How to develop a proper theory of infinitesimals I | Famous Math Problems 22a | N J Wildberger

Infinitesimals have been contentious ingredients in quadrature and calculus for thousands of years. Our definition of the term starts with the Wikipedia entry, modified a bit to reduce the dependence on "real numbers", which is actually quite unnecessary--- but as a logical definition it is still clearly unsatisfactory. A quantity which is positive and non-zero but smaller than any other strictly positive rational number: does this make any sense?? Is there a modern way to establish these mysterious quantities without resorting to philosophical or logical hand-waving? Yes there is, and it involves yet another application of the remarkable Dihedron algebra that we introduced in the previous Famous Math Problem 21 on the true complex numbers. This first video sets the stage, reviewing in some details Archimedes' approach to the quadrature of the parabola using The Method of infinitesimal balancing based on his Principle of the Lever. Then we move to the 16th century with the work of Cavalieri and the Leibniz with the foundations of Calculus. And then to the 1960's with the introduction of non-standard analysis of Laugwitz and Robinson. Our approach is based on the dual complex numbers, originally introduced by Clifford in the 1870's. ************ 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)