p adic numbers. Part 1 of 3: 10-adic integers. 

When you take -(-2^63) you get -2^63 because you're not working in the integers NOR are you working in the 2-adics; you're working in Z/(2^64)Z, the integers mod 2^64, which is an approximation of the 2-adics. Here, -2^63 is its own additive inverse, and why you get itself when you negate it.

This has instantly become my favourite introduction to p-adic numbers :)

Adding 10-adic numbers seems to make more sense than adding real numbers, especially the part about adding from right to left

Thanks Professor Borcherds, these are great. Will you be doing a course on Lie groups and Lie algebras?

If instead of using a little digit clicker he had used a Curta mechanical calculator, he could then have gone straight on to 10s complement subtraction, because that's how the Curta does it. By shifting the winder, to subtract 716 it adds 2 in the 7 column, 8 in the 1 column and 3 in the 6, and then there's a very clever bit of the mechanism which adds one in the least column.

Thank you very much for introducing the concept through this sort of discovery approach. I had come across p-adics briefly before, but this gave me a much larger intuition for how n-adic numbers are represented and why p-adic is the naturally more interesting system to study.

This is wonderful. Wish I had this a few years ago, but very glad to have it now. P-adic long division truly is a Herculean task.

Really good video. A child can understand this.

his hair matches his sweater ;o great video!

it's nice to see such a complicated topic exlained in an easy to understand way.

I appreciate the analogy of an odometer a lot and the use of the counter tool; it helped me get a better mental model of these numbers. Thank you! 🙏

Hello Professor Borcherds, In case you have not already heard the song, I highly recommend "New Math" by Tom Lehrer. I was reminded of this song at 7:37 in your talk. Thank you, Zubin

Thank you for sharing your knowledge. Very interesting!

P adic numbers are the missing link for understanding the universe. For me I like to call them just Infinite Numbers. Just when I go into detail i think calling them p-adis is better

Thank you very much professor for this beautiful introduction :)

You should take 10's complement, not 9's complement.

Crystal clear explanation of a subject that can be rather confusing. Thank you very much.

Thank you for a wonderful lecture

25 is also a 2-adic number

Thank you

I love your series (as usual) but must add a correction. At 34:37 you state that (...357418751)^2 is a square root of 1. When this is squared it actually gives a value of (...400001). The value(s) I get are (...574218751) and its negation (...425781249).

Thank you

a brilliant person clearly explaining a difficult concept. Thank you

I love your videos Dr. Borcherds, please keep them coming.

how do know if the number neg or pos?

I'm confused why you don't seem to be aware that all decimal expansions are 10-adic. sure, pi = 3.141592653589793238462643383279502884197169399375105920974944U but it's also R00000000000000000000000003.141592653589793238462643383279502884197169399375105920974944U where U indicates the next digit is unknown, and R indicates a repeating pattern in the next digits similarly, 1/3 yields a decimal expansion of R0.3R you might object by saying that the 10-adic representation of 1/3 is R3.0R -R9.0R, and thus 1/3=R6.0R, but note what's actually happening here. R0.3R is the mirror image of R3.0R, and they're also vectors with the same magnitude but opposite sign. that is, the orthography indicates their symmetrical opposition without need for introducing a concept of positive and negative. so it becomes clear that R6.0R=R0.3R, just as 2/6=1/3. they are merely different names for the same value, and once this is recognized it becomes obvious why p-adic notation is not at odds w/ conventional arithmetic. the reason is we always use p-adic notation, we simply don't use it fully.

Woah, that's cool

Great video!

This is really so extremely good. Thank you prof. Also you seem like such a nice man. Such a nice person. So calm and patient. I used to be a 70-year-old college freshman and now today, because of your generosity, I am a 70-year-old high school senior. Thank you!