Undergrad Math Club, Keith Conrad (UConn): 'Continued Fractions'

Thursday, April 23, 2015
12:00 PM - 1:00 PM (ET)
ESC 109
Event Type
David Constantine, dconstantine@wesleyan.edu

Abstract: The standard way to represent a number is through its decimal expansion. Decimals are easy to compare (to see which one is larger) and easy to add and multiply. This talk will be about a different way to write a number: its continued fraction expansion. Continued fractions are not easy to add and multiply, but they are far superior to decimals for the task of finding good approximations. For example, the continued fraction expansion of the square root of 2 leads to the rational approximation 99/70, which has just 2 digits in its denominator but approximates the square root of 2 with an error less than 1/10000.Continued fractions have applications to number theory, dynamical systems, hyperbolic geometry, knot theory, cryptography, and even music theory (why are there 12 notes in the Western musical scale?) and the Gregorian calendar (why is there a leap year every fourth year except if the year is divisible by 100 except if the year is divisible by 400?). In this talk we will learn what continued fractions look like, how to compute them, some of their properties, and we'll learn how to answer seemingly unanswerable questions like this: if an unknown fraction is roughly 2.32558, what is it? (The answer is not 232558/100000.)

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