For almost 300 years, continued fractions—that is, numbers representable as the sum of an integer and a fraction whose denominator is itself such a sum—have fascinated mathematicians with both their remarkable properties and their myriad applications in such fields as number theory, differential equations, and computer algorithms. They have been applied to piano tuning, baseball batting averages, rational tangles, paper folding, and plant growth … the list goes on. This course is a rigorous introduction to the theory and mathematical applications of continued fractions. Topics to be discussed include quadratic irrationals, approximation of real numbers, Liouville’s Theorem, linear recurrence relations and Pell’s equation, Hurwitz’ Theorem, measure theory, and Ramanujan identities.
Mike is recommending the Continued Fractions text by Aleksandr Yakovlevich Khinchin. I found a downloadable digital copy of the 1964 edition (which should be ostensibly the same as the current Dover edition and all the other English editions) at the Internet Archive at Based on my notes, it looks like he’s following the Khinchin presentation fairly closely so far.
If you’re interested, do join us on Tuesday nights this fall. (We’ve already discovered that going 11 for 37 is the smallest number of at bats that will produce a 0.297 batting average.)
If you’re considering it and are completely new, I’ve previously written up some pointers on how Dr. Miller’s classes proceed: Dr. Michael Miller Math Class Hints and Tips | UCLA Extension
The Collatz Conjecture is the simplest math problem no one can solve — it is easy enough for almost anyone to understand but notoriously difficult to solve.
You don’t make a bagel by first baking a bialy and then punching out the center. No—you roll out a snake of dough and join the ends together to form the bagel. If you denied that a bagel has a hole, you’d be laughed out of New York City, Montreal, and any self-respecting deli worldwide. I consider this final.
The offering is naturally dependent on potential public health measures in September, which may also create a class limit on the number of attendees, so be sure to register as soon as it’s announced. For those who are interested in mathematics, but have never attended any of Dr. Miller’s lectures, I’ve previously written some details about his stye of presentation, prerequisites (usually very minimal despite the advanced level of the topics), and other details.
A few of us have already planned weekly Thursday night topology study sessions through the end of Spring and into Summer for those interested in attending. Just leave a comment with your contact information and I’ll be in touch with details.
I hope to see everyone in the fall.
“we hold these truths to be self-evident” wasn’t Jefferson’s line; his first draft of the Declaration has “we hold these truths to be sacred & undeniable.” It was Ben Franklin who scratched out those words and wrote “self-evident” instead, making the document a little less biblical, a little more Euclidean.
Me: We don’t Mississippi in this house! Maybe we should Tennessee since that’s where Grandma and Grandpa live?
Evie: I’ve Mississippi’ed since I was three.
Me: Maybe since we’re Welsh we should Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch? You know: 1-Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch, 2-Llanfairpwllgwyngyllgogerychwyrndrobwllllantysiliogogogoch, …
Evie (interrupting): Wait, what number are we on now???
The Chudnovsky brothers yearned to probe the mystery of pi, so they built their own supercomputer out of mail-order parts.
He was named professor emeritus after teaching in the Department of Philosophy for nearly four decades