Mathematics
The Newton-Raphson Method for Finding Roots of Equations
Quick fact
Newton-Raphson exhibits quadratic convergence: each iteration roughly doubles the number of correct digits, so starting with a reasonable guess often yields full precision in just a handful of steps.
Why this is interesting
You try to solve an equation like x^3 - 2x - 5 = 0 by hand and feel stuck. But what if there's a shortcut that, with a few steps, gives you a very accurate answer?