To find where a curve crosses zero, stand at a guess, slide down the TANGENT line to where IT hits zero, and stand there instead — repeat. Each step roughly DOUBLES the number of correct digits, so a handful of steps nails a root to machine precision. It’s how calculators do square roots and how solvers find equilibria. Slide to step and watch the guess rocket onto the root.
Newton–Raphson finds a root of f by linearizing: from a guess xₙ, follow the tangent to its own zero, xₙ₊₁=xₙ−f(xₙ)/f′(xₙ). Near a simple root convergence is QUADRATIC — the error squares each step, doubling the correct digits — so √2 (root of x²−2) reaches full double precision in ~6 steps. It generalizes to systems (with the Jacobian) and underlies most nonlinear solvers and optimizers. A fail-loud self-check throws unless iterating on x²−2 reaches √2 to 1e−12. ◆ real numerical methods, node-verified.
Newton needs the derivative and a decent starting guess: it can diverge, cycle, or stall at f′=0, and convergence drops to linear at multiple roots. The quadratic-convergence and tangent-step math are exact.