Calculus solves hard problems by imagining them built from infinitely many simple pieces

Strogatz identifies a single unifying strategy behind calculus's astonishing range of applications: take a problem that's difficult because something is continuously curving or changing, and approximate it instead using a large number of small, simple, straight or flat pieces that are individually easy to handle — then examine what happens as the number of pieces grows toward infinity and each piece shrinks toward zero.

This approach dates back to ancient efforts to calculate a circle's area by approximating it with many thin triangular slices, refining the approximation as the number of slices increased. Calculus generalized this insight into a systematic method applicable to an enormous range of problems involving curves, rates, and continuous change.

What makes the method work is a leap of imagination: trusting that the limit of an infinite process of approximation gives an exact answer, even though no finite number of pieces ever quite gets there. That leap, once justified rigorously, opened up problems that had resisted solution for millennia. Takeaway: many seemingly impossible problems become tractable once broken into infinitely many simple, solvable pieces.