How real are the real numbers, really?

We usually say that infinity isn't real, but here we'll see how crucial it is to have one very big infinity for the real world; there is an infinite number of numbers. But why do we need real numbers at all? Aren't rational numbers enough? And what about hyperreal numbers?

What we'll see in this video is that discovering or defining the real numbers is what allowed calculus to be made rigourous- and without it, we'd need to divide by 0 every time we took a derivative.

This video is about the seeming mathematical paradox that arises to get Archilles from A to B (that isn't Zenos paradox!).

[tone]

"This is crazy, but don't worry, there's way more" is my new favorite introduction.

[ScienceAsylum]

I just wanted to say this collaboration was super fun. So many good discussions in the making of these videos! ...and I think they both turned out better because of it.

[adityakhanna]

When there's a problem, I think quite a lot about it. My friends remark that I'm being *irrational.*

I often reply "I just need to fill the gaps"

It mostly takes a *real* effort, (hyperreal effort, I might add) to think up a solution.

But once the *limits* of my capabilities are reached, the problem is over in a *fraction* of a second.

[upandatom]

I like how you take seemingly simple concepts that we've all glided over and present them in a creative and unique way while outlining that they are not as simple as what we first thought :)

[LookingGlassUniverse]

A lot of people are commenting about the Planck length as 'the smallest scale'. This isn't true. In physics we still use continuous variables (so Achilles still runs along smoothly)- rather than a discrete grid. But I would love to discuss the possibility of having a grid at some point. It's a very interesting idea.

[jesscool]

Really brilliant video. As a Physics graduate myself, I've taken Advanced calculus and Fundamental concepts of Algebra this semester, two math courses, and it completely revolutionized everything I knew! My respect for math grew. Also, 3Blue1Brown is a great channel for understanding math well. Math is just brilliant! I wish I had 3 lives so that I can properly do Physics, Math and Philosophy in each of those, haha!

[christopherblare]

Achilles doesn't run over the fractions. Not because he's smooth and fractions are jumpy like first proposed at a glance, but the opposite. The smallest physical scale that maintains any coherent meaning of distance is the Planck scale, he's slowly incriminating up his distance traveled by one Planck length and in doing so jumping over all the irrational, all the transcendentals, all but tiny fraction of the fractions.

[kumarpranshu]

But few say complex numbers are just as 'real' as real numbers. It's as strange as negative number is to positive

[kapick]

Awesome video!
I'm no expert, but the Achilles you drew looks very much like Hermes :D

[krzyszwojciech]

If there were infinite amount of places between two points in space and reaching any point in between would take a moment, then traversing that length would take infinite amount of moments. Infinite time. So it seems there must be a limit to the divisibility of space.
The problem is - we don't know where is that limit, so we can't calculate in wholes, instead we must in parts. Real numbers, while I don't think they are in fact real [in the sense of representing how the world really is], are still very useful, because they allow for divisibility that's arbitrarily small. They allow us to accommodate the holes in our knowledge.

[Kraflyn]

hyperreals next! And alephs! And continuum! Cheers! :D <3

[Nickelnine]

But what space and time was fundamentally discrete? That would get around all these problems right? If space and time really was continuous wouldn't you need an infinite amount of information to describe anything? This troubles me :/

[adityakhanna]

Well, I doubt that the resolution was necessary to do Physics.
At that time, I don't know, but can't we call the planck length our finite tiny number?

It's mostly a mathematical problem pertaining to function continuity.

As far as I go. :/

[nateg]

I like how they both plugged for eachothers videos. I watched his first which peaked my interest in this video.

[adarshkishore]

I really like how you researched into the origin of real numbers way back to the past to make this. Its like the numbers were saying all the time 'To infinity and beyond'.

[BangMaster]

Some infinities are bigger than other infinities. A circle is not real, because it requires infinitely many points to be formed in the real world, but, with the concept of the circle, we get Pi, and with the Unit Circle, we get Sin, Cos, Tan, all of Trigonometric identities, which is amazing because it helps us model many real world phenomenons in Physics,
something that can not physically exist in the real world helps us explain the real world, if that isn't crazy magic, then i don't know what is

[chrisg]

I get on a train and ask a fellow passenger if it goes to Cambridge. Yes comes the reply. Later on I look up from my paper to see us speeding smoothly through Cambridge without even slowing down. My protest is met with "Sure it goes to Cambridge, it just doesn't stop there". As soon as we mark any point along the path taken by a moving object, , whether with a name, a fraction or a furlong post, we're inevitably treating motion as if it were a series of stops or final destinations. The smooth is broken up into the rough. But it's a useful convention.

[JoJoModding]

I think that Newton and Leibiz said: "Lets look what happens when that time gets closer and closer to zero". Thats a huge difference when compared to ".. that time is small, really small".

[theultimatereductionist]

I love our modern (over 100 years old) well-defined ideas of infinity & set theory & measure theory & the real & complex numbers & other number fields. All the stuff undergraduate & graduate math majors get. I have NEVER understood nor cared for nor loves this archaic Newtonian idea of "infinitesimals". What practical good can one do with it? What can one compute with it?


I love how we now can define Lebesgue integration over subsets of the reals such as "the set of all numbers whose decimal expansion contains no 3s, 7s, or 9s, and 1s only at a prime-number position".

[chiepah]

The only other person who was able to explain concepts as well as you was my kindergarten teacher. She was able to teach a room full of 5 year olds the concept of multiplication (which for the longest time I thought was impressive) and the concept of zero as well as multiplying by it (which I have only recently realized how big a deal that was).

Anyway all that to say, thanks for another great video.