Re: irrational number continuum



slartibartfast wrote:

we don't accept the existence of the square root of -1, yet we use it
usefully.

The square root of -1 does exist; see below.

we don't have to accept the actual existence of such a number as the
square root od 2 in order to do useful calculations with that "notion"
either. I've never once had to write it out in full!

I *define* the *real* number 1 as the set of *rational* numbers q
such that q is less that the *rational* number 1.

I *define* the *real* number 2 as the set of *rational* numbers q
such that q is less than the *rational* number 2.

etc.

I *define* the *real* number sqrt(2) as the set of *rational* numbers q
such that q is less than *rational* 0 or q^2 is less than *rational* 2.

I *define* the sum of two *reals* r1 + r2 as the set of *rational* numbers q
such that for q1 in r1 and q2 in r2, q = q1 + q2.

etc.

Now if you consider that such sets of *rational* numbers actually exist,
then the the corresponding *real* numbers actually exist, *by definition*.

If you don't consider such sets of rational numbers as actually existing,
well, I don't care to argue the matter.

For more about constructing the real numbers, see: http://tinyurl.com/kt53wy .

Having constructed the real numbers, we can then construct the complex numbers
simply as ordered pairs of real numbers:

http://en.wikipedia.org/wiki/Complex_number#Formal_development .

--
hz
.