123.3 + 0.1 is 123.3999999999 ?

Andrew Dalke adalke at mindspring.com
Thu Jun 5 04:06:21 CEST 2003


Erik Max Francis:
> >I suspect he's being facetious, since 0.999... = 1.

Tim Rowe:
> ... usually :-)
>
> (It's true in the definition of real numbers mathematicians /usually/
> use, but there are alternative definitions available -- and used --
> when it's convenient to make the distinction)

I seek enlightnment.

I remember back in high school reading a proof of Cantor's diagonalization
which showed the cardinality(Z+) != cardinality(reals).  It was the table
description, which looked like

1 |  0.01312314553754698372...
2 |  0.87348204895729483218...
3 |  0.74195034785983576422..
 ...

then take the diagonal to get a new number, which is 0.071...
add one to each digit to get 0.182... and hence a construction
of a number which is not matched to a Z+.

Only years later did I learn that that's an approximation, in that
if the diagonal happens to have 888888888.... then the number
generated is 99999999.... which may construct a number which
isn't in [0, 1).

I only recently read the Cantor's original proof was much more
elegant than this, using the intersection of an infinite number of
closed segments instead of the actual digit representation, and
showing that the intersection must be non-empty, hence more
reals than integers.  (Elegant because I don't like proofs which
depend on the representation of a number - I would say axiom
of choice, but I was more of an analysis person, and not an
algebra weenie ;)

Anyway, so I wish to know which definition of the reals you
refer to, in the sense that I though all the definitions used (like
continued fractions) were equivalent to the common notation
where "0.99999...." does equal 1.

"Die ganze Zahl schuf der liebe Gott, alles Ubrige is Menschenwerk."
(God made the integers, all else is the work of man.)
But then again, Kronecker didn't like Cantor :)

                    Andrew
                    dalke at dalkescientific.com






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