08-04 Best Approximations of the First Kind

This one's a doozy.
We characterize the best approximations of the first kind for an irrational number x.
They include all the continued fraction convergents, but also *some* of what are called semi-convergents.
We start (00:00) with a definition of what semi-convergents are and (05:52) how they approach x in relation to the convergents used to create them.
Next (19:08) we use semi-convergents to answer the question "up to a certain size of denominator, which rational number is closest to x while being larger, and which is closest while being less than x?" That's not exactly the same thing as best approximations of the first kind, but the answer is nice to state.
Then (28:28) we prove that a semi-convergent made with more than half the next partial quotient is a best approximation of the first kind, while a semi-convergent made with less than half the next partial quotient is not.
The special case of exactly half the next partial quotient is then studied (38:15), shown to be equivalent to a specific inequality.
We close with a few examples (50:40), including a discussion on why 355/113 is such an incredibly good approximation to pi, even with a denominator that isn't too large.