Veritasium - The Discovery That Transformed Pi

Short Summary

Main recommendation: Compute pi with the binomial series that Isaac Newton derived in 1666, rather than the polygon-bisection method that dominated for about 2,000 years. The video shows that Newton extended the binomial theorem beyond positive integers, integrated the resulting series for a quarter circle, and tuned the integration limit to speed convergence.

Key Points

• Polygon bisection starts from a hexagon bounding pi between 3 and 4, then refines with 12-, 24-, 48-, and 96-sided polygons; Archimedes reached pi between 3.1408 and 3.1429 in 250 BC.
• Later bisection efforts were pure precision contests: Ludolph van Ceulen used a polygon with 2^62 sides over 25 years to get 35 correct decimal places, later inscribed on his tombstone.
• Newton, quarantining during a 1666 bubonic plague outbreak at age 23, noticed that Pascal's triangle coefficients could be generalized.
• Newton applied the binomial theorem to negative and fractional exponents, producing infinite series such as 1/(1+x) and (1+x)^(1/2).
• Newton's series for the unit circle, integrated and evaluated at a limit of 1/2, gives a rapidly converging formula, so about 50 terms match van Ceulen's polygon accuracy.

Details Worth Noting

• Concrete instructions: To build Pascal's triangle, start each row with 1 and add each pair of neighboring numbers from the row above to get the number below; the coefficients of (1 + x)^n are the numbers in row n.
• Concrete instructions: Use the binomial expansion (1 + x)^n = 1 + n·x + [n(n−1)/2!]·x² + [n(n−1)(n−2)/3!]·x³ + ..., which terminates for positive integer n but is infinite otherwise.
• Concrete instructions: Derive the circle formula by solving x² + y² = 1 for y, giving y = (1 − x²)^(1/2), then expand with n = 1/2 and replace x with −x².
• Concrete instructions: Integrate the series term by term from 0 to 1/2, which shrinks each term by an extra factor of x² = 1/4, and use the sector-plus-triangle area (π/12 + √3/4) to isolate π.
• The first five terms yield π ≈ 3.14161, about two parts in 100,000 off, and the video notes that a comparable result needs only about 50 terms.
• Newton used the same series to compute √3 quickly by writing 3 = 4(1 − 1/4), so √3 = 2·(1 − 1/4)^(1/2), giving a rapidly converging expansion.
• The video notes Newton's calculus was called "the theory of fluxions" in his own terms, and that the binomial pattern for negative integers matches Pascal's triangle rotated on its side.

Bottom Line

Newton's method replaced thousands of years of polygon-counting with an infinite series that converges far faster, showing that pushing patterns beyond their original domain can yield large computational gains.