci sequence, sequences of rational numbers, etc. Here is the abstract of the article: