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You add infinite number of positives numbers to each other and you get -1 as a result. p-adic numbers are adding some sense to this seemingly contradictory result. Great intro: https://www.youtube.com/watch?app=desktop&v=XFDM1ip5HdU

⬐ padics

Just to give this some more context, since I feel Grant sort of rushes at the end a bit when he covers this and goes from powers of two to 1,3,7,15..

Grant shows:

    1=1
    3=1+2
    7=1+2+2**2
    ..

And so utilises a bit of syntactical sugar that I think can be a bit confusing where he calculates for the viewer only 2**0 and 2**1 and requires the viewer to do rest of the power calculations and the sums in their head to see the relation.

    1+2+4+8+16+..==-1
    is the same as saying
    2**0+2**1+2**2+2**3+..==-1

    If we look at the partial sums, successively adding new powers of two:
    
    power          reals         2-adics distance metric
    2**0           ==  1         == -1
    2**0+2**1      ==  3         == -1
    2**0+2**1+2**2 ==  7         == -1
                   (DIVERGES)    (CONVERGES)
    and on ad infinitum

This has an interesting characteristic that an infinite sum in the reals that diverges, "gets larger", converges, "settles on a number".

What is special about it?

In physics you will hear complaints about equations that have infinities that diverge, or "blow up". This can cause many problems in physical models.

This toy example shows one such infinity that blows up in the reals, but converges in the 2-adics.

I think this leaves mathematicians and physicists wondering if this technique can be used to remove, or better understand, the problematic infinities they encounter.

What are we missing?

That's a great question, and one that mathematicians and physicists are trying to answer. Jump in there and give answering it a try! We could always use more help!

https://en.wikipedia.org/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B...

3lue1brown has a great visualization of constructing p-adic numbers and calculating their metric through the lens of 2-adic numbers:

https://m.youtube.com/watch?v=XFDM1ip5HdU

This article fills in some bits omitted from the video: https://www.quantamagazine.org/how-the-towering-p-adic-numbe...

So many things I could say about this, but I"ll let Alan Kay speak for me - https://youtu.be/p2LZLYcu_JY Alan talks about how ideas in Calculus could be manifested from an early age and a richness in understanding built up as they aged up:

- at very young ages, kids really respond well through "doing" / the enactive channel. When asked to draw a circle, kids in Papert's group would first emulate what a LOGO turtle would do by rotating their body in a circle (making tiny increments in x and y).

- as they got a bit older, the visual / iconic channel was more developed and they could understand the abstraction of a circle on pencil/paper and how the concepts carried over there

- closer to early teens, symbols were much easier to grasp and relate to, etc.

With this context in mind, there have been some cool efforts to mix the second and third channels I just mentioned to communicate advanced math concepts. Vi Hart and Grant Sanderson's youtube channels come to mind. Here are my favorite videos by Grant:

- "What does it feel like to invent math": https://www.youtube.com/watch?v=XFDM1ip5HdU

- On the visual intuition behind a hard problem on the Putnam exam - "The hardest problem on the hardest test": https://www.youtube.com/watch?v=OkmNXy7er84

- From vectors to matrices to vector spaces to higher-level ideas like the link between linear algebra and calculus: "Essence of linear algebra" series: https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQ...

What I especially like about Grant's videos, is that he often walks through what a mathematician would do, the questions she would ask, etc.

⬐ allengeorge

Note that he does have a webpage [0] and you can support his continued work via Patreon [1].

[0]: https://www.3blue1brown.com/

[1]: https://www.patreon.com/3blue1brown

⬐ risefromashes

I didn't understand the proof for 0.5 + 0.25 + ... = 1 itself. By his visual number line analogy, wouldn't 2/3 + (2/3)^2 + ... = 1 too? Visually, that too seems to "approach" 1.

⬐ ssivark

Coming from someone who is "used to" this idea for a long time, I'm not sure how helpful my explanation would be to a learner, but I decided to take a shot at it, just in case it helps someone. That said, in my experience, a significant part of "understanding" involves getting used to things and expanding the scope of what you consider "natural".

--

Suppose you wish to walk from 0 to 1. You'll have to walk half the distance. Then half of the remaining half. Then half of the remaining quarter. Then half of the remaining eighth. If you keep doing this "asymptotically", you will get very very close to 1 and (almost) reach there. That's the basic idea.

This is also known as Zeno's paradox: https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Paradoxes_o...

Instead of halving step sizes on each iteration, you could reduced by a different fraction. In general, this is called a geometric series: https://en.wikipedia.org/wiki/Geometric_series

When the ratio is (2/3), after two steps you overshoot 1. (You can see that (2/3) = 0.66, (2/3)^2 = 0.44 )

⬐ leereeves

The number line analogy does work for 2/3, in the same way it worked for 9/10.

It represents the repeating trinary number 0.22222... = 1.

But the analogy only corresponds to p + p^2 + p^3 ... = 1 when p = 1/2 because that's the only time the remainder = p.

For 2/3, the number line analogy instead gives 2/3 + 1/3 * 2/3 + (1/3)^2 * 2/3 ...

⬐ evanb

It approaches 2. 2/3+(2/3)^2 = 2/3 + 4/9 = 6/9+4/9=10/9 > 1, and all the terms are positive, so as you add more terms it will get farther from 1. If you do 10 terms you get to 1.96532somethingsomething. If you do 100 terms you get 1.[seventeen 9s]508somethingsomething.

⬐ BillBohan

leereves is correct.

The sequence is 2/3 + 2/9 + 2/27 + 2/81 + 2/243 + ...

and it does approach 1

In your second term you should take 2/3 of the remaining 1/3, not (2/3)^2

⬐ Houshalter

I like the algebraic approach:

    x = 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ...
    2x = 1 + 1/2 + 1/2*1/2 + 1/2*1/2*1/2 + ...
    2x = 1 + x
    x = 1

⬐ afarrell

Does anyone know of a book, channel, or subreddit which is focused on critiquing the explanations in videos and blogs like this and talking about the art of explaining technical concepts?

⬐ posterboy

I don't think that would be straight forward, because of competing ideas.

⬐ afarrell

Hence why I suspect it would make more sense as a subreddit or other forum, where you have people debating what does or does not make a good explanation.

I guess it would be kinda like literary analysis, but with a much clearer practical point.

⬐ posterboy

for children or adults, general or math specific, school/university/doctorate context or just plain how to make things understood? Because the latter is basically just what mathematics should be, see http://etymonline.com/index.php?term=mathematic

⬐ allemagne

This isn't directly what you're asking for, but the book "How To Solve It" by George Polya is a great tour of how to be a teacher of mathematical concepts and how to "simulate" that teacher in your mind if you're encountering a problem no one else has.

https://www.amazon.com/How-Solve-Mathematical-Princeton-Scie...

⬐ Thoreandan

I especially like the 3Blue1Brown videos on Hanoi/binary-counting and explaining (visually) Euler's Identity.

⬐ khana

Been there. It's beautiful. Do it now.

⬐ kbd

As seems typical, this video goes on for a while and is straightforward until it completely loses you all of a sudden. I have no idea what would even motivate the separation of numbers into "rooms" like the video shows, let alone how that explains why an infinite positive sum = -1.

⬐ taneq

"Step 2: Draw the rest of the owl."

⬐ espeed

The infinite sum makes sense when using the p-adic number system [1] -- and measuring distance using the p-adic metric -- which forms an ultrametric space [2]. An interesting related concept is the Bruhat–Tits building [3].

For an intro on p-adic numbers, read these two short articles "A first introduction to p-adic numbers" [4] and "A Tutorial on p-adic Arithmetic" [5] or see the short video "Introduction to p-adic Numbers" [6]:

[1] https://en.wikipedia.org/wiki/P-adic_number

[2] https://en.wikipedia.org/wiki/Ultrametric_space

[3] Bruhat–Tits building https://en.wikipedia.org/wiki/Building_(mathematics)

[4] A first introduction to p-adic numbers http://www.madore.org/~david/math/padics.pdf

[5] A Tutorial on p-adic Arithmetic https://koclab.cs.ucsb.edu/docs/koc/r09.pdf

[6] Introduction to p-adic Numbers https://www.youtube.com/watch?v=vdjYiU6skgE

⬐ cderwin

I think the problem is that the introduction of the p-adic metric was poorly motivated. It's not clear why we would want 1+p+p^2+p^3+... to converge to -1, and introducing the p-adic metric to do so doesn't show why the p-adic numbers are useful in general (all it's really saying is that the partial sums are 2^(n+1)-1). I'm sure you can derive all sorts of weird metrics so that various weird identities are true; that alone fails to make them interesting. Based on this video alone it's not clear that either the identity or the p-adic norm are interesting in any non-trivial sense of the word.

The result is that the introduction of the p-adic metric is hard to follow and the resulting identity seems arbitrary, even if you manage to follow the bit about the metric.

(And these combined with a lack of rigor where it's needed seem to be recurring problems in 3Blue1Brown videos.)

⬐ stablemap

My short advertisement is that it has proven very useful to study Diophantine equations by reducing mod n. The Chinese remainder theorem [1] tells you to focus on reducing mod p^r, where p is a prime. If r = 1 then you are working in a field but in the ring Z/4, for example, I know 2 ≠ 0 and yet 2·2 = 0.

To make the p-adics Z_p I stitch all of these Z/p^r together: an element is a choice of a_r in each Z/p^r and these have to be compatible: a_2 reduces mod p to a_1 and so on. The resulting Z_p has no "zero divisors", and if I allow myself to invert p I get a field Q_p.

This is a huge improvement, a foundation on which to build analysis and geometry as we did over R.

[1] https://en.wikipedia.org/wiki/Chinese_remainder_theorem

⬐ posterboy

>doesn't show why the p-adic numbers are useful in general

well, you certainly are neither more help either ;)

⬐ t-ob

> I'm sure you can derive all sorts of weird metrics so that various weird identities are true

On the rational numbers, at least, the p-adic metrics are more or less your whole lot, according to Ostrowski's Theorem [1].

There is a kind of cognitive hurdle everyone who studies these numbers has to clear, in that things that should be "large" turn out to be very small indeed, when viewed under a p-adic lens. I think it's more instructive to build up the ring of p-adic integers first [2, chapter 2], and construct the p-adic numbers from there. I can assure you they are very useful, though! A general theme in number theory is to take a "global" problem, defined over the integers, and to translate it into infinitely many "local" ones (over the p-adics, for each prime p). These are sometimes easier to solve and, if you're lucky, offer insight into the global solution you're looking for.

[1]: https://en.wikipedia.org/wiki/Ostrowski's_theorem

[2]: http://www.springer.com/gb/book/9780387900407

I loved this video https://www.youtube.com/watch?v=XFDM1ip5HdU "An exploration of infinite sums, from convergent to divergent, including a brief introduction to the 2-adic metric, all themed on that cycle between discovery and invention in math."