The rational numbers occupy the number line densely, but they do not account for every point on it. Numbers such as $\sqrt2$ and $\pi$ cannot be written as ratios of integers; these are irrational numbers.

The rational and irrational numbers together form the set of real numbers, written $\mathbb R$.

The number systems

The number systems introduced so far are nested:

$$\mathbb N\subseteq\mathbb Z\subseteq\mathbb Q\subseteq\mathbb R.$$

The real numbers fill the number line continuously, and their order matches position on that line: if $x<y$, then $x$ lies to the left of $y$.

Absolute value and distance

The absolute value $|x|$ is the distance from $x$ to zero. More generally, the distance between two real numbers $x$ and $y$ is

$$|x-y|.$$

Intervals

Often we are interested not in one real number but in a whole subset of $\mathbb R$. An interval contains all real numbers between specified endpoints. Parentheses exclude an endpoint and brackets include it:

$$(a,b),\qquad [a,b],\qquad [a,b),\qquad (a,b].$$

Intervals can also be unbounded, such as $(-\infty,a)$ or $[a,\infty)$. Inequalities and interval notation describe the same regions of the number line; for example,

$$a\le x<b \qquad\Longleftrightarrow\qquad x\in[a,b).$$