quasi
Posts:
12,067
Registered:
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Re: Can a fraction have none noneending and nonerepeating decimal representation?
Posted:
Aug 2, 2013 2:56 PM


jonas.thornvall wrote:
>But one 1/phi is not a rational ratio, but the 1/6 hexagon >(center to vertex/sum of sides) is and so is all polygons >derived from multiplying the vertices.
Let n be an integer with n >= 3, and let p be the perimeter of a regular ngon inscribed in a circle of radius r. Then p/r is rational if and only if n = 6.
In other words, the regular hexagon is the _only_ regular ngon for which p/r is rational.
Thus, your claim that the ratio p/r remains rational for regular polygons derived from "multiplying the vertices" (which presumably just means "increasing the number of vertices") is a false claim.
With the exception of the case n = 6, all of those ratios are irrational.
But as has been pointed out by others, even if those ratios were all rational, that would _not_ automatically imply that the limit is rational.
For example, let x be an irrational number between 0 and 1.
Define x_1,x_2,x_3,... by
x_1 = (x rounded to the nearest 1/10) x_2 = (x rounded to the nearest 1/10^2) x_3 = (x rounded to the nearest 1/10^3) ... x_k = (x rounded to the nearest 1/10^k) ...
The numbers x_1,x_2,x_3,... are clearly rational.
It's also clear that the limit of the sequence x_1,x_2,x_3,... exists and equals x.
Thus it's possible for a sequence of rational numbers to have a limit which is irrational.
quasi

