Errors in handling real numbers ?

• To: mathgroup at smc.vnet.net
• Subject: [mg8660] Errors in handling real numbers ?
• From: "Piotr G. Kin" <Piotr.Kin at per.clw.csiro.au>
• Date: Mon, 15 Sep 1997 02:48:49 -0400
• Organization: CSIRO
• Sender: owner-wri-mathgroup at wolfram.com

```I was trying to develop a function that would return number of digits
after the decimal point in a real number. The function would work as
follows:

In[]:=	a = 1. 5.1234;
b = 1. 0.00123;
c = 1. 0.5;
digitsafterdot[{a,b,c}]

Out[]=	{4,5,1}

(
The results would differ from those obtained through application of
RealDigits[] i.e.,

In[]:=	Apply[Length[#1] - #2 &, RealDigits[{a,b,c}], {1}]
Out[]=	{15,18,16}
)

I was REALLY surprised to discover that there is a serious error in
handling real numbers. Consider the expressions:

In[]:=	x = 0.11;
NestList[FractionalPart[10#] &, x, 5]
NestList[(10. # - IntegerPart[10. #]) &, x, 5]
Out[]=	                      -16            -15           -14
-13
{0.11, 0.1, 8.88178 10   , 8.88178 10   ,8.88178 10   , 8.88178
10   }

-16            -15
-14            -13
{0.11, 0.1, 8.88178 10   , 8.88178 10   ,8.88178 10   , 8.88178
10   }

The results are as expected and are identical for both expressions.
Remarkably, when value of x changes to (say) 0.12 the following happens:

In[]:=	x = 0.12;
NestList[FractionalPart[10#] &, x, 5]
NestList[(10. # - IntegerPart[10. #]) &, x, 5]

Out[]=	{0.12, 0.2, 1., 1., 1., 1.}
{0.12, 0.2, 1., 1., 1., 1.}

The results are identical again but hardly expected. For visual
demonstration run the expression:

In[]:=	Plot[Function[x, Chop[Nest[FractionalPart[10 #] &, x ,2]]][y],{y,
0.00, 0.10}, PlotPoints -> 100]

It gets even funnier with numbers with precision higher than
\$MachinePrecision:

In[]:=	x = 0.20000000000000000;
NestList[FractionalPart[10#] &, x, 5]
NestList[(10. # - IntegerPart[10. #]) &, x, 5]

Out[]=	17

{0.2000000000000000,1.000000000000000,1.00000000000000,1.0000000000000,1.000000000000,1.00000000000}
{0.2000000000000000,0.,0.,0.,0.,0.}

The results of two mathematically equivalent expression are
significantly different.

Am I to sensitive or is this a serious problem?

Peter.

P.S. All the above results were obtained on INTEL Pentium 166 PC.

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Piotr G. Kin
CSIRO Land & Water              e-mail: Piotr.Kin at per.clw.csiro.au
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___________________________________________________________________________

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NOT
commit me to anything or any place or even CSIRO. All views stated here
are
wholly or partially my own or conveniently borrowed or vaguely
remembered.
___________________________________________________________________________

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```

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