FYI (Y=list), my Theorist program will take 0.4714045207910317 and
"Uncalculate" it to tell me it's sqrt(2)/3. In fact, that's how I
obtained the decimals.
But it won't uncalculate (1 + sqrt(2))/3. Indeed if it did I'd be
surprised.
It will take the decimals for 5sqrt(2) and uncalculate that to sqrt(50).
It will take 0.9428090415820635 and uncalculate that to 2 sqrt(2)/3.
And that's mildly surprising.
So, quick or not, someone has done some work on this subject.
Of course I agree that one makes a lot of assumptions when one takes a
finite sequence of decimal digits and says they are the same as some
irrational number. Or, even a rational. Theorist will also tell me
that 0.6666666666666666 uncalculates to 2/3. ... and that's true, if
uncalculate is defined as making a best guess at where a sequence of digits
may have "originated."
>In fact, there is no way to _communicate_ an arbitrary irrational number.
Certainly true in the context of decimal expansions, and
I basically agree, but it seems to me that the nature of number is even
more mysterious, since most irrational numbers can't even be named, in
any sense of the verb "to name." I would think that Cantor's proof of the
uncountability of the reals would prove this also.
But I qualify my agreement for two reasons. First, how does one even talk
about an arbitrary real number. Second, because the word "communicate" may
leave enough wriggle room for some algorithm, on the other side of the
axiom of choice, which purports to somehow name all the irrationals.
I think that the student wondering about 4.66666667 could be led to study
more math, to investigate the complexity (pun intended) of the real numbers.
But most people don't have this kind of interest, and the real world runs on
rational numbers (a small, finite subset of them), so I guess thats why
the fact that decimals are used more, and "closed form" expressions are used
less, doesn't bother me.
Phil Mahler
Middlesex CC
Bedford, MA
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