Math

How Newton Changed the Way We Calculate Pi

For millennia, mathematicians calculated π using geometry and slow convergence. Then Newton introduced a revolutionary approach using calculus, drastically accelerating the process.

References:

• Books and Academic Sources:
  • Arndt, J., & Haenel, C. (2001). Pi-Unleashed. Springer Science & Business Media. Link
  • Dunham, W. (1990). Journey Through Genius: The Great Theorems of Mathematics. Wiley. Link
  • Borwein, J. M. (2014). The Life of π: From Archimedes to ENIAC and Beyond. In From Alexandria, Through Baghdad (pp. 531–561). Springer. Link

Special Thanks:

• Alex Kontorovich – Professor of Mathematics at Rutgers University and Distinguished Visiting Professor for the Public Dissemination of Mathematics at the National Museum of Mathematics (MoMath)

Credits:

• Written by: Derek Muller & Alex Kontorovich
• Animation by: Ivy Tello
• Filming: Derek Muller & Raquel Nuno
• Edited by: Derek Muller
• Music: Jonny Hyman & Petr Lebedev
• Additional Music: Epidemic Sound
  • "Particle Emission"
  • "Into the Forest"
  • "Stavselet"
  • "Face of the Earth"
  • "Firefly in a Fairytale"
• Thumbnail by: Gianmarco Malandra & Karri Denise

How Giving Up Reality Solved the Cubic Equation

For centuries, a general solution to the cubic equation was considered impossible—until mathematicians embraced complex numbers, abandoning the strict requirement that math reflect physical reality.

References:

• Videos:
  • 500 Years of Not Teaching the Cubic Formula – YouTube
  • Imaginary Numbers are Real – YouTube
• Books and Papers:
  • Dunham, W. (1990). Journey Through Genius: The Great Theorems of Mathematics. Link
  • Toscano, F. (2020). The Secret Formula, Princeton University Press. Link
  • Bochner, S. (1963). The Significance of Some Basic Mathematical Conceptions for Physics, Isis, 54(2), 179–205. Link
  • Muroi, K. (2019). Cubic Equations of Babylonian Mathematics, arXiv:1905.08034. Link
  • Branson, W. (2014). Solving the Cubic with Cardano. Link
  • Rothman, T. (2013). Cardano v Tartaglia: The Great Feud Goes Supernatural, arXiv:1308.2181. Link
  • Siadat, M. V., & Tholen, A. (2021). Omar Khayyam: Geometric Algebra and Cubic Equations, Math Horizons, 28(1), 12–15. Link
  • Merino, O. (2006). A Short History of Complex Numbers, University of Rhode Island. Link
• Historical Works:
  • Cardano, G. (1545). Ars Magna (Dover reprint, 1993). ISBN: 0-486-67811-3
  • Bombelli, R. (1579). L’Algebra. Link
• Software:
  • Manim Community Developers. (2021). Manim – Mathematical Animation Framework (v0.13.1). Link

How an SAT Question Became a Mathematical Paradox

A fascinating breakdown of an SAT question that led to widespread confusion and uncovered a deeper paradox in mathematics.

References:

• MindYourDecisions. Summary of this problem: YouTube
• Kyle Hill. More cool math about this problem: YouTube
• MinutePhysics. Discussion of a solar day: YouTube
• Murtagh, J. (2023). The SAT Problem That Everybody Got Wrong, Scientific American: Link
• United Press International (1982). Error Found in S.A.T. Question, New York Times: Link
• Yang (2020). Hardest SAT math problem?, Quora: Link
• Coin rotation paradox, Wikipedia: Link
• Simmons, B. (2015). Circle revolutions rolling around another circle, Math StackExchange: Link
• Sidereal time, Wikipedia: Link
• Solar Time vs. Sidereal Time, Las Cumbres Observatory: Link
• Zotti, G., et al. (2021). The Simulated Sky: Stellarium for Cultural Astronomy Research: Link
• Newspapers from 1980s–1990s, Newspapers.com: Link
• SAT Practice Test, College Board: Link
• Revolution Definition, NASA: Link
• Revolution Definition, Merriam-Webster: Link
• Earth motion animation, NASA: Link
• Satellite animation, NASA: Link