Irrationals that you cannot count on

I am fascinated by just looking at the cover of this book.

I flipped through inside... a lot of fascinating things that will take some time to go through... This would be a great book to read if I have some spare time. Irrationals numbers are useless you say? I have to say.... yes there perhaps is not too many practical uses but they are fascinating.

This book cover already listed the most interesting irrational numbers: π e, and the golden ratio φ. There aren't many irrational number given a special name. That spiraling triangle.... continuously make use of the Pythagorean Theorem to construct lengths of the radicals.

And why you cannot count on? That's because irrationals just cannot be represented by countable things such as fractions (there is something not so rational about these radicals).

That Amazon link... talks about Zeta(3). You would not realize how cool is this... perhaps unless you know about Zeta(2)'s jaw-dropping relationship with π.

Ok, Zeta(s) is actually straight forward, the sum of the inverse of integers to the s-th power:

And that it can be proven:

Ok I guess I have to buy the book to see what's up with Zeta(3). Oh, irrationality of e and π... is not for faint of heart. Google search for the proof to see what I mean.