Re: Rationalize
- To: mathgroup at smc.vnet.net
- Subject: [mg83758] Re: [mg83665] Rationalize
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Thu, 29 Nov 2007 06:42:39 -0500 (EST)
- References: <200711281023.FAA17533@smc.vnet.net>
On 28 Nov 2007, at 19:23, Scott wrote: > Does anybody wish to hazard a guess as to the formula or algorithm > that is implemented by Rationalize? Alternatively, can anybody > recommend a formula or algorithm with the same functionality? > > Thanks much. > > Cheers, Scott > The classical approach to this is based on the following well known theorem: Let e be an irrational number and N some given positive integer. Then there is a rational number p/q with denominator q<N such that Abs[e-p/q]<1/(q N). (In fact it is possible to replace N by N+1). The proof is easy and can be found among other places in Chandrasekharan "Introduction to Analytic Number Theory", Chapter III. The proof can be converted into an algorithm for computing suitable rational approximations to irrational numbers. It is also the basis for representation of irrational numbers as continued fractions. For example, one can get the following representation for Pi: Pi = 3 + 1/ (7+1/(15 + ... The method (due to Adams) is described in Serge Lang, Introduction to Diophantine Approximations. The actual method used by Mathematica may be somewhat different as these books were all written before computational number theory became popular so possibly modern algorithms are more efficient. Andrzej Kozlowski
- References:
- Rationalize
- From: Scott <sguthery@gmail.com>
- Rationalize