# Interval Notation

**Interval notation** is a way to describe continuous sets of real numbers by the numbers that bound them. **Intervals**, when written, look somewhat like ordered pairs. However, they are not meant to denote a specific point. Rather, they are meant to be a shorthand way to write an inequality or system of inequalities.

## Writing Interval Notation

Intervals are written with rectangular brackets or parentheses, and two numbers delimited with a comma. The two numbers are called the **endpoints** of the interval.
The number on the left denotes the least element or lower bound. The number on the right denotes the greatest element or upper bound.

The rectangular bracket symbols, $[\ ],$ are used to describe sets with a "less than or equal to" or a "greater than or equal to" element, respectively. They correspond to the $\ge$ and $\le$ symbols:

$\begin{array}{lc} \text{Inequality:} & 3 \le x \le 9 \\ \text{Interval:} & [3,9]. \end{array}$

In this case, $x$ could equal $3$ or $9$. When both of the endpoints are included in the interval, the interval is a **closed interval**.

The parentheses symbols, $(\ ),$ are used to describe sets with a lower bound or upper bound, respectively. They correspond to the $>$ and $<$ symbols:

$\begin{array}{lc} \text{Inequality:} & -1 < x < 4 \\ \text{Interval:} & (-1,4). \end{array}$

In this case, $x$ does not equal $-1$ or $4$. When both of the endpoints are excluded from the interval, the interval is an **open interval**.

The different types of brackets can be used in the same interval:

$\begin{array}{lc} \text{Inequality:} & -3 \le x < 5 \\ \text{Interval:} & [-3,5) \end{array}$

In this case, $x$ could equal $-3$ but it cannot equal $5$. When one of the endpoints is included in the interval but the other is not, then the interval is a **half-open interval.**

If an interval has no lower bound or upper bound, then the $-\infty$ or $\infty$ symbols are used. These symbols are always used with a parentheses bracket, because infinity is not a number that can be included in a set:

$\begin{array}{lc} \text{Inequality:} & x \le 7 \\ \text{Interval:} & (-\infty,7] \end{array}$ $\begin{array}{lc} \text{Inequality:} & x >2 \\ \text{Interval:} & (2,\infty). \end{array}$

## Use interval notation to represent the interval notation shown on the number line below.

The interval includes values between -6 and 3, but does not include 3. Therefore, the correct interval notation is $[-6,3).$

Intersections and unions of intervals can be written with the $\cap$ or $\cup$ symbols:

$\begin{array}{lc} \text{Inequality:} & x \le -4\ \cup\ 0 < x < 6 \\ \text{Interval:} & (-\infty,-4]\ \cup\ (0,6) \\ \end{array}$ $\begin{array}{lc} \text{Inequality:} & x \ne 1\\ \text{Intervals:} & (-\infty,1)\ \cup\ (1,\infty). \\ \end{array}$

## Alternate Notation

In open and half-open intervals, the parentheses can also be replaced with reversed brackets, for example $(0, 1)$ can also be rewritten as $]0, 1[$.

In some countries where comma (,) is used as decimal points, a semicolon (;) may be used in place of a comma as a separator to avoid ambiguity: for example, $(0; 1)$.

**Cite as:**Interval Notation.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/interval-notation/