Definition and real straight line

We define real numbers as the union of the rational and irrational numbers. We call the set of real numbers $$\mathbb{R}$$. Then, we have $$\mathbb{Q}\subseteq \mathbb{R}.$$

Real numbers completely fill the straight line, so all the real numbers can be represented in the straight line and, reciprocally, a real number can be assigned to all the points of the straight line. That is the reason why we call it the straight line of real numbers, or to simplify, real straight line. We use it by referring to the straight line as a set of numbers.


In the mentioned straight line, between two rational numbers there is always an irrational one, and rationals and irrationals are always as close as we want.