# 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.”