How 9801 generates a nice integer sequence

An explanation of how a fraction like 1/99^2 generates the following sequence:
1/9801 = 0.00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 ...

SOCIAL:

[lexfridman]

1/9801 = 0.00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 ...

[mockingbird]

Make more of these Math videos! Loved this one!

[preciousakpata]

I should be studying Statics II, but instead I'm watching this . . . because I set standards for my methods of procrastination!

[MrChoppertime]

I need to learn this. I like these little videos

[luismiguel.b]

Man, Im loving these videos.
This is interesting.
It reminded me when I used to be into competitive programming.

[azai.mp]

Oh wow I hadn't thought you could use the derivative like that. My first thought was 1/99 = in base 100. Squaring it then gives you
0.01111... +
0.00111... +
0.00011... +
0.00001... + ... which can be seen to equal
0.01234... by simply summing across the columns.
Using similar reasoning gives you the case for (1/999)^2 or (1/9999)^2 etc. since the idea works in any base.

This idea can be used to quickly verify a series like the sum of n * 2^-n from n=1 to n=infinity. The value of this infinite sum in binary is 0.1 + 0.02 + 0.003 + 0.0004 + ... (If we allow bigger digits than 1 in binary). This sum is equal to 0.1234... = 0.1111... + 0.0111... + 0.0011... + 0.0001... + ... = 1 + 0.1 + 0.01 + 0.001 + ... = 1.111... = 2. (I think it's nicer to visualize this as a grid.)

The approach is give is less rigorous than yours. It might be interesting to formalize why all the steps in my approach are valid (and under which circumstances they wouldn't be), but I'm on summer break right now.

[jacobkeeney]

Very useful video, just ran across this number myself!

[austinfrancis]

Huge fan of these small interesting videos. Pls keep making them Lex!

[JAVS]

1/(99^2) is how simple life problems seem to be from outside and the infinite sequence of decimals is how they seem to be in your head. This would be a way to explain to an AI how the human brain works sometimes, especially when it struggles with emotions lol

[qualia]

I found out by messing around that 1/81 is neat a long time ago, but to hear why now is really cool.

[jacobbradtmueller]

I've been looking for this kind of math content for so long

[pavel]

Man, that Grant Sanderson podcast episode sure left an impression

[EAG]

It’s much to do with the ‘12 60’ counting system and Tesla’s 369, particularly the number 9 grid. It’s interesting stuff with amazing underlying patterns.

[TonydeKaro]

I would love to see this as an Mandelbrot set!

[hschofield]

Great explanation! What software (or programming language) was used to create the "animation" of this video?

[imranq]

This was a great video! If you could do a quick video on how NLP transformers work that would be awesome.

[avisternlieb]

Great video! Wondering what you used to animate it! (Reminds me of 3B1B’s open-source Python one. Could that be it?) Thanks a mill :)

[subhasish-m]

I think you made a mistake from about 0:45 onwards. The decimal expansion has 5 zeros before the 1 instead of 3.

[judgeomega]

id like to request a video on modern tools and libraries which are useful for ML, perhaps with commentary on which you might want to use in what situation.

[muhammettekin]

1/(99**3) is also interesting.. with 0.03061015... looks ilke the sequence is incremental by (3, 4, 5...).