Algebra Lessons


Interval Notation (parentheses & brackets)

Interval Notation

In mathematics, interval notation is a way to write an interval using parentheses, brackets, and commas. But first, let's define an interval.


An interval is a range of real numbers; it contains the set of real numbers lying between two numbers on a number line. An interval can be written in several ways but it is important to know how to recognize one. In the following example, we can see an interval starting on -3 and ending at 1. However, notice the circles. On -3, we observe a filled circle while on 1 we observe in open circle. Circles represent whether or not that number is being included in the interval. A closed circle means we include the number while an open circle means we do not.

So, this example, -3 is included in our interval while 1 is not. To represent this, it is common for people to use inequalities. For this example, we can observe the inequality: -3 is less then or equal to x which is less than 1. However, there is another way to write this—that's right, interval notation!

Interval Notation

For interval notation, rather than using inequality signs, we will be using parentheses and brackets. There are two key things you need to know:

  1. Open circles are represented with parentheses
  2. Filled circles are represented with square brackets

Let's take a look at an example.

In this example, we can see the interval beginning at -2 with an open circle and ending on 3 with a closed circle. Remember the key points from above! To represent this in interval notation, we will write (-2, 3]. -2 is written with a parenthesis because it contains an open circle and is therefore not included in the interval. Meanwhile, 3 is written with a square bracket because it contains a filled circle and is therefore included in the interval.

What if the interval is infinite?

If the interval is infinite, you will observe an arrow on the number line rather than a circle. The arrow represents that the set includes an infinite amount of real numbers in the direction it is pointing on the number line. Let's look at an example.

In this example, we observe an open circle on -1 and an arrow pointing towards all positive real numbers. To write this in interval notation, we will write (-1, ∞). -1 is represented by an open circle so we will use a parenthesis. When using infinity in your interval, you will always represent it with a parenthesis as well. Because ∞ is not a real number, it cannot be "included" in the number set so we must always represent it with a parenthesis, not a square bracket.



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