# 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/