Re: The IDS, the EDS and the DBMS

Laconic2 wrote:
> First off, the common fraction is really an expression. If I express a
> number as 10/3, I've expressed the number in a few symbols, but what I've
> expressed is "the result of dividing 10 by 3". This number can't be
> expressed exactly as a finite decimal fraction, or as a finite binary
> fraction.
>
> In this sense "sqrt(2)" is not all that different.

Every rational can be written in a canonical form using two integers - if you add or multiply two rational numbers this is still the case.

This closure under addition and multiplication makes them a lot easier to deal with.

But irrationals, or even the subset of them that is the nth roots of integers, don't have such a canonical form - for example consider sqrt(2) + sqrt(3). So after a few simple addition or multiplication operations the representation will rapidly get very unwieldy.

Also I think there are almost certainly an uncountably infinite number of irrationals that can't be expressed as a finite expression of nth roots of integers with addition and muliplication. For example the ones commonly called "pi" and "e", but these just have names because they are somehow special. What about logarithms? Trigonometric functions?

I think the fundamental reason why rationals are so much easier is that they are countably infinite, whereas the irrationals are uncountable. (in a mathematical sense).

Paul. Received on Tue Sep 14 2004 - 17:56:51 CEST