Which of the following statements are true? a. Algebraic numbers over Z are dense in R. b. Transcendental numbers over Z are dense in R.
Let me write what I did. For a), the concerned set is a subset of Q and so is dense in R. For b), I've constructed a set A={n/(π^m) : n is in Z ans m is in N} (mimicking the way we construct dyadic rationals). Then since π is transcendental over Q, it is so over Z. And so the elements of A are transcendental over Z (I've a little doubt here). The way we show that the set of dyadic rationals is dense in R, we can show that A is dense in R. Since A is a subset of the concerned set, it is also dense in R. Am I correct? I'll be very glad if someone verifies it. Thank you. :-)