Comments on Approximation by rationals

Karl (2003-12-10T15:53:11Z)

8502 tropical years, 105155 synodic months, and 3105289 solar days, are very close.

This is too precise for practical use in a calendar because of the variation in the length of the tropical year and synodic month in days and also there are other complications with the tropical year.

I'm aware of the following approximations (years,months,days):

334, 4131, 121991
649, 8027, 237042
725, 8967, 264801
1040, 12863, 379852.

The 725-year cycle is interesting, because it consists of 183 cycles of 49 months equal to 1447 days. This is the best lunar approximation short of 400 months.

Karl (2003-12-10T15:37:18Z)

I've discovered that 6 to the power of 9 is within one percent of 10 to the power of 7.

This means that a base-six Metric system would have a Metre within 1cm of the decimal Metre.

Anonymous Coward #272 (cossaw) (2003-12-10T13:16:50Z)

For the uninitiated, arithmetics tends to be considered the "worst" (that is most difficult and least understood) field of maths. I tend to agree…

Anonymous Coward #313 (ines) (2003-12-10T08:44:34Z)

The problem of approximating several real numbers by rationals of the same denominator is known as simultaneous diophantine approximation. It can be solved efficiently using the LLL algorithm (Factoring Polynomials with rational coefficients, Lenstra, Lenstra, Lovasz, Math. Ann. 261., 515-534, 1982).
Another application of LLL is to find Q-linear relations among given real numbers.

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