p adic numbers. Part 1 of 3: 10-adic integers.
Channel: Richard E Borcherds
Date: 2020-09-03

This is the first part of a 3-part talk on p-adic numbers for advanced high school students. It is part of a series organized by the Berkeley mathematics circle.

We define the 10-adic integers, which are similar to ordinary integers except their decimal expansion can be infinitely long. We show that one can do addition, multiplication, and subtraction with them. However they have some problems because it is possible for two nonzero 10adic integers to have zero product. We will see how to fix this in the next video.

Links related to the video:
Berkeley math circle: https://mathcircle.berkeley.edu/
Handout for talk: https://drive.google.com/file/d/1-UUjAqOIPK-PWUElAXWTdXJZVq7LFGAe
Part 2 of talk: https://youtu.be/PeM5Hp0gWf4
Part 3 of talk: https://youtu.be/FWE3mA2otTU

Further reading:
Borevich and Shafarevich, Number theory, chapter 1.3 (advanced)
J.-P. Serre, A course in arithmetic, chapter II (more advanced)
N. Koblitz, p-adic numbers, p-adic functions, and zeta functions (very advanced, for anyone who is really ambitious)