Rational numbers
Definition
These are all the numbers that can be written as a fraction:
$$\mathbb{Q} = \left\{\, \dfrac{a}{b} \;\mid\; a,\, b \in \mathbb{Z},\; b \neq 0 \,\right\}$$
Example
$$\dfrac{1}{3}, \qquad \dfrac{1}{2}, \qquad 5.\overline{3} = \dfrac{48}{9}, \qquad 1.0\overline{52} = \dfrac{521}{495}, \qquad 1.3 = \dfrac{13}{10}$$
The bar over the digits means that those digits repeat for ever. Every terminating or repeating decimal can be written as a fraction, and so it is rational.
Irrational numbers
Definition
$\mathbb{I}$ is the set of numbers that cannot be written as a fraction.
Example
$$\pi, \qquad e, \qquad \sqrt{2}, \qquad 1.23456789101112\ldots, \qquad 1.010010001\ldots$$
The decimal expansion of an irrational number is infinite and non-repeating: the digits never settle into a repeating block. The last two examples are built on purpose to show this.
Two irrationals with a name
$\pi \approx 3.14159\ldots$ is the ratio of a circle’s circumference to its diameter.
$e \approx 2.71828\ldots$ is the base of the natural logarithms.
The real numbers
The real numbers ($\mathbb{R}$) are the rationals together with the irrationals.
- $\mathbb{N}$, the naturals: $0,\, 1,\, 2,\, 3,\ \ldots$
- $\mathbb{Z}$, the integers: $\ldots,\, -2,\, -1,\, 0,\, 1,\, 2,\ \ldots$
- $\mathbb{Q}$, the rationals: those that can be written as a fraction.
- $\mathbb{I}$, the irrationals: those that cannot.
- $\mathbb{R}$, the reals: the rationals and the irrationals together.