# Is 0 a rational or irrational number?

Apr 1, 2016

Rational

#### Explanation:

Rational numbers $\mathbb{Q}$ are basically all your fractions in that they can be written as a ratio of integers $\mathbb{Z}$.
By definition, $\mathbb{Q} = \left\{\frac{m}{n} | m , n \in \mathbb{Z} , n \ne 0\right\}$

Now $0$ can be written as $\frac{0}{n}$ for all $n \in \mathbb{Z} , n \ne 0$, and hence $0 \in \mathbb{Q}$.

Irrational numbers $I$ cannot be written in this form as a ratio of integers and include numbers such as $\pi , e , \frac{1}{\sqrt{2}} , \ln 2 ,$ etc.
By definition $I = \mathbb{R} - \mathbb{Q}$, where $\mathbb{R} = \left(- \infty , \infty\right)$ is the set of all real numbers.
But the set of rational and irrational numbers are disjoint, ie. they have empty intersection, and $\mathbb{R}$ is a topological space.
In other words $I \cup \mathbb{Q} = \mathbb{R} \mathmr{and} I \cap \mathbb{Q} = \phi$.