I believe:
Calculators typically use base 10 calculations, computers typically
employ base 2. In base 2 the numbers here can't be represented
exactly because each term in the binary sequence (1/(2^n)) can only
get arbitrarily close to the number. In base 10 the number can be
represented exactly.
When you limit the precision of binary calculations to some number of
places n, the binary machine code likely automatically rounds at the
n + 1 digit, like the format() function.
PCs use IEEE calculations and truncate without rounding, giving the
binary result. One of the reasons the IEEE was requested to
standardize was to eliminate the differences between results given by
different OS's.
Cheers,
Art
On Jul 28, 2006, at 11:58 AM, realbasic-nug-
[EMAIL PROTECTED] wrote:
Subject: Re: Re: Math Error?
From: "Peter K. Stys" <[EMAIL PROTECTED]>
Date: Fri, 28 Jul 2006 10:53:51 -0400
On 7/28/06, Phil M <[EMAIL PROTECTED]> wrote:
On Jul 28, 2006, at 10:35 AM, Peter K. Stys wrote:
Interesting discussion. To muddy the waters further, using
Mathematica:
N[72/2000, 100] (ie calculate to 100 significant digits of
precision)
you get:
0.036000000000000000000000000000000000000000000000000000000000000000
00
000000000000000000000000000000000
No argument here, 72/2000 is really 0.036 exactly.
As people said, this real number may not be representable exactly
as a
binary floating point number, however:
N[72/2000] (calculate to machine precision)
gives 0.036, implying that the machine is indeed able to represent
0.036 exactly.
Yes, Calculator.app shows exactly "0.036" even with 16 digit
precision. Now try "0.036 * 750".
In[26]:= N[72/2000*750,100]
Out[26]=
27.0000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000
In[27]:= N[72/2000*750]
Out[27]= 27.
Still suggests all these numbers and intermediate results are
representable exactly at machine precision.
Odd indeed.
P.
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