# Interval Notation Calculator

The interval notation calculator is a free online tool that displays the number line for the interval input. Our interval notation calculator tool is an online tool that makes calculations faster and shows the number line in a fraction of a second.

## How to Use the Interval Notation Calculator?

To use the interval notation calculator, follow these steps:

Step 1: Fill out the input fields with the interval (closed or open)

Step 2: Click the Calculate button to obtain the results

Step 3: Once the new window is opened, the number line representing the given interval will be displayed

## What Is An Interval In Mathematics?

A mathematical interval is a set of real numbers that falls between two given numbers called the endpoints of the interval. The set contains all the numbers lying between the two endpoints of the set. They may or may not be included in the set, depending on the interval’s “type” or notation.

Before we discuss these notations, let’s look at an example of an interval: the set of numbers x satisfying -1 ≤ x ≤ 1 is an interval that includes -1, 1, and all the numbers between them. In our example, the interval could have included the endpoints, but not in our example. Therefore, the interval would be -1 < x < 1. As a result, the interval only contains zeroes and ones within the range. The real intervals play an important role in integration theory because their “size” or “measure” is the simplest and easiest to define.

• Inequalities
• Interval Notation
• The Number Line

### Inequalities

With Inequalities we use:

• > greater than
• ≥ greater than or equal to
• < less than
• ≤ less than or equal to

Like this:

Example: x ≤ 20
Says: “x less than or equal to 20”

And means: up to and including 20

### Interval Notation

In “Interval Notation” we just write the beginning and ending numbers of the interval, and use:

• [ ] a square bracket when we want to include the end value, or
• ( ) a round bracket when we don’t

Like this:

Example: (5, 12]

From 5 to 12, do not include 5 but do include 12

Depending on the characteristics of their two endpoints (a and b), intervals are notated in different ways, known as interval notation. Including a and b, we note the interval as [a,b], and excluding a and b, we note the interval as (a,b). For countries that use commas to write decimal numbers, we can replace the comma with a semicolon.

The Symbol for Interval Notation

The notations we use for different intervals are:

• [ ]: Square brackets are used when both endpoints are in the set.
• ( ): The round bracket is used when both endpoints are excluded.
• ( ]: This is a semi-open bracket that excludes the left endpoint, while the right endpoint is included in the set.
• [ ): The left endpoint of this bracket is also included, while the right endpoint is excluded.

An open interval

An open interval with endpoints a and b does not include the endpoints in the interval. This means that the interval ]a,b[ is formed by all the numbers of the interval that are found between a and b. Formally, we will write that x belongs to the interval if a<x<b.

Graphically, an open gap is represented by a segment whose ends are formed by hollowed-out points.

To write this interval in interval notation, you must use parentheses: (a,b).

A half-open interval

Semi-open intervals can also be semi-closed, which is what we call half-open intervals. Intervals with no endpoints include only one of them in the interval. Right or left, a semi-open interval exists.

]a,b] includes all the numbers either greater than a or less or equal to b regarding the endpoints of the interval to the left. The interval contains x if a < x ≤ b. Semi-open intervals are represented graphically by segments whose left end is hollowed out and the right end is solid.

A half-open interval to the right with endpoints a and b [a,b[ includes all the numbers greater than or equal to a and strictly less than b. We will write that x belongs to this interval if a ≤ x < b. Half-open intervals on the right are represented graphically by segments with solid left ends and recessed right ends.

Use parentheses to show whether an endpoint is included or excluded from this interval, and square brackets to show whether it is included (a,b]; use parentheses to show whether it is excluded [a,b).

A closed interval

A closed interval with endpoints a and b includes both of them in the interval. Meaning that the interval [a,b] is formed by all the numbers between a and b. We write that x belongs to the interval if a ≤ x ≤ b. Graphically, a closed interval is represented by a segment whose two ends are filled.

To write this interval in interval notation, you must use square brackets: [a,b].

### Number Line

With the Number Line we draw a thick line to show the values we are including, and:

• a filled-in circle when we want to include the end value, or
• an open circle when we don’t

Look at the handy table that distinguishes between all the types of intervals.

## How To Convert Inequality To Interval Notation?

Follow the steps mentioned below to convert an inequality to interval notation.

• Graph the solution set of the interval on a number line.
• Write the numbers in the interval notation with a smaller number appearing first on the number line on the left.
• If the set is unbounded on the left, use the symbol −∞ and if it is unbounded on right, use the symbol ∞.

Let’s take a few examples of inequality and convert them to interval notation.

## Important Points To Remember

• Interval notation is used to express the set of inequalities.
• There are 3 types of interval notation: open interval, closed interval, and half-open interval.
• The interval with no infinity symbol is called a bounded interval.
• The interval containing the infinity symbol is called an unbounded interval.

## Conclusion

The article focused on the fascinating concept of interval notation. The students begin by applying what they already know about Interval Notation, and then creatively frame a novel concept in their minds. The message must be relevant, easy to understand, and stick with them for a lifetime.