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Re: Is there a simple way to transform 1.1 to 11/10?

• To: mathgroup at smc.vnet.net
• Subject: [mg92980] Re: Is there a simple way to transform 1.1 to 11/10?
• From: Alain Cochard <alain at geophysik.uni-muenchen.de>
• Date: Wed, 22 Oct 2008 05:36:28 -0400 (EDT)
• References: <gdkajv$4r2$1@smc.vnet.net>
• Reply-to: alain at geophysik.uni-muenchen.de

Szabolcs Horv=E1t writes:
 > On Tue, Oct 21, 2008 at 14:49, Alain Cochard
 > <alain at geophysik.uni-muenchen.de> wrote:
 > > Szabolcs Horvat writes:
 > >  > Alain Cochard wrote:
 > >  > > The obvious
 > >  > >
 > >  > >     In[1]:= x=1.1`Infinity
 > >  > >
 > >  > > is not syntactically correct.
 > >  > >
 > >  > > I understand that SetPrecision[1.1,Infinity] does not work ei=
ther:
 > >  > >
 > >  > >     In[3]:= SetPrecision[1.1,Infinity]
 > >  > >
 > >  > >             2476979795053773
 > >  > >     Out[3]= ----------------
 > >  > >             2251799813685248
 > >  > >
 > >  > >     In[4]:= N[%,20]
 > >  > >
 > >  > >     Out[4]= 1.1000000000000000888
 > >  > >
 > >  > > I searched the newsgroup and thought I had the solution with =
Rationalize:
 > >  > >
 > >  > >     In[5]:= Rationalize[1.1,0]
 > >  > >
 > >  > >             11
 > >  > >     Out[5]= --
 > >  > >             10
 > >  > >
 > >  > > But
 > >  > >
 > >  > >     In[9]:= Rationalize[1.000000001,0]
 > >  > >
 > >  > >             999999918
 > >  > >     Out[9]= ---------
 > >  > >             999999917
 > >  > >
 > >  > >     In[10]:= N[%,20]
 > >  > >
 > >  > >     Out[10]= 1.0000000010000000830
 > >  > >
 > >  > > So any simple way=3F
 > >  > >
 > >  >
 > >  > Hello Alain,
 > >  >
 > >  > Rationalize is the way to go.  Floating point numbers are usual=
ly stored
 > >  > in a binary (not decimal) representation on computers.  1.00000=
0001 is
 > >  > not exactly representable in binary (in the same way as 1/3 ==

 > >  > 0.3333333... is not exactly representable in decimal).  Note th=
at in
 > >  > your example you start with a MachinePrecision number (approxim=
ately 15
 > >  > digits), and then convert back to a number with 20 digits of pr=
ecision.
 > >  >   If you start with 1.000000001`20 then everything will be fine=
=2E
 > >  >
 > >  > In[1]:= Rationalize[1.000000001`20, 0]
 > >  > Out[1]= 1000000001/1000000000
 > >  >
 > >  > In[2]:= N[%, 20]
 > >  > Out[2]= 1.0000000010000000000
 > >  >
 > >  > Another example:
 > >  >
 > >  > In[1]:= Rationalize[N[Sqrt[2], 30], 0]
 > >  > Out[1]= 1023286908188737/723573111879672
 > >  >
 > >  > In[2]:= N[%, 30]
 > >  > Out[2]= 1.41421356237309504880168872421
 > >  >
 > >  > In[3]:= % - Sqrt[2]
 > >  > Out[3]= 0.*10^-30
 > >
 > > I don't really convert back to a number with 20 digits of precisio=
n.
 >
 > You used N[%,20] in your example where % was the rationalization of =
a
 > MachinePrecision number.  If you convert it back to a MachinePrecisi=
on
 > number (simply N[%]), then you'll get the same number that you start=
ed
 > with, and not a different one.
 >
 > > It was just to convince myself that it was not 1000000001/10000000=
00
 >
 > Actually 1.000000001 isn't 1000000001/1000000000 either ... see belo=
w.
 >
 > > The solution you give will not be fully general (as that by bob) s=
ince
 > > you have to adapt the '20' or the '30' for each case.  This is a b=
it
 > > painful for a given number and impossible to do in a program, it s=
eems
 > > to me.
 > >
 >
 > No, you misunderstood me.  I did not offer a solution.  I just tried=

 > to explain why the result returned by Rationalize[1.000000001,0] is
 > correct, and why the "problem" does not exist at all.

Sorry that I misundertood you.  Also, I wasn't implying the Mathematica does
not give the correct result. 

There is a problem for me because when I play with mathematical
numbers like 1.608910743981704391 with an infinite number of zeroes
afterwards, it is painful to manually count the figures and write it
as 1608910743981704391/1000000000000000000

 > You asked how you can transform 1.1 to 11/10.  This is not a
 > precisely defined question.  Why do you prefer 11/10=3F  Why not
 > 1100000000000000088/1000000000000000000, for example=3F

Because I'd say that for pure mathematics, 11/10 is closer to 1.1 than
1100000000000000088/1000000000000000000 is...

 >
 > [...]
 >
 > What I meant was that you have to decide what precision to use.  [...]

I want infinite precision, juste like when you enter 1, Pi, or 11/10.