Complex exponential map restricted to the rational numbers

Question

Consider the mapping $$\varphi: \mathbb{Q} \to \mathbb{C}\\ \quad \qquad x \mapsto e^{2\pi i x}$$ What is the image $\varphi(\mathbb{Q})$?

Intuitively I would say that it is a countable subset of $\mathbb{S}^1$.

Answer 1

You are correct. Firstly it's manifestly countable since $|f(X)|\le |X|$, you cannot increase cardinality using a function. As for density, just note that any element of $S^1$ is of the form $e^{i\theta}$ for some $0\le \theta <2\pi$, and so if you take a sequence of rational numbers $q_n\to\theta$ then by continuity of the exponential map you know that $e^{iq_n}\to e^{i\theta}$ proving density. In fact, you can say even a little more, namely that the image is the set of all roots of unity, i.e. elements of $S^1$ satisfying $x^n=1$ for some $n$, since $e^{ip/q}$ raised to the $q$ is $1$.