Math

Here is the ultimate study guide for anything and everything you need to know about quadratics. Go to jensenmath.ca for free math resources for all math topics. Hope you enjoy the video and that it helps your understanding. Please consider subscribing. It really helps the channel. 0:00 - What is a Quadratic Relationship 2:09 - Standard Form 3:53 - Vertex Form 7:01 - Factored Form 9:33 - Factoring 12:18 - Solving by Factoring 14:37 - Solving by Completing the Square 16:17 - Quadratic formula 18:32 - The Discriminant 20:41 - 3 Ways to Find the Vertex

What are these funny little brackets? They represent absolute values. What's that? Well are you here to read this, or watch the clip? Let's go?

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 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

Einstein was wrong about black holes — what else?

This video was inspired by conversations with leading physicists and key scientific literature. Below are the references:

References

• Thorne, K. (1995) – Black Holes & Time Warps: Einstein's Outrageous Legacy
• Hamilton, A. J. S. (2021) – General Relativity, Black Holes, and Cosmology – Link
• Relativity Playlist by ScienceClic – Link
• Black Hole Events by PBS Space Time – Watch
• Newton’s Letters via The Newton Project – Link
• Einstein, A. (1915) – Die Feldgleichungen der Gravitation – Link
• Schwarzschild, K. (1916) – Über das Gravitationsfeld eines Massenpunktes – Link
• Wali, K. C. (1982) – Chandrasekhar vs. Eddington—An Unanticipated Confrontation – Link
• How to Build a Black Hole – Watch
• Oppenheimer, J. R., & Volkoff, G. M. (1939) – On Massive Neutron Cores – Link
• Oppenheimer, J. R., & Snyder, H. (1939) – On Continued Gravitational Contraction – Link
• Schwarzschild Geometry by Andrew Hamilton – Link
• Why Time and Space Swap by ScienceClic – Watch
• Why All World Maps Are Wrong by Vox – Watch
• Hamilton, A. J., & Lisle, J. P. (2008) – The River Model of Black Holes – Link
• Mapping The Multiverse by PBS Space Time – Watch
• Rotating Black Hole via Wikipedia – Link
• Wormhole Travel by PBS Space Time – Watch
• Morris, M. S., & Thorne, K. S. (1988) – Wormholes in Spacetime and Their Use for Interstellar Travel – Link

🖼 Visuals & Simulation References

• ScienceClic – “What would we see if we fell into a black hole” – Watch
• Kazmierczak, J. et al. (2021) – NASA’s NICER Tests Matter’s Limits – Link
• Bridgman, T. et al. (2024) – M5.1 Flare at Active Regions 13559 & 13561 – Link
• Schnittman, J. et al. (2019) – Black Hole Accretion Disk Visualization – Link
• Wiessinger, S. et al. (2020) – A Decade of Sun – Link
• Skelly, C. et al. (2017) – What is a Neutron Star? – Link
• Earth texture – Link
• First image of Sgr A* – Link
• Image of M87 – Link
• Polarized light image of Sgr A* – Link