The Distance Between Numbers - Numberphile

The one thing that I think would make this more compelling is if there was some explanation of why we would ever want to use something like the 2 adic for distance. There was some hunting toward it being relevant due to p adic being generalizable and better fitted, but starting with that I think would have brought some more context to why we inventing this in the first place. It kind of feels like we are creating this method of finding distances in order to show this strange result rather than this strange result being a product of something that has more obvious uses.

I think the biggest problem with this video is that Tom didn't fully explain what the p-adic numbers actually are, which makes this distance function seem arbitrary without context. It's not just that this function is technically a distance function because it follows rules x, y, and z; this function literally defines what distance means for the p-adic numbers. Eric Rowland posted a great video on the p-adics a while back that I think gives much needed context here.

This has -1/12 vibes EDIT: Alright everybody chill, I know it's p-adic distances, I just said it has the vibes of the "-1/12" video because of the original silly statement. Of course you can also say that 5+5=12 but in the octal number system, relax please

P-adic numbers are cool because they don’t just define a new distance, but an entirely new calculus where you can still take derivatives and infinite series, but limits which didn’t exist in the real numbers suddenly exist in p-adics. There’s even a sense in which e and pi can be found in certain extensions of p-adic numbers.

My favourite thing about p-adic numbers (and ultrametric spaces in general) is that every triangle is isosceles. Fun stuff.

The missing part is that there isn’t just one way to organize numbers on a number line. If you reorganize them to adhere to a p-adic system, they will now be in a point cloud where the distances you measure between them is now aligned with the p-adic formula.

"We're going to look at a sequence and show that it converts to a limit you weren't expecting." He forgot the "...in this space that you had no reason to think about."

For all you programmers out there, this is very related to “two’s complement”!

Sneakily glossed over proving that d(x,y) = 0 iff x = y. I guess you’d need to separately define this as true for the 2-adic metric, since 1/2^m can’t ever be zero.

1:00 Numbers getting bigger converging to a negative number? Oh no, here we go again!

Additional fun fact about the p-adic metric: In some ways is better than the usual distance of d(x,y) = |x-y| because for any p-adic distance we have the strong triangle inequality d(x,z) <= max{d(x,y),d(y,z)}, which is (obvious by the name) stronger that the triangle inequality mentioned in the video.

I feel like one thing that's missing in the explanation of "why are these the rules for what a distance function is", is that these are exactly the rules we need in order to be able to speak about convergence, and have nice properties like for example uniqueness of limit points.

It's the -1/12th thing all over again!

Agreeing with a lot of the comments here. Some understanding of what p-adic numbers are used for, either in the real world or some basic understanding of what they're used for in math, would have gone a long way to dispelling "this is a cool limit off a technicality"

I worried that this was going to go approximately like the -1/12 video but this was very well-explained. Nice!

Worth mentioning: The reason these "distances" matter isn't just "pure math." There are different number systems (not everything is Base-10, such as Binary, making this math valuable for computer science) and when things enter into "real world" numbers (like, say physics equations), it can be very difficult to see how things relate in "normal" number-space because they seem to have no similarities, but if you transform them into "arbitrary" number-spaces, you can find relationships and trends and such that can help you work BACKWARD to find something that WASN'T related, but all of a sudden can be shown to matter in whatever thing you're doing. I'd hazard you might find a ton of this math used in "real world" applications via things like String Theory or Quantum Mechanics, or, as I'm sure Tom is familiar, Navier-Stokes work.

I notice that Tom uses the word "modulus" for what I have always heard referred to as "absolute value" (unless I misunderstand what he's doing). Is this a difference between American and British mathematical terminology? (I'm in the US.)

I think Tom missed the chance to glimpse that 'distance' could mean the 'proximitiness' or 'likeliness' between two numbers in any sense whatsoever, one of which the shared presence of them in a given interval of the real line (which is the 'distance' we are used to call by this name, and which has a physical correspondance to our world). This also explains why the distance between a number and itself must be zero (a number shares all of its properties with itself), as well as the commutative property remains valid (they don't need to be ordered to be compared). For the triangular property, it seems they want to restrain the comparison to non-cyclic variations.

Brady’s skepticism here is all of us

Eric Rowland has a beautiful video on explaining p-adic numbers using 3b1b animations