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Topic: Wikipedia and fractions in other base then decimal
Replies: 16   Last Post: Feb 11, 2013 9:43 PM

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Virgil

Posts: 9,012
Registered: 1/6/11
Re: Wikipedia and fractions in other base then decimal
Posted: Feb 11, 2013 9:43 PM
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In article
<05226575-5837-41af-9489-89936267dcb5@j4g2000vby.googlegroups.com>,
JT <jonas.thornvall@gmail.com> wrote:

> On 11 Feb, 23:28, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <3c4b196b-1998-412f-94aa-be6b71bd8...@k14g2000vbv.googlegroups.com>,
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >  JT <jonas.thornv...@gmail.com> wrote:

> > > On 11 Feb, 09:00, Virgil <vir...@ligriv.com> wrote:
> > > > In article
> > > > <03a59b13-1fbd-4098-91d7-f093d7cf5...@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 one-half 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/(1-r) so
> >    r + r^2 + r^3 + ... = r/(1-r)
> >
> > So 0.111... base three = 1/3 + 1/3^2 + 1/3^3. + ...
> >                        = (1/3)/(1-1/3)
> >                        = (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.
--





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