Offprint · Wondering About Pi · No. 3.141592653

Patience

One, minus a third, plus a fifth, and so on.

Start with one. Take away a third. Add a fifth. Take away a seventh. Keep going forever, multiply what you have by four, and you have π exactly.

Madhava found this in fourteenth-century India; Leibniz found it again in 1674. It is exact, and it is slow. A thousand steps in, you are still off by about a thousandth.

Exact is not the same as quick.

π

After 20 steps: 3.09162. Off by 0.04997.

Figure Each step overshoots, then undershoots, by a little less each time. The line is π.

“Patience.” Wondering About Pi, no. 3.141592653 (24 September 2026). wap.mom/patience/