DMC with Walter Stromquist
Join Us for a Talk: Fractions, Fareys, and Fords!
Join us for a talk presented by Research Associate Walter Stromquist titled "Fractions, Fareys, and Fords." If the fractions 𝑎/𝑏 and 𝑐/𝑑 are each in lowest terms, then their Farey Sum is given by 𝑎𝑏 ⨁𝑐𝑑=𝑎+𝑐𝑏+𝑑. This simple operation has some surprising properties. We will use it first to construct a Farey Diagram, which is a graph with exactly one node for each rational number. The graph leads to some pretty pictures, and we will make a connection to Ford Circles, which leads to even more pretty pictures. In this case, beauty goes well with practical applications, and we will use the Ford Circles to prove a theorem about how well we can approximate irrational numbers by fractions. You may know that 1/π ≈ 7/22 and that 113/355 is a better approximation to 1/π. Can you think of a still better one? The distributive law is a wonderful tool in arithmetic and algebra. We all know that multiplication of fractions distributes over addition of fractions. Can you think of an operation that the addition of factions distributes over? I’ll bet you can!
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