Question 1

Question

What distinguishes the real numbers from the rational ones?

Answer

The main way to distinguish these is by knowing how they are constructed. And we get a set of these numbers based on how they are constructed.
But the basic structure of what real numbers encompass is: $\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}\subset \mathbb{C}$. and $1\subset \mathbb{R}$.

Rational Numbers

Rational numbers are numbers that you can express in a fraction.
For example $\left\{1\equiv \frac{1}{1},0.5\equiv \frac{1}{2},2\equiv \frac{2}{1}\right\}$.

Irrational Numbers

They are numbers that cannot be expressed as a fraction. These are numbers such as pi, which as of now has only been calculated to 300 trillion digits.