Re: BigDecimal and trigonometrics

Oliver Wong wrote:
"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 

Yes, of course they exist.  Name one.

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).

Nicely done!!! So there certainly are unrepresentable irrational numbers. Let me "clarify" my position: Any irrational number that can be clearly represented in a traditional mathematical formula or proof, can also be represented in a computer program.
.

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