Re: Re: Math Error?

[Robert Delaney <delaneyrm at earthlink dot net>]

At 12:35 PM -0400 7/28/06, Peter K. Stys wrote:
> Actually we're both right and wrong:
>
> In[55]:=RealDigits[0.036,2,100]
>
> Out[55]={{1,0,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,1,1,1,
>    1,1,0,1,1,1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
>    0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0},-4}
>
> Shows that 0.036 can be represented exactly in binary, but requires 51
> significant digits (if I counted correctly to the last 1).  So with a
> 64 bit double, it probably just missed the # of digits in the mantissa
> to represent 0.036 exactly.

I couldn't reply in this thread in a timely manner because of a 
change in earthlink's SMTP server which prevented me from sending any 
email.
The above binary representation of .036 is incorrect. My Base 
Converter application shows and verifies that to 100 binary point 
places the correct representation is:
..000010010011011101001011110001101010011111101111100111011011001000101101000011100101011000000100001

So the first part of Out[55] is correct, but the final group of all 
zeroes is not. I saw that someone else has proven that .036 requires 
an infinite number of binary point places for an exact representation 
..
Base Converter is a Universal Cocoa application for Mac OS  X only. 
It's freeware and can be found at:
http://homepage.mac.com/delaneyrm/BaseConverter.html

I wrote it to develop my Cocoa skills. If there's any demand I'll 
write a REALbasic version.
Bob
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