Is there some theorem that shows that rational and irrational numbers together can be used to completely occupy a continuous infinite line? i.e. that the real numbers are the union of the two sets.
Is there some theorem that shows that rational and irrational numbers together can be used to completely occupy a continuous infinite line? i.e. that the real numbers are the union of the two sets.
Since the irrationals are defined to be those real numbers that are not rational, we are forced to the conclusion that the deals are the union of the rationals and irrationals.
OK, I understand that, but the book threw me off a bit when it said "it can be shown", as if it required an actual proof rather than just a definition. I think they should have made it clear.