Do we even need the real numbers??

We explore the possibility of a calculus only over the rational numbers. This leads us to the important role of the axiom of completeness in the important theorems from first semester calculus.

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[a-manthegeneral]

I guess the real numbers were the friends we made along the way

[barutjeh]

For the first example, you could use x^2 < 2 instead of x < sqrt(2); that way the nay-sayers have nothing to complain about. (You could include a 'or x < 0' if you want to keep your function exactly the same.

[mohammedjawahri]

They always ask "do we need the real numbers" but never "how are the real numbers" :, (

[toast_recon]

What an excellent video. I've always been curious where the rationals aren't good enough. They always seemed plenty abundant to me, but wow, they're missing some key features we take for granted in the reals.

[SlidellRobotics]

The original function can be defined without referencing anything irrational: 1 if x^2 < 2 or x<0. 0 otherwise.

[MyOneFiftiethOfADollar]

It may have been a real analysis book by Rudin that defined real numbers as Cauchy sequences of RATIONAL numbers.

I think I actually understood that at some point earlier in life, but always wondered how by hand we could do even the standard arithmetic operations on a pair of “unreal” I mean real numbers!

[JansthcirlU]

Oh, I didn't realize I was watching an NJ Wildeberger video ;)

[atonaltensor]

When I first saw the title of this video, I thought you were going to use the fact that the rationals are totally disconnected and zero-dimensional. You should do a sequel on calculus on the irrationals. The topology of the irrationals is homeomorphic to the Baire Space (N^N), not to be confused with spaces which satisfy the baire category theorems. I’ve heard descriptive set theorists refer to N^N as “digital calculus”. It might mean it’s more amenable to calculus-like theorems.

[alnitaka]

Also I note that there is a 3-dimensional field over Q, by adjoining roots of x^3-3x+1=0, but it is well known that there is no 3-dimensional field over R.

[arnaudparan]

I already knew all of that and I am glad that I have had great math teachers on such topics, but that video is great. I am glad there are people here doing youtube videos trying to give others mathematical intuitions and I hope it serves many :)

[sanjursan]

All that in under half an hour. Amazing. Super intro to Real Analysis, or maybe mini-course.

[conorbrennan]

Could you do a video on the set of constructible real numbers and then maybe calculus on the constructible reals?

[sowoul_]

how many integrals i gotta do to get dat big arms??

[sea]

It was interesting, thanks. On a second thought, some of the issues you raised disappear if we use uniform continuity instead, ie a function which is uniformly continuous over ℝ is also uniformly continuous over ℚ.
Also if f is uniformly continuous over ℚ and for every point in ℝ f has a right limit and a left limit then f is uniformly continuous over ℝ (I added the right and left limit hypothesis to exclude pathological functions like indicator of ℚ).

[Entropize]

Sure. You can define Calculus over anything, without limits even, thanks to Algebraic Geometry. The results don't have to be pretty in all cases, just consistent.

[elidamon]

Your counterexample to the Mean Value Theorem is also a counterexample to the Extreme Value Theorem.

[edusoto]

Note that the first step function can be defined without use of real numbers (or sqrt 2). Take f(x) = 1 if x<0 or x^2 < 2; and f(x) = 0 otherwise.

[mxpxorsist]

The problems pointed out in the video already disappear if you consider the algebraic numbers and algebraic functions.
So you'd have to cook up a natural example where a transcendental number pops out e.g. integral of 1/x is ln(x).
Are there any nice examples that arent integration and arent trying to take an arbitrary limit that doesnt exist?

[kumoyuki]

This is actually really important for Machine Learning applications. Really any Numerical Methods course should be teaching this because when we actually calculate (by hand or computer) we use rationals.

[BethKjos]

Turn it around and say that a function is continuous precisely when the intermediate value theorem holds in the limit coming from both directions. E.G. for Y=X^2-2, you can get Y as close to zero as you like from either positive or negative direction with rational choice of X; you need not necessarily have a rational root to accomplish this. And then you can talk about Dedekind.