Let's $\mathbb{S}$ the set of numbers in $\mathbb{R}$ that can be expressed by :
$$\frac{\sqrt{a}}{\sqrt{b}} \: \text{or}\: -\frac{\sqrt{a}}{\sqrt{b}}$$
Where $a$ and $b$ are integers.
What numbers fall in this category ?
For example for $n_1$ in $\mathbb{N}$ :
$$n=\frac{\sqrt{n^2}}{1}$$
for $n_2 < 0$ in $\mathbb{Z}$ :
$$n=-\frac{\sqrt{n^2}}{1}$$
for $\frac{p}{q} $ in $\mathbb{Q}$ :
$$\frac{p}{q} = \frac{\sqrt{p^2}}{\sqrt{q^2}}$$
for $\frac{p}{q} \lt 0$ in $\mathbb{Q}$ :
$$\frac{p}{q} = -\frac{\sqrt{p^2}}{\sqrt{q^2}}$$
But what other combinations of addition, multiplication and radicals, can be expressed as this ?
For example is the golden ratio in $\mathbb{S}$ ?