LIVE! IN REAL TIME! Dedekind Cuts - Constructing the Real Numbers (Step 5 Part 4) #4.3.1.4h

I read through Landau's proof of multiplicative inverses and try to make sense of it while the cameras are rolling. I have not read it before, so I decided to read it live rather than reading it first then presenting what I learned to you after.

At last we are proving that real numbers exist! Using Dedekind Cuts. In this video we continue proving that every cut has a multiplicative inverse.

After constructing the real number field, we will move on to the climax of this mini-series, proving the uniqueness of the real number field. We will then lay the foundation of multivariable calculus before moving on to complex numbers.

It turns out Elliptic Curves over Complex Numbers are central to Wiles's proof of Fermat's Last Theorem. With the first videos on this channel, we managed to keep the math very simple, almost exclusively nothing more advanced than algebra. Now, to complete our brief tour of elliptic curves, we have to take an ENORMOUS leap forward in the level of the mathematics involved.

I want to make sure we are all on the same page before we begin learning the more advanced stuff in earnest. I am going to assume we are all familiar with high school math and college math through first or second year calculus. I'm going to introduce complex numbers and complex integration in some detail with the next several videos. We need to examine multivariable calculus as well.

This series will culminate with the Weierstrass Equation, which is the thing that connects elliptic curves to complex numbers, and thus allows us to connect them to modular forms, which is what Wiles's proof is all about.

Here is the outline of this series on complex numbers:
0. Introduction
1. The Real Number System (We are here)
2. Complex Numbers
3. Complex Functions
4. Exponential and Trigonometric Functions
5. Complex Integration
6. Cauchy's Integral Theorem
7. Cauchy's Integral Formula
8. Laurent Series
9. Complex Residues
10. Lattices and Doubly Periodic Functions
11. Lattices and Tori and Groups
12. The Weierstrass p-Function
13. The Weierstrass Equation: Complex Functions and Elliptic Curves

Please leave any questions, comments, or suggestions in the comments below!

Credits:
Music: "Oberkorn" and "St. Jarna" by Depeche Mode.

Bibliography:
LANDAU, Edmund. "Foundations of Analysis."
RUDIN, Walter. "Principles of Mathematical Analysis."