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Re: Pi == 3 for imprecise values of 3?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg22951] Re: Pi == 3 for imprecise values of 3?
*From*: Robert Knapp <rknapp at wolfram.com>
*Date*: Fri, 7 Apr 2000 02:54:33 -0400 (EDT)
*Organization*: Wolfram Research, Inc.
*References*: <8chbil$9ei@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Gavin Scott wrote:
>
> In playing with my spiffy new copy of 4.0.2 for Windows, I'm curious
> why:
>
> In[1]:= N[Pi] == 3`0
> Out[1]= True
>
> It seems that a value compared to a number with very low precision
> will result in Equal[] returning True.
>
> In[2]:= 1 == 2`0.00001
> Out[2]= True
>
> In[3]:= 1 == 2`0.0001
> Out[3]= False
>
> but then
>
> In[4]:= 10^10 == 2`0
> Out[4]= False
>
> I suppose that when you have zero digits of accuracy then you have no idea
> what the value is, so it certainly *could* be equal to any value, but I
> find it surprising that == chooses True over False in these cases.
>
In Mathematica, Equal (and Less and Greater) use a tolerance so that two
numbers do not have to be identical to be deemed Equal. THe motivation
behind this is beyond the scope of a simple reply, but if you are
interested, I suggest section 4.2.2 on Accuracy of floating point
numbers in Knuth's "Seminumerical Algorithms"
Given a tolerance, t, two numbers a and b are considered Equal to that
tolerance if
Precision[a - b] < t
A way of interpreting this is that a and b must be the same except in
the t least significant digits since the other digits must cancel in the
subtraction. So in your example:
(Set $MinPrecision here so it is not at all restricted)
In[1]:=
Block[{$MinPrecision = -Infinity}, Precision[Pi - 3`0, Round -> False]]
Out[1]=
-1.32608
The tolerance Equal uses is
In[2]:=
$EqualTolerance
Out[2]=
2.10721
which corresponds to 7 bits. In fact, with this tolerance,
In[3]:=
Block[{$MinPrecision = -Infinity}, Precision[Pi - 3`3.3, Round ->
False]]
Out[3]=
1.97392
so...
In[4]:=
Pi == 3`3.3
Out[4]=
True
Rob Knapp
Wolfram Research, Inc.
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