Help me to prove that $(\mathbb R,+)$ is not cyclic?

Question

Can you help me to prove $(\mathbb R,+)$ is not cyclic?

Answer 1

Suppose, seeking a contradiction, that $(\mathbb{R}, +)$ were a cyclic group. Then there exists a generator, $a \in \mathbb{R}$, such that $$\mathbb{R} = \langle a \rangle = \{a^n \mid n \in \mathbb{Z} \}.$$ Note: the operation in the group is addition, so when I write $a^n$, I really mean $\underbrace{a + a + \ldots + a}_{\text{$n$ times}}$. You might even write $na$ instead.

Clearly, if I can disprove the above equality by finding an element of $\mathbb{R}$ that cannot be written as $a^n$ for some $n$, I've established that no generator can exist.

There are a lot of different possibilities I can choose, but take as a simple one $\frac{1}{2} a$. If $\frac{1}{2} a \in \langle a \rangle$, then there exists $m \in \mathbb{Z}$ such that $$\frac{1}{2} a = ma.$$ Clearly $a \neq 0$, so I can divide by $a$: $$\frac{1}{2} = m,$$ but $\frac{1}{2}$ is not an integer. We have reached a contradiction.