Newton, the towering scientific figure of the 17th century, discovered a lovely method for finding approximate solutions to equations, involving iterated constructions of tangent lines and their intersections. We describe this method in general and then apply it to the simplest and most familiar example; the standard quadratic polynomial x^2. To calculate tangent lines we use the algebraic calculus, and focus on approximating sqrt(2). This gives, remarkably, the same algorithm as the ancient Babylonians had for approximating sqrt(2), and so another alternative to the Vedic procedure.This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
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