This is a very nice and compact overview. Thanks for sharing this!

This is really good! I had a feeling it would be from the thumbnail/name combo, for some reason. The cadence you tell this story with feels very natural and compelling. Also, I had never heard of the Buckingham pi theorem before, and it is super interesting!! I'm more on the computational side of things, and dimensionality reduction is always a really big deal there, so I'm sure I'll get to use it at some point.

nice compact and useful video

Solid video man

Like a lot of other commenters here, I independently thought of "factor space", and to me it's the most natural way to understand what multiplication actually is from an abstract algebraic point of view. I majored in math but I was kind of disappointed that it was never taught in my courses. Cool vid overall

really interesting topic, do you have additional ressources to go more in depth in this subject ?

I disagree, if we didn't have multiplication, we'd end up with Presburger arithmetic, it'd be complete and we would have algorithms to prove any formula we wanted! Jk, cool video!

Lets write 1=[ ], and 2=[1]. Then 2 = [ [ ] ]. It's like the integer definition all over again.

My ears can now rip, bye, didn't even continue the vid

this factor-space is ... very logarithmic number theory isn't really my thing, but i am fascinated by certain kinds of rotational algebra, and at least one algebra for rotations along the surface of a paraboloid (embedded in the null-cone of a minkowski-space, encoded in a geometric algebra) has interesting properties allowing it to model co-ordinate translations with the same multiplicative structure as rotations, instead of addition (but still being additive in a linear projection) ... i actually really doubt this would make number theory any easier, but one made me think of the other, and i'll have to contemplate a bit on _why_ it wouldn't help ... and i don't actually remember how to represent scaling operations in this model, which would be essential here edit: apparently dilation involves two successive inversions with the same center but different radii - that would definitely complicate things a little

Bonus points if you remake this video without the obnoxious music and link it in the description.

All 4 numbers are divisible by 30, was my answer to your first question.

Yet another illustration that everything is linear algebra

You need to balance your audio levels. Ruins an otherwise good video

great example of a video that would be a shallow and straightforward 8/10 dropping to 2/10 due to not being edited properly

Isn't this also equivalent to: "if a,b,c,d exist such that det([a!,b!],[c!,d!])=0" ?

314th view 👍

"Hard" is only that you cannot say "this theorem is true bc bla bla", bc numbers is not finite set I'm in number theory tho

I really enjoyed hearing Verdi's requiem out of nowhere. You don't here that everywhere. However it almost woke up my brother xD

i immediately disagree cuz addition is kind of the base for everything but yt shoved this vid infront of me so many times that it cant be that bad

3n+1. Whoever develops the theoreticap framework that solves this will be laying the groundwork for understanding addition and succession in this space.

0:17 - The narration leaves out the key phrase "two different ways." The point the narrator is trying to make doesn't make sense unless the viewer is reading the text on the screen rather than only listening. 0:24 - The phrase "multiplying 720 by 7 to both factorizations" doesn't make sense for at least two reasons: (1) Since the wording "two different ways" was left out (see previous comment), the idea that there are two factorizations of each number has not yet been introduced by the narrator. (2) I don't know what it means to multiply a number by another number "to" a factorization. Bottom line: The ideas in the video are good, but the presentation needs some work.

3:40 WHY???? Are you purposefully talk very silently for the whole video, so that you can later rupture people's eardrums with super loud music??

I opened this video to find out why number theory is hard. I must have missed something.

It doesn't beg the question, it raises the question. You're a mathematician, you shold know better.

So you watch loads of numberphile and think you have a fair handle on number theory and then you see the prime factors as vectors for the first time

What, if the dimension of the vector itself is prime? Then we can add /subtract the projection of the units vectors and they end up at the unit circle? just be accident?

What was that random Quidditch World Cup theme at 3:41 about?

Extending the allowed components, the vector [1,1,1,1,1,1,1,1,1,1,1,...]=4π² (the product of all the primes). So [-1,1,1,1,1,1,1,1,1,1,1,...]=π² and [-½,½,½,½,½,½,½,½,½,½,½,...]=π.

0:11 - They all end in zero. Done! Next video.

This video was recommended to me out of the blue. Not my regular piece of feed but I was intriguided by the title and also have some interest in number theory. But I was a bit disappointed by this because this was like an extremely high level overview of a lot of math jargon. Like vector, linear algebra, vector space basis, mobius function, totient function. Felt like a half-hearted attempt. Would have loved some details and intuition behind the theorems, application etc. rather than just taking the name of some random math concepts. You earned a subscriber and a like, hoping for better content.

If you have n vectors {0}.{1.{2}.{3}.{4}.{5}.{6}.{7}.{8}. ..... and so on, then in factor space the vectors {0}.{1}. {2}. {3}. {5}. {7}.. .... will be orthogonal, the others not?

What is the name of the vector with prime inside i want to know more,does this prime vector have a magnitude?

It's a trivial proof, but my favorite rendition of the irrationality of root 2 is what introduced me to p-adic valuation. Suppose for contradiction that root 2 is rational -> there exist n,m in the naturals s.t root(2)=n/m 2 * m^2 = n^2 v_2(2 * m^2) = v_2(n^2) 2 * v_2(m) + 1 = 2 * v_2(n), which are Odd and Even respectively. The log-like behavior of v_p is because it's working in "factor space" - ie, the scaling operation at 2 minutes.

Alternatively, maybe multiplication ruins everything.

Great video!

Man I'm an english major to be, I got lost at the number 3

"Now that we have established the definition of beauty,". Cope and seethe, philosophers.

At 3:23 is there an error in the typing? Should it be "x" in the brackets not "+" ?

Absolutely outstanding way of narration 😊😊😊

Over the years I've tried to get a very basic understanding of number theory...this has opened the door for me to start again.

I'm in the math Olympiad world and number theory is the hardest one for me because of how technical it is, you can't even understand the problems or even have an idea of what to do if you don't have experience nor any solid basics but when you get used to it's pretty manageable, you can solve problems and keep learning more without as much difficulty as as when you started. Also I love how mysterious it is sometimes but anyways good video ❤.

Wow its crazy that I got recommended this video just after having the idea of factor space myself. Very cool to see how its used :)