Omar Khayyam and the Binomial Theorem | Famous Math Problems 5 | NJ Wildberger

Today's famous problem is how to expand a power of a binomial, in modern terms how to write out explicitly (a+b)^n when n is a natural number, n=1,2,3,.... This is a familiar topic to all students of mathematics, highlighting its central importance and interest. Omar Khayyam, the Persian mathematician, astronomer, poet and philosopher who lived in Nishapur (in modern Iran, not Iraq as I mistakenly stated in the video!) around 1100 A.D. was probably the first to solve it in general, although the case n=2 was known to the ancient Greeks, and the case n=3 was familiar to the ancient Indian mathematicians, in particular Aryabhata around 500 A.D. A Chinese mathematician Chu Shi Chieh wrote down what we now call Pascal's triangle of Binomial coefficients around 1300. In this video we explain the modern approach to this problem, along with the less well known extension called the Trinomial theorem. We mention a few applications to calculus. And we read a few quatrains from Khayyam's epic poem The Rubaiyat, translated into English from Arabic by Edward Fitzgerald. Video Content: 00:00 Introduction 7:08 Omar Khayyam's "Treatise on Algebra" (1070) 8:52 Chinese discovery: (1300) Chu Shi-Chieh 9:58 Expansion of (a+b)^n and coefficients 21:21 B. Pascal and his triangle 27:02 The Binomial theorem and applications ******* Research Gate page: https://www.researchgate.net/profile/... Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon:   / njwildberger   Your support would be much appreciated. Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriF... Insights into Mathematics Playlists:    • The Algebra of Boole, Logic and Circuit An...   (31 videos)    • Box Arithmetic: a new framework for Mathem...   (18 videos)    • Hypergroups and Diffusion Symmetry: an int...   (6 videos)    • Rational Trigonometry for maths, physics a...   (4 videos)    • Sociology and Pure Maths   (44 videos)    • Old Babylonian mathematics and Plimpton 322   (8 videos)    • Math Foundations   (226 videos)    • Math Seminars N J Wildberger   (26 videos)    • Math History (ancient to modern)   (45 videos)    • Geometric Linear Algebra   (43 videos)    • Algebraic Topology   (40 videos)    • Universal Hyperbolic Geometry   (55 videos)    • Differential Geometry   (34 videos)    • Elementary Probability and Statistics   (8 videos)    • Math Terminology for Incoming Uni Students   (9 videos)    • Famous Math Problems   ( 46 videos)    • Elementary Mathematics Explained (K-6)   (40 videos)    • Ancient Mathematics and insights of Howard...   (7 videos)    • Wild West Banking: A mathematician goes We...   (7 videos)    • Playing Go: the ancient oriental board game   (19 videos)    • Maths and Music   (21 videos)    • Year 9 Mathematics (review fractions, deci...   (10 videos)    • Wild Trig: An introduction to Rational Tri...   (94 videos) Wild Egg Maths Playlists:    • Intro to Algebraic Calculus with Box Arith...   (4 videos)    • Classical to Quantum (for Members)   (64 videos)    • Solving Polynomial Equations and the Geode...   (45 videos)    • De Casteljau Bezier curves and associated ...   (20 videos)    • Exploring q-series (for Members)   (8 videos)    • Six: A mathematical exploration   (9 videos)    • Algebraic Calculus One: a new foundation f...   (52 videos)    • Advice to prospective research mathematici...   (9 videos)    • The Hexagrammum Mysticum: a Geometric Gem ...   (14 videos)    • Algebraic Calculus Two   (8 videos)    • Special (Polynomial) Functions and Maxel N...   (25 videos)    • Dynamics on Graphs: The ADE Phenomenon & B...   (30 videos)