A formal justification of the Ancient Chinese Method of Computing Square Roots

: In this paper a formal justification of the ancient Chinese method for computing square roots is given. As a result, some already known properties of the square root which is computed with this method are deduced. If any other number base is used, the justification given shows that the method is applied in the same way and that the deduced properties are still being fulfilled, facts that highlight the importance of positional number systems. It also shows how to generalize the method to compute high orders roots. Although with this elementary method you can compute the square root of any real number, with the exact number of decimal places that you want, the mathematicians of ancient China were not able to generalize it for the purpose of computing irrational roots, because they did not know a positional number system. Finally, in order for high school students gain a better understanding of number systems, the examples given in this paper show how they can use the square root calculus with this method to practice elementary operations with positional number systems with different bases, and also to explore some relationships between them.