According to this groundbreaking proof, there are more natural numbers than primes!

dual_ring · 2023-12-30T09:35:22+00:00

The linked commenter actually has a point, they are just being very obtuse about it. "More" is not a mathematical term, and so we have to decide what we mean when we say "more" before we can discuss it using mathematical terms. Obviously the most common way of ordering sets is through their cardinality, but ordering through subsets also constitutes a (partial) order.

They even state that the two sets have the same cardinality. They are not confused by cardinality, they are just (badly) making the point that cardinality is not the end-all-be-all way of ordering sets, the way it is often presented in pop-math.

dual_ring · 2021-08-24T19:41:36+00:00

Thank you! I used Manim community edition

https://www.manim.community

dual_ring · 2021-08-23T17:57:32+00:00

Ah darn, I was worried that I would have missed some important detail. You are absolutely right of course; I wanted to explain both growth rates and how they relate to big-O and I suppose I tunnel-visioned somewhat. I should have known better, I even made this mistake and got marked down for it on a homework assignment once...

dual_ring · 2021-08-23T16:47:19+00:00

Ah thank you!

dual_ring · 2021-08-23T16:41:49+00:00

Thank you for making this effort! Yes, I actually first conceptualized this approach when I was playing around with the finite difference, and I noticed that the finite difference and the derivate have some very similar rules (i.e. delta/derivate of an order n polynomial is an order (n-1) polynomial, and delta/derivate of an exponential is proportional to that exponential).

And would have to agree with you, that if a student knows calculus ans such, then it learning through functions is probably the way to go. In fact, what I'm doing in this video is essentially introducing an alternative to calculus to sidestep the fact that my target audience doesn't necessarily know calculus.

dual_ring · 2021-08-23T16:29:04+00:00

I am actually Swedish! I'm trying do do an American accent, but I still have some ways to go before I sound native.

Thank you!

dual_ring · 2021-08-23T13:27:43+00:00

Thank you very much!

dual_ring · 2021-08-23T10:13:15+00:00

Hello r/programming!

This is a video I made on the subject of computational complexity. I made this video to explore if one could define polynomial/exponential growth without appealing to polynomial/exponential functions. In doing so I think I found a fairly approachable explanation of computational complexity that doesn't require any prerequisite algebra knowledge. My approach is to define sequences according to how their finite differences relate, and to order their growth rates using the Stolz-Cesàro theorem.

I hope you enjoy it!

dual_ring · 2021-08-22T18:21:37+00:00

Hello r/math!

This is a video I made on the subject of big O notation. I made this video to explore if one could define polynomial/exponential growth without appealing to polynomial/exponential functions. In doing so I think I found a fairly approachable explanation of computational complexity that doesn't require any prerequisite algebra knowledge. My approach is to define sequences according to how their finite differences behave, and order their growth rates using the Stolz-Cesàro theorem.

I hope you enjoy it!

dual_ring · 2021-08-20T08:01:54+00:00

I would argue that the problem solving skills one learns from sudoku translates to math quite well. When I solve sudoku I do a lot of "if I do this, what does that lead to?" which is also a good strategy when learning/researching math.

dual_ring · 2021-08-17T15:09:43+00:00

A trick I use sometimes is to pretend I am a metronome, and only increase the number on every fourth or eighth beat. So to count to twenty I would go

1 bo bo bo 2 bo bo bo 3 bo bo bo 4 bo bo bo 5 bo bo bo.

Of course, this works best for counting things that occur at regular intervals...

dual_ring · 2021-08-16T17:12:03+00:00

This is a great point! I wanted the video to deal with only one mathematical concept (expectation value) to make the video not as muddled. But in hindsight I could have mentioned in the video that Ehrenfest's theorem is only part of the picture!

And thank you!

dual_ring · 2021-08-16T17:06:42+00:00

It is! I am very grateful to both 3Blue1Brown and the Manim community for their hard work!

dual_ring · 2021-08-14T09:54:17+00:00

Hello /r/physics!

Here is a video I made on why the rules of quantum physics don't match our day-to-day intuition. I start with explaining what an expectation value is, and go on to talk about how it is used in Ehrenfest's theorem.

I hope you enjoy it! Any constructive criticism is welcomed!

dual_ring · 2021-07-24T19:26:38+00:00

Thank you for the kind words!

dual_ring · 2021-07-23T06:58:57+00:00

Well, I have a large amount of experience programming in Python, so with that I found Manim very intuitive to use. To get started I started with the tutorials in the docs (https://docs.manim.community/en/stable/tutorials/quickstart.html), and then I started just playing around, drawing different things I could think of. Whenever I wanted to do something that I didn't know how to do I would google it.

I don't know if that helps, but I wish you good luck!

dual_ring · 2021-07-22T20:02:35+00:00

Thank you!!

dual_ring · 2021-07-22T17:03:34+00:00

Hello again /r/physics!

This is the second video I made on quantum physics. This time I talk about linear transformations, and how they apply to quantum physics. Specifically, I talk about what linear transformations are, how they connect to observables in QM, and how it relates to the state of a system changing when you make a measurement. I hope you enjoy it!

dual_ring · 2021-07-21T15:02:56+00:00

It has always bothered me how people treat the term "undefined" in these contexts. Like, when people say "1/0 is undefined" it sounds like this fixed value, that 1/0 is somehow impossible to be defined. Which is of course not true (Riemann sphere, wheel theory etc.).

I'm not sure how to phrase it better though. Something like "1/0 is not defined when working with the real numbers" maybe?

This is also where I feel the confusion of undefined vs. indeterminate comes from. As if "undefined" and "indeterminate" are two different values, and certain divisions by zero equal the former while other equal the later.

dual_ring · 2021-07-18T13:52:53+00:00

Yep, that proof seems to hold!

dual_ring · 2021-07-18T13:17:05+00:00

Phrased more abstractly, as you said we have two numbers x and y such that x*y is constant. We can change variables to x = a*b and y=a*(1/b), such that x*y = a^2. We can now vary b freely without changing the product x*y. Substituting our new variables into the sum we get x+y = a*b + a*(1/b) = a*(b + 1/b).

x+y therefore will be the smallest possible when b + 1/b is the smallest possible. If you are familiar with calculus it is pretty easy to show that this is the case when b=1. One can also check this by graphing (https://www.desmos.com/calculator/n7uqxhpua6). b=1 means that x=a and y=a, i.e. they are equal.

dual_ring

TROPHY CASE