Irrational Numbers

Are rational numbers the only numbers? The continuity of real numbers is an assumption in the Cantor-Dedekind axiom [1]. The idea of continuity brings up the possibility of a different type of number called an irrational number that is, simply, not a rational number. So, if a number is not a rational number, it is called an irrational number–this definition of an irrational number may need to modified later if new classes of numbers arise. An irrational number is denoted by a Greek letter such as $ \sigma$ in contrast to the lowercase italicized letters for rational numbers. The author of reference [1] claims that it possible to define any rational or irrational number in terms of open intervals about a single point on a number line. This collection of intervals is called a nest of intervals $ J_n$ or $ (x_n | y_n)$ in which the rational number $ x_n$ is one of the lower bounds and the rational number $ y_n$ is one of the upper bounds for the rational or irrational number that is being considered [1]. Here, $ y_n – x_n$ forms one null sequence [1]. See https://derive-it.com/2020/12/19/definition-of-a-sequence/ for a definition of a null sequence. Furthermore, the set of all rational and irrational numbers comprise the so-called system of real numbers [1]. The system of real numbers is given the symbol, $ \mathbb{Z}$. I now include an interesting statement from Reference [1].

“This system of real numbers is in one-one correspondence with the whole aggregate of points on the number-axis.”

-Konrad Knopp

This statement implies that there are no other numbers that do not fall into the categories of rational and irrational numbers. It is interesting that the real numbers are introduced in this book by considering the geometrical concept of a number line. Is it possible to define real numbers without geometry? I do not know. Anyway, the important points from reference [1]’s Section 3, Irrational Numbers, have been covered here.

References