Question : Prove that if $y,z \in Q$ then $y^z \in A$ My attempt: Definition 2.7.8 states that a number s is an algebraic number when there exists some $p \in Z[x]$ such that $p(s) =0$. Let us denote the set $ A = [x \in C: x $ is algebraic] The set of all polynomials in x with coefficients fr...

Let $a,b \in R$ where $ a < b$. Prove that there exist a rational number $c$ and an irrational number $d$ such that $ a <c<b$ and $ a<d<b$. Hint: consider decimal expansions of $a$ and $b$ Theorem 2.7.5 states that a real number is rational if and only if its decimal expansion terminates or has ...