NEW: Follow-up video with puzzle solution is here:    • Ramanujan's infinite root and its cra...  
In this video the Mathologer uses infinite fractions to track down the most irrational of all irrational numbers. Find out about how the usual suspects root 2, e, and pi stack up against this special number and where the irrationality of this special number materialises in nature.

Another video to check out is this leisurely lecture by Professor John Barrow:    • Continued Fractions - Professor John ...   and his write-up in Plus+ magazine: https://plus.maths.org/content/chaos-...
If you are reasonably clued up mathwise have a look at the following VERY nice textbook chapter on infinite fractions by Professor Paul Loya from Binghampton University: http://www.math.binghamton.edu/dikran... In particular, check out section 7.5.1. The mystery of π and good and best approximations. I use the definition of "best rational approximation" given there. And if you are okay with all this and are having transcendental numbers for breakfast, definitely also don't miss out on the last section 7.10. Epilogue: Transcendental numbers, π, e, and where’s calculus?

Enjoy :)