Interval Notation
Pre-AlgebraInterval notation is a way to describe a set of numbers on the number line using brackets and parentheses to indicate whether endpoints are included or excluded.
Definition
Interval notation is a shorthand way to write a range of numbers. Square brackets [ ] mean the endpoint is included. Round brackets (parentheses) ( ) mean the endpoint is not included.
Example
$[3, 7]$ means all numbers from $3$ to $7$, including $3$ and $7$. $(3, 7)$ means all numbers between $3$ and $7$, but not $3$ or $7$ themselves.
Key Insight
Use infinity for ranges that continue forever: $[2, \infty)$ means all numbers $2$ and above. Always use parentheses with infinity because you can never reach it.
Definition
Interval notation describes the solution set of an inequality using pairs of endpoints and bracket types. Square bracket: endpoint included (closed). Parenthesis: endpoint excluded (open). Infinity always uses a parenthesis. The union symbol ($\cup$) connects disjoint intervals.
Example
$x \ge -4$ is written $[-4, \infty)$. $-2 < x \le 5$ is written $(-2, 5]$. $x < 1$ or $x > 3$ is written $(-\infty, 1) \cup (3, \infty)$.
Key Insight
Interval notation is more compact than set-builder notation and more informative than just the inequality. It clearly shows boundedness and whether endpoints are included.
Definition
An interval in $\mathbb{R}$ is a connected subset: a set $I$ such that for any $a, b \in I$ and any $c$ with $a < c < b$, we have $c \in I$. The nine types of intervals are: $(a,b)$, $[a,b]$, $(a,b]$, $[a,b)$, $(a,\infty)$, $[a,\infty)$, $(-\infty,b)$, $(-\infty,b]$, and $(-\infty,\infty) = \mathbb{R}$. Intervals are exactly the connected subsets of $\mathbb{R}$ under the standard topology.
Example
The solution set of $x^2 < 9$ is $\{x : -3 < x < 3\} = (-3, 3)$, an open interval. It is connected and open in the standard topology on $\mathbb{R}$.
Key Insight
The Heine-Borel theorem characterizes compact subsets of $\mathbb{R}$ as exactly the closed bounded intervals $[a, b]$. This underpins why closed intervals guarantee the existence of maxima/minima for continuous functions, a fact essential throughout analysis.