Estimating π with “fake circles” using Python. This short shows how to simulate the geometry, calculate the approximation step by step, and see how accuracy improves as the model gets refined. #Python ...

GPT-5.2 Pro delivers a Lean-verified proof of Erdős Problem 397, marking a shift from pattern-matching AI to autonomous ...

Discover how Markov chains predict real systems, from Ulam and von Neumann’s Monte Carlo to PageRank, so you can grasp ...

J. William Carpenter is an economist who writes financial topics. He is an author of published works for higher education and business clients. Katie Miller is a consumer financial services expert.