If asked to approximate a number, most people think about the initial piece of its decimal expansion, like the square root of 10:

This talk is about a different way to approximate numbers, using continued fractions. A continued fraction approximates a number in a more subtle (some would say better) kind of way than a decimal, with a few surprises. For instance, while the decimal digits above look random, the continued fraction for the square root of 10 is periodic!

Continued fractions provide a convenient explanation of the leap year rules (an extra day for each year divisible by 4, unless the year is a multiple of 100, except when the year is a multiple of 400...). They are also the main tool needed for exact rational number recognition on a computer. For instance, what is the “most likely” fraction whose decimal expansion begins with 2.34579439252? (Hint: the answer has a much smaller denominator than you would expect.)

USG funded