Are the real numbers... REAL?

The first video in a series which will build up to defining the p-adic numbers. In this episode, we break down our intuitive understanding of the real numbers, and build it back up again using the formality of Cauchy sequences.
#some2 #SoME2

Introduction (0:00)
Non-definitions (0:50)
Playing with sequences (3:35)
The definition of Cauchy sequences (5:31)
Important side note! (8:35)
Quotient your problems away! (9:11)
The definition of the real numbers (14:00)
0.999... = 1 (14:57)
Denouemont (16:35)

[pra.]

Wish my analysis class started with this video

[hybmnzz]

This doesn't scratch the surface of the wildness of the real numbers but I'm glad someone is calling out the real numbers instead of "did you know imaginary numbers aren't that imaginary?"

[thestemgamer]

The animations definitely played a role in the understanding of the video.
For myself, I struggled to initially understand Cauchy Sequences until I had finally been able to draw that visual in my mind where you have a distance that is slowly decreasing to zero. It's not entirely obvious when you just read the rigorous definition, especially in plain text.

[lexinwonderland]

It still amazes me that this didn't absolutely KILL it with SoME2. The visualizations are so incredibly well made, and you know I'm already a fan of your explanation of very advanced topics in an accessible way. Such a great rewatch! <3

[lexinwonderland]

Excitedly waiting for the future videos you hinted at! What a wonderful video!

[PhysicsBeyond]

Absolutely brilliantly done! I really like how you narrate the thinking process of how one could discover the Cauchy-completion construction of the rationals oneself by asking natural questions about the reals and then embarking on using the concept of approximation, which is at the heart of Analysis. Really great animations, too! I love how you pose problems to engage the audience in thinking along and ahead. Guess the end will be a bit harder to digest for someone unfamiliar with quotients by equivalence relations, but you refer to other videos to catch up on that. Very well done!

[ryanjbuchanan]

VERY well made! The animations are stunning here.

[MATHsegnale]

Very nice video! It becomes very clear with the graphics and the animation, also I liked a lot finishing the video with the = 1 fact. Thank you very much!

[musictheorytree]

So happy to see you taking on this format, Kris!! Bravo!

[paulinahering]

What a beautiful animation and lovely way of presenting and explaining! :)

[gergelybencsik]

Great video! Incredibly underrated too! I like the style of this low frame-rate animation, reminds me of stop-motion. Love the R transition to 3D in the beginning, the PI rotation at 2:09, squiggles! at 4:20. And the colors! Would you share what tools and techniques you used to make this? :)

[Mathinity]

What a great video! I feel like you've done a really good job explaining how the real numbers are constructed. I've been struggling to understand that concept while doing Analysis 1 three years ago and I think that this video would've helped me a lot because of the great visuals. Good job! Hope your video does well in SoME2! #peer_review

[gustavosalessi]

Hi, i have a question, sorry if it is too basic. Every time i see that “0, 999.. = 1” i get a bit confused when i think about the cantor’s diagonal argument, because for me he uses this common sense definition of the real numbers, i don’t know, what am i missing here? Anyway thank you for the video.

[callanmcgill]

Interested to know what software you used to make the video?

[PhilipSmolen]

I love the math, but "real" numbers always feel real to me. When I take a measurement in the "real" world, I know that it's just an approximation, and I could get better and better approximations if I had better equipment, and the "real" answer is what I'm getting closer and closer to. I learned about ∂ and ε and rational vs irrational much later, in school. But that was just a way for formalize these notions.

[hawbster]

Beautiful video but actually 0.9 is not equal to 1 they are very different