(Difficult) I understand how the continued fractions are used to get decimal approximations of real numbers, but the Theorem in this section was hard to follow. If I understand it correctly, if we have a rational number (r/q) as an approximation of a real number, if the difference between the rational approximation and the real number is less than 1/(2q^2), then (r/q) is a partial quotient of continued fractions. It seems it would be a simple proof to show that any partial quotient will be within a specific range of the real value, but the theorem seems to imply that continued fractions are an optimal approximation when limiting the order of the numerator or denominator. That is, if we want to optimize are approximation with a limited number of digits in the numerator/denominator, continued fractions will guarantee optimization. The first paragraph of the sections seems to hint to this, since 22/7 is a better approximation of pi then 314/100, but I am still not sure if that is what the theorem is stating, or what other purpose the theorem may have in this section.
(Reflective) I had never thought of how continued fractions were related to the euclidean algorithm before. We could in fact calculate the gcd by using continued fractions, although the euclidean algorithm appears to be easier to use. However, for a computer, continued fractions may be an easier way to calculate the gcd. The process involved in both formats are very similar. I am curious as to which would be easier to implement on a computer, and if there are any advantages/disadvantages to either one. At the end of the section when it talks about partial quotients, its seems as though this method would have an advantage when implemented on a computer because it says that the partial fractions can be calculated without starting a new computation as we do with each layer of the Euclidean Algorithm.
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