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The p-adic numbers are bizarre alternative number systems that are extremely useful in number theory. They arise by changing our notion of what it means for a number to be large. As a real number, 1 billion is huge. But as a 10-adic number, it is tiny! ---------------- References: A result known as Hensel's lemma gives conditions for Newton's method to work in the p-adic numbers. Bézout's identity can be used to prove that the numbers from 2 to p-2 pair up perfectly, and the partner of a given number can be computed using the Euclidean algorithm. The 2-adic limits arising from the (2^n)th Fibonacci numbers were established in (page 216). ---------------- 0:00 Introduction 2:16 Properties of the real numbers 3:19 10-adic integers 6:55 Properties of the 10-adic integers 10:06 Division? 12:47 Limit points 13:50 5-adic limit 15:36 Fibonacci numbers 16:31 Square roots of -1 18:25 What are p-adics good for? ---------------- Animated with Manim. Music by Marc Rowland and Cody Leavitt. Thanks to @catpfaff for helpful feedback on an earlier version. Web site: Twitter:
