Question

Are all real numbers either rational or irrational?​

Answer 1

Answer:

Yes. All real numbers are either rational or irrational.

Step-by-step explanation:

The set of all irrational numbers is commonly defined as the set of all real numbers that are not rational.

Let $\mathbb{R}$ denote the set of all real numbers. Let $\mathbb{Q}$ denote the set of all rational numbers.

For any given real number $x$ ($x \in \mathbb{R}$,) either $x\!$ is rational ($x \in \mathbb{Q}$) or $x\!\!$ isn't rational ($\lnot (x \in \mathbb{Q})$, such that $x \not\in \mathbb{Q}$ and thus $x \in (\mathbb{R} \backslash \mathbb{Q})$.)

Assume that $x$ isn't a rational number. By the definition of rational numbers, since $x\!\!$ is a real number but not a rational number ($x \in (\mathbb{R} \backslash \mathbb{Q})$,) $x\!$ would be an irrational number.

Therefore, either $x$ is rational or $x\!$ is irrational. Every real number would either be rational or irrational.