1. Experiment name : Best fractional approximation
2. Experimental environment : Mathematica 10.3 software
3. Purpose of the experiment : To study how to use fractional approximation to make the best approximation to a given irrational number. "Best" means that the error must be small and the denominator must be small. We first need to set specific and clear criteria for "best", and we also need to find a simple and easy algorithm to find the best approximation of the score.
4. Experiment content:
1. Fractions are the best approximation to irrational numbers
Take n=50, let the denominator take
the integer values from 1 to , and for each denominator
,
round to an integer
as the denominator, so as to obtain the
closest
approximation of the fraction with the denominator
.
1. The continued fraction expansion of π
2. Continued fraction expansion of real numbers
Continued fractions will be expanded and verified.
5. Experimental results and result analysis
1. The best approximation of fractions to irrational numbers
Result analysis : By observing the experimental results, the closest fractional approximation with the denominator q can be obtained
.
2. The continued fraction expansion
Result analysis : The results we obtained are already relatively close , but if we continue this process, the approximate values obtained will become closer and closer
.
3. Continued fraction expansion of real numbers
