Lebl-Basic-Analysis-I · Problem 1.4.9

Real Numbers

Mathematics · intervals-cardinality

Problem

(Challenging) A number xx is algebraic if xx is a root of a polynomial with integer coefficients, in other words, anxn+an−1xn−1+⋯+a1x+a0=0a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 = 0 where a0,a1,…,an∈Za_0, a_1, \ldots, a_n \in \mathbb{Z}. a) Show that there are only countably many algebraic numbers. b) Show that there exist non-algebraic (transcendental) numbers (follow in the footsteps of Cantor, use the uncountability of R\mathbb{R}). Hint: Feel free to use the fact that a polynomial of degree nn has at most nn real roots.

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