JT
Posts:
1,448
Registered:
4/7/12


Re: Can a fraction have none noneending and nonerepeating decimal representation?
Posted:
Aug 1, 2013 4:45 AM


Den torsdagen den 1:e augusti 2013 kl. 10:35:52 UTC+2 skrev jonas.t...@gmail.com: > Den torsdagen den 1:e augusti 2013 kl. 04:18:23 UTC+2 skrev Virgil: > > > In article <699bb484e79d4c05863e415a2e6a4d68@googlegroups.com>, > > > > > > jonas.thornvall@gmail.com wrote: > > > > > > > > > > > > > The question was if a fraction (rational) expressed in one base, must > > > > > > > necessarily have terminating and nonerepeating pattern in another base. > > > > > > > > > > > > If a fraction is expressed as a fraction, it does not matter what base > > > > > > is used, its expression is then a quotient of two integers. > > > > > > > > > > > > If you are talking about the basal (e,g, decimal or octal) > > > > > > representation of such a quotient the if such a representation > > > > > > terminates in one base it will also terminate in any multiple of that > > > > > > base. > > > > > I am talking about that 1/3 do not terminate in base 10 but in base 3. > > And i also say it is ***obvious***, that any multiple of a hexagon have a rational radius to perimeter ratio. > > > > And having all multiples of six having rational perimeter, it follows that 3 also will have it. But not only that, it becomes clear that if you build polygons around primemultiples you can multiply the vertices by 3. To have a least common multiple, from there you can work backwards to find the ratio for the desired primenumber based polygon. > > > > This means that any polygon has a rational perimeter to radius ratio. > > But using a hexagon is just simpler. > > > > The question really is if you iteratively double up the number of vertices in infinity, is it a circle. > > > > And i have no doubt it is, and the same is true for any primebased polygon where you double up the vertices infinitly. And it is because they all have least common multiple, so this is true for all polygons. > > > > > > > > > > For example, any rational whose basal expansion in base 2 terminates > > > > > > will also have terminating basal expansions in base 4, 6, 8, 10 12, and > > > > > > so on. > > > > > > > > > > > > Similarly any rational whose basal expansion in some integer base does > > > > > > not terminate, will also not terminate with base any integer factor of > > > > > > that original base. > > > > > > > > > > > > For example, a rational whose expansion does not terminate in base 10 > > > > > > will also not terminate in either base 2 or base 5. > > > > > > > > > > > > Anyone of reasonable competence should have been able to work out those > > > > > > simple rules for themselves. > > > > > > 
The thing is that i can express Pi as an exact fractional ratio of the perimeter of a polygon, without using trigonometry just continued fractions.
And that is both rational and a fact.

