| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
Problem Zero:
x is irrational. y is a number, 0 <= y < 1. frc(x) is the
fractional part of x obtained by subtracting from x the
greatest integer not greater than x. For example, frc(5.7)
= .7. frc(-3.6) = .4.
Given that there exists an integer t such that frc(tx) = y,
is there a better way to find t than trying integers one
by one?
Is there a way to determine whether or not such a t exists?
Problem One:
Given x, y, and frc as above, let e be a positive number.
Is there a better way to find an integer t such that
|frc(tx)-y| < e than trying the integers one by one?
-- edp
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 473.1 | How do you know when you have found t? | METOO::YARBROUGH | Fri Apr 25 1986 10:13 | 9 | |
re problem 0: x irrational and (integer)t*y = (integer)R+y implies
that y is also irrational. This leads us to a philosophical problem:
we have no way of representing irrational numbers except as a member
of a class of functions of integers. Therefore, in a sense, to say
that there exists a t with the defined properties implies that we
already know what t is (in terms of the functions that describe x
and y), since otherwise we have no way of identifying y. That is,
either t can be derived from the functions defining x and y, or
else there is no way of verifying the equality of x*t and y, modulo 1.
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| 473.2 | CLT::GILBERT | Juggler of Noterdom | Fri Apr 25 1986 10:54 | 5 | |
This problem looks similar to the problem of rational approximation.
That is, given a real number, find a good rational approximation to it.
I believe that some very good solutions to this problem are known
(especially since MACSYMA does it), but I haven't checked the literature.
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| 473.3 | see 475: "rational approximations" | SIERRA::OSMAN | and silos to fill before I feep, and silos to fill before I feep | Wed Apr 30 1986 16:10 | 0 |