No, the point is that no matter how many digits you carry out the square
root of two it is still an approximation. It is not the square root of
two. Using such a number as if it were the square root of two means you
should understand that at some point you will get inaccuracies in your
calculations if you try to carry them out to too great a precision. I
took exception when you took a number that you assumed was the square
root of two but was really an approximation and then tried to use it as
if it were the square root of two. As expected, it gave erroneous
results when you carried out calculations beyond the precision of the
number. Don't blame the software or the computer. Blame the nut behind
the wheel (or keyboard, in this case).
All too often we start with measurements of something of limited
precision and try to glean out information beyond the accuracy of the
data. Statisticians give answers in terms of standard deviations yet
people who don't understand statistics take their answers as absolute.
Computers use base two arithmetic because it is efficient and gives
practical solutions to real problems. The fact that it can't even
represent the number one-tenth exactly usually doesn't cause problems.
But I'm sure it does in the banking industry.
Handling instabilities in calculations is still an art, not yet a
science. Recall the discussions on LAPACK in the newsgroup recently.
I worked in an oil exploration group for a number of years and there was
a joke that floated around.
Ask a mathematician, "What's two plus two?"
He says, "Four".
Ask an engineer, "What's two plus two?"
He says, "It's 3.999999999."
Ask a geologist, "What's two plus two?"
He says (with hesitation), "It's somewhere between three and five."
Ask a geophysicist, "What's two plus two?"
He leans over close to you and says very quietly, "What would you like
it to be?"
Unfortunately there is some truth to that joke. We see it on the news
and hear it from politicians every day.
p j wrote:
which matches the web reference I made
(http://www.rossi.com/sqr2.htm)... at least the first
2 lines of the maple result does.
If I understand the criticism of the web pasting I
made, it is that I only pasted approx 50 digits of the
solution, so I think the point was that even the right
50 digits is an innacurate approximation :(
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