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Re: Output Precision Exploration

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  Re: Output Precision Exploration Sseziwa Mukasa <mukasa@gmail.com> - 2011-04-28 10:33 +0000

#1942 — Re: Output Precision Exploration

On Apr 27, 2011, at 5:39 AM, Rafael Dunn wrote:

> Mathematica 8.0.1.0, Mac OSX x86
>
> In:=
> Log[173.5/173.5]
>
> Out:=
> -1.11022*10^-16
>
> I expect an output of exactly 0.  Although 10^-16 is small, it turned
> out to be the largest factor in a chemical equation I was attempting to
> compute.
>
> I discovered this is because Mathematica does not actually evaluate
> 173.5/173.5 = 1.  The output is actually some number 0.9999999999...
>
> However, for most decimal constants x/x produces an exact output of 1. 
> By entering a few decimals off the top of my head I also found 1733.5, 
> 26.44, and 27.44 do not produce an output of 1 when divided by
> themselves.
>
> Why?  I understand Mathematica's algorithms for working with decimals
> must make approximations, but why is there so much variance among
> decimal calculations?  173.49/173.49 = 1, while 173.5/173.5 != 1. 
> Furthermore, I find:
> x=173.49999999999999
> x/x = 173.5/173.5, with infinite precision.  If you add or remove a
> single 9 to the end of x, this ceases to be true.
>
> Furthermore, this looks like a contradiction to me:
>
> In:=
> 173.5/173.5 = 1
> Log[1] = 0
> Log[173.5/173.5] = 0
>
> Out:=
> True
> True
> False
>
> I have learned a lot about Mathematica's precision and approximation through the help documentation, but I still can not explain this or see how I can expect Log[x/x] = 0 for the sake of calculations on the 10^-16 scale.

My advice is to use exact numbers for x where possible, which it is in this case.  Machine precision is only good to 16 digits, if you need more you need to specify it explicitly.  Alternatively, if you don't actually need the precision but are annoyed by extraneous small values you can use Chop.

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