On Tuesday 11 October 2005 14:59, Bill Hooper wrote: > A question recently raised here is how to show the results on non-even > division (like 20 divided by 3) without using common fractions. It's > interesting to note that there are other numbers that can't be > described using common fractions. > > How would one use common fractions to cope with numbers like the square > root of 2 or the square root of 3 etc., or the value of pi, or e (the > base of the natural logs). Common fractions give one additional > (superfluous?) way to describe one kind of number (results of non-even > division) but they fail completely in describing other kinds of numbers > (like the irrational numbers mentioned above). > > The best that can be done is to give approximations like > pi = 22/7 > > or > square root of 2 = (1 and 2/5) > or (1 and 3/7) > or (1 and 13/32) > > All of these are approximations, but approximation are no better or > worse here than are approximations like > 20 / 3 = 6.67 > which can be used for expressing the result of dividing 20 by 3.
Transcendental numbers can be approximated much better by rationals that are not constrained to power-of-ten denominators (see Liouville's theorem), and even algebraic numbers usually have better approximations. A particularly good approximation to pi is 355/113, and 99/70, which I'm sure you'll recognize better as 297/210, is a better approximation to sqrt(2) than 1.414. (1189/841 is the next best one, 41/29.) phma
