Real_numbers

Real number axioms:

:Let R denote the set of real numbers, and w,x,y,z be elements of R. Then:

The set R is a Field
The field R is ordered i.e.

if x >= y then x + z >= y + z
if x >= 0 and y >= 0 then x * y >= 0

If S is non-empty subset of R and S has an upper bound in R, then there exists some z in R that is a least upper bound for S.

A real number is one that can be expressed in the form

'DDD.ddd'.

DDD is zero or more decimal digits
ddd is zero or more decimal digits

::Of course, DDD must be finite in length. This restriction does not apply to ddd.

Special cases:

ddd is 0 - the format is 'DDD' and the decimal point is not included

Note: There are many subsets of the real numbers, including:

Natural numbers
Rational numbers
Whole numbers
(??? name) - infinitely repeating fractions (include in rational numbers???)

These are the rational numbers (if you allow 12 = 11.9... If you don't allow this, the set is base dependent, which is undesirable.) This may be proved by summing as a geometric progression e.g

: 0.121212... = 12/100 + 12/100^2 + 12/100^3 + ...

irrational numbers -- Any number which cannot be represented as the ratio of two integers. These have infinitely long, non recurring decimal representations.

*transcendental numbers -- those that are not the roots of any polynomial with integer coefficients.