Re: BigDecimal and trigonometrics


"Jeffrey Schwab" <jeff@xxxxxxxxxxxxxxxx> wrote in message
news:CwJef.502$xD5.738229@xxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> Yes, that's true. But can you please point me to an irrational number
> that cannot be derived by a sequence of mathematical operations on
> rational numbers?

No, but I think I can prove their existence if I can prove that all
sequences of mathematical operations on rational numbers is countable (since
the there are uncountably many irrational numbers).

Every sequence of mathematical operations on rational numbers can be
represented by some ASCII string (e.g. "1+1")

You can order them by using Java's standard string sorting algorithm.
Associate the first such legal string with the integer 1.
Associate the second such legal string with the integer 2.
And so on.

You now have a 1 to 1 mapping between sequences of mathematical
operations on rational numbers and the set of natural numbers, thus showing
that there are only countably many sequences of mathematical operations on
rational numbers.

Note that I'm assuming the ASCII representation is finite, which I think
is true as long as the number of operators and the number of arguments to
each operator is finite in the sequence (and as long as each operator and
each term can be represented by a finite number of characters, which is true
for rational numbers).

- Oliver


.

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