What does it feel like to invent math?

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.

Music: Legions (Reverie) by Zoe Keating

Thanks to these viewers for their contributions to translations
Italian: Marco Fantozzi
Thai: @korakot, own doggoV●ᴥ●V

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Various social media stuffs:

[scottanderson]

Whenever I invent math, my teacher marks it wrong.

[lucaslzt]

"You are a mathematician [...], so you don't let the fact that something is nonsensical stop you" A true mathematician spirit

[nitinnilesh]

Warning - This background music with his voice can lead to a state of mind where you can invent anything. Thank you 3b1b for this high-quality introspection of math.

[P-]

I remember when I was trying to solve a problem for a while, and had an epiphany when I was trying to fall asleep one night. I started writing down some ideas until I came to a conclusion about the problem. Not a full solution, but a big step. Later on, I found a paper published in 2008 about the problem, and halfway through the paper they used the same process I did. So I can say that it did feel awesome to come up with that in my own 😊

[Azmidium]

Even the universe has integer overflow :o

[fqidz]

How it feels to invent math


5 math, stimulate your senses

[TemperThetaDelta]

As a programmer the 2^n example is easy to answer: the infinite-precision integer storing whole numbers overflowed into the negatives

[osotanuki]

literally a few hours ago i forgot the formula to find the infinite sum of a converging series for a precalculus test, but it was the last question and i still had 40 minutes left, so i basically reinvented the formula exactly like this and got the right answer. this is what growing up on 3b1b does to you.

[Mathologer]

Love this video. I keep getting asked by (numberphiled) students why 1+2+3+... = -1/12 and I usually end up telling them about analytic continuation, etc. From now on I'll also refer them to your video to expand their minds in a different direction :)

[Jaymac]

Ever since I took calc II, I basically treated “approaching” and “equalling” as the same thing. It’s honestly made things seem less ridiculous. For example, I essentially treat 0 and infinity as reciprocals because of how y = 1/x looks on a graph. It doesn’t entirely work because the limits don’t technically exist, but it still makes the universe seem less ridiculous.

[shawon]

I love how you can tell how good grant has gotten with his videos. The voice over, the designs and what not... kudos to you!

[cyclingcycles]

I can see the fabric of space-time now.

[pennrogers]

It’s important to understand that the theories of p-adic numbers (for each p) and the theory of real numbers are distinct theories. Otherwise, such statements lead to obvious ambiguity. So, the statement “1+2+4+... diverges” is true in the theory of the real numbers, while, independently of this fact, the statement “1+2+4+...=-1” is true in the theory of the 2-adic numbers.

On the other hand, field extensions lead to extended theories. For instance, the theory of complex numbers is an extension of the theory of real numbers, or, similarly, for any field extension of some p-adic field. So, in other words, every statement of equality that holds in the theory of real numbers still holds in the theory of complex numbers.

These two concepts, along with the distinction between them, seem to be lost on a good deal of commenters. The first creates a distinct theory with a distinct metric, while the second creates an extended theory with an extended metric.

[pedroivog.s.]

13:58 "Now this sum makes totally sense"
Me, still stuck on why the powers of 2 are approaching to zero: O_o

[Meow_yj]

I'm grateful that I found your channel ! It makes math ideas look so beautiful and elegant. Especially linear algebra series.

[kcz]

Everybody: What was first chicken or egg?
Mathematics: 1-1+1-1+...=1/2 so it was half egg and half chicken.

[avikdas]

Expectation: Determined to fully understand a 3b1b video
Reality: Facepalm

[shubhamsharma-cpte]

This channel's production quality is better than Netflix

[satyenpandita]

Thank you for this brilliant illustration. My first instinct was this is completely wrong but I never thought about that I had been constrained in think about distance between numbers in the traditional linear fashion and that if we change the notion of distance, some very counterintuitive results make sense.

[soniczdawun]

This is undoubtedly becoming my favorite mathematics channel on YouTube. While I love Numberphile a lot, you give your viewers' level of understanding a lot more credit, and you explain these concepts beautifully. I remember Vi Hart mentioning the p-adics briefly in one of her videos, but you took on the task of actually explaining them in a way that makes sense, and tied it all into the arbitrary (although consistent) notions behind metrics, and how we use them to think of an "organization" to the rational numbers. You just flipped the idea of "closeness" on its head, and I love it!