# How are rational and irrational numbers related?

Apr 9, 2016

Rational and irrational numbers are mutually exclusive but jointly or collectively exhaustive set of real numbers.

#### Explanation:

The set of rational numbers and irrational numbers are unique The two most important features are:

(1) They are mutually exclusive i.e. a number from one set say Rational numbers $\mathbb{Q}$ cannot be a member of another set that is irrational numbers $\mathbb{Z}$ and vice-versa. In set theory, we say that the intersection of $\mathbb{Q}$ and $\mathbb{Z}$ is $\phi$, the null set or $\mathbb{Q} \cap \mathbb{Z} = \phi$.

This is because while rational numbers can be written as a ratio of two integers say $\frac{p}{q}$, where $p$ and $q$ are integers and $q \ne 0$, irrational numbers cannot be written as such.

In decimal notation, while rational numbers are terminating after decimal sign or have non-terminating but repeating (or recurring decimals), irrational numbers have non-terminating non-repeating (or non-recurring decimals).

(2) However, together they form the set of Real numbers $\mathbb{R}$ and both rational and irrational numbers can be represented on real number line and in set theory we say that $\mathbb{Q} \cup \mathbb{Z} = \mathbb{R}$ and there are no real numbers, which do not fall in one or he other category.

In short, we can say that rational and irrational numbers are mutually exclusive but jointly or collectively exhaustive set of real numbers.