What does it feel like to invent math? (2015) [video]

[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.

[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

[taneq]

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

[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.

[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

[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.

[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?

[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...

[posterboy]

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

[Thoreandan]

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