Virgil
Posts:
9,012
Registered:
1/6/11


Re: Wikipedia and fractions in other base then decimal
Posted:
Feb 11, 2013 9:43 PM


In article <05226575583741af948989936267dcb5@j4g2000vby.googlegroups.com>, JT <jonas.thornvall@gmail.com> wrote:
> On 11 Feb, 23:28, Virgil <vir...@ligriv.com> wrote: > > In article > > <3c4b196b1998412f94aabe6b71bd8...@k14g2000vbv.googlegroups.com>, > > > > > > > > > > > > > > > > > > > > JT <jonas.thornv...@gmail.com> wrote: > > > On 11 Feb, 09:00, Virgil <vir...@ligriv.com> wrote: > > > > In article > > > > <03a59b131fbd409891d7f093d7cf5...@w4g2000vbk.googlegroups.com>, > > > > > > JT <jonas.thornv...@gmail.com> wrote: > > > > > Wikipedia claims that 1/2 decimal 0,5 is 0,1 in ternary... and also > > > > > 0,1 in binary..... > > > > > > > In reality it is 0.12 in ternary and 0.101 in binary > > > > > > Shouldn't 0.12 in base three be the same as > > > > 1/3 + 2/3^2 = 3/9 + 2/9 = 5/9? > > > > > > I make onehalf expressed in base 3 to be 0.1111... > > > >  > > > > > No both you ,wikipedia 0,5 decimal= 0,1 ternary and William Elliot and > > > you are wrong 0.111..., while i am correct. > > > > How is 0.12 base three = 1/3 + 2/3^2 = 3/9 + 2/9 = 5/9 wrong? > > > > Also, for r < 1, the infinite series 1 + r + r^2 + r^3 + ... > > converges to 1/(1r) so > > r + r^2 + r^3 + ... = r/(1r) > > > > So 0.111... base three = 1/3 + 1/3^2 + 1/3^3. + ... > > = (1/3)/(11/3) > > = (1/3)/(2)3) = 1/2 > > > > So it appears that, at least for those who can work it out for > > themselves, you are the one who is wrong! > >  > > No the algorithm working perfectly correct, for anybase.
If your algorithm produces a terminating representation for one half in base three then it is wrong.
For any positive integers m an n greater both than one, 1/m can have a finite expression in base n only if m divides exactly into some power of n.
For example in base 10,fraction 1/2, 1/4, 1/8, and 1/5, 1/25, 1/125 and so on all have exact decimal expressions because 2 and 5 both go into 10 exactly, 4 and 25 go into 10^2 exactly, and 8 and 125 go into 10^3 exactly.
Since 2 does not go into 3 exactly a whole number of times with no remainder, 1/2 cannot have a finite expansion in base 3.
And any algorithm that says otherwise is wrong. 

