Van Aubel's Theorem has a Beautiful and Fun Proof Using Complex Numbers (3Blue1Brown SoME1) скачать в хорошем качестве

Van Aubel's Theorem has a Beautiful and Fun Proof Using Complex Numbers (3Blue1Brown SoME1)

In this video, we prove Van Aubel's Theorem in a fun and beautiful way. We use the algebra and geometry of complex number addition, subtraction, and multiplication to prove a surprising relationship between squares on the sides of any quadrilateral, which is Van Aubel's Theorem. The lines connecting midpoints of opposing squares are perpendicular and have the same length, no matter what the quadrilateral is. See: "Visual Complex Analysis", by Tristan Needham: https://amzn.to/3zY7B7N. My math blog: https://infinityisreallybig.com/ This is my submission for 3Blue1Brown's SoME1: https://www.3blue1brown.com/blog/some1 Van Aubel's Theorem: https://en.wikipedia.org/wiki/Van_Aub... Original Title: The Beauty is in the Proof - Complex Numbers and Quadrilaterals (3Blue1Brown SoME1). Changed the name in the morning on Sunday, November 21, 2021. Second Title: The Beautiful Geometry of Complex Numbers and Quadrilaterals (3Blue1Brown SoME1) #3Blue1Brown #SoME1 #SummerOfMathExposition 🔴 "Visual Complex Analysis", by Tristan Needham: https://amzn.to/3zY7B7N 🔴 Complex Numbers are Real at Infinity is Really Big: https://infinityisreallybig.com/2021/... The line segments between midpoints of squares on the opposite sides of a quadrilateral are perpendicular to each other and have the same length. The geometry of complex number of arithmetic can be used to prove this. If we represent complex numbers as vectors, then there are four key geometric interpretations that are useful: 1) the parallelogram law for vector addition, 2) vector subtraction v - w represented by a vector from w to v, 3) multiplying a complex number by a positive number (scalar) rescales the complex number, 4) multiplying a complex number by the imaginary unit i rotates the vector counterclockwise (anticlockwise) by 90 degrees. Mathematical Beauty in Truth and Proof - Complex Numbers and Squares on Quadrilaterals Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter:   / billkinneymath   🔴 Follow me on Instagram:   / billkinneymath   🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ ⏱️TIMESTAMPS⏱️ (0:00) Why math is beautiful (0:13) Draw squares on a quadrilateral and connect the midpoints (1:21) What are complex numbers? (1:54) Complex plane and complex vectors (2:47) Complex number addition and parallelogram law for vector addition (3:53) Head to tail addition (5:00) Complex number subtraction and geometric interpretation (5:53) Multiplication by a positive real number (scalar) (6:54) Multiplication by i is a counterclockwise rotation by 90 degrees (7:55) The Proof AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.