A Simple Method to Approximate Square Roots

The number \(\sqrt{2}\) is famously irrational: it cannot be expressed as a fraction (ratio of two integers). However, we can use fractions to approximate it. Here is one simple algorithm to do so, which uses only elementary operations. Let our initial estimate be \(x_0 = 1\). From there, we generate the next estimate using the following procedure: Add one to the current estimate. Invert the…