Math

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?

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

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

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