Most definitions of irrationals that I've seen, basically boil down to merely stating that they're real, but not rational. This is a definition in negative terms: stating what irrational numbers are NOT. According to what little formal logic I've been told, formal, mathematically-logical, consistently well-behaved definitions are supposed to be stated in the positive. I 'd like to know whether irrationals can be defined positively: stating what they ARE, in general, universally-applicable terms; not just giving some specific, limited examples. If irrationals cannot be defined positively, how can we then define reals without referring to irrationals, i.e.: without saying that reals are the union of rationals and irrationals? ...And, furthermore, how can we then define complex numbers? How can we continue using Math, if such fundamental definitions cannot be stated in formal, mathematically-logical, consistently well-behaved terms?
2006-12-12
17:57:11
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7 answers
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asked by
Shem ben Av
1
in
Mathematics