(Solved): Let \( G=\mathbb{R} \) and \( H=\mathbb{Z} \) be the groups of real numbers a ...

???????

Let \( G=\mathbb{R} \) and \( H=\mathbb{Z} \) be the groups of real numbers and integers under addition. Consider the factor group \( G / H=\mathbb{R} / \mathbb{Z}=\{a+\mathbb{Z}: a \in \mathbb{R}\}=\{\bar{a}: a \in \mathbb{R}\} \). Warm-up questions. What are the orders of \( \overline{5}, \overline{0.5}, \overline{2 / 3}, \overline{7 / 43} \) in \( \mathbb{R} / \mathbb{Z} \) ? What is the order of \( \overline{\sqrt{2}} \) ? Don't write the answers to the warm-up questions in your answer sheet. Answer only the questions below. Find all elements \( a \) of \( \mathbb{R} \) such that \( o(\bar{a}) \) in \( \mathbb{R} / \mathbb{Z} \) is finite. Find all elements \( a \) of \( \mathbb{R} \) such that \( o(\bar{a}) \) in \( \mathbb{R} / \mathbb{Z} \) is infinite. Prove your answers.