What is an Interval in Math? Meaning, Definition, Examples

Interval in Math

Imagine if you had to sit in class continually without a recess or lunch break. What would that be like?

Most things cannot go on without break. There is usually a gap that comes in between. These gaps are known as intervals.

Did you know there are intervals in math as well? They help us measure these gaps in terms of numbers. Let’s understand what is an interval in math in detail. 

What is an Interval in Math?

Let’s look at the definition of interval in Math.

An interval is a gap between two things or points. Let’s understand this better. 

How are these two lines different? 

The first line is continuous, and the second line has gaps.

What is the difference between the emergency flashing light of an ambulance and its headlamp?

The headlamps flash continuously. However, the emergency flashing lights flash at intervals. Continuous means without any gap whereas interval means with gaps. 

Let us see another example to understand intervals:

When a car moves on the road, it has to stop when the traffic light flashes red. However, if the traffic light is green, the car would continue moving without having to stop. 

Interval in Math

An interval in math is measured in terms of numbers. An interval includes all the numbers that come between two particular numbers.

This range includes all the real numbers between those two numbers. Real numbers are any kind of number you can think of. 

For example, an interval between 0 and 10 would include not just 1, 2, 3, 4, 5, 6, 7, 8, and 9 but also the other kinds of numbers between them. These would include decimal numbers like 1.3 and fractions like 1/10 as well. 

Interval Notation

Interval notation is a simplified way of describing a particular interval. Let’s see how it’s done. 

Step 1: We write the first and last number of the interval, which are the endpoints of the interval. For example, if the interval is from 6 to 20, we write 6, 20. 

Step 2: We use a round or square bracket on each side of the two numbers. We use:

• A square bracket [ ], if we want to include the endpoints 
• A round bracket ( ), if we don’t want to include the endpoints

So in this example, we use: 

• [6, 20] if the interval includes 6 and 20
• (6, 20) if the interval excludes 6 and 20 
• (6, 20] if the interval excludes 6 but includes 20
• [6, 20) if the interval includes 6 but excludes 20 

Types of Intervals in Math

So, we can see that some intervals include the endpoints, some partially include them, and some do not include them. Based on this, there are three types of intervals in math.

These are:

• Open interval
• Closed interval
• Half-open and half-closed interval

Open Interval

This type of interval does not include the endpoints. For example, (5, 10) does not include endpoints 5 and 10. 

Closed Interval

This type of interval includes the endpoints. For example, [4, 9] includes the endpoints 4 and 9. 

Half-Open and Half-Closed Interval

This type of interval includes only one of the endpoints.

For example, (6, 16] includes 16 but does not include 6, whereas [6,16) includes 6 but does not include 16.

This is what the different types of intervals look like on number lines: 

Time Interval

Let us learn what a time interval is.

Suppose you are walking from your home to your school. You walk continually and reach school in 10 minutes. 

Now, while walking back home, you get tired and decide to sit for 2 minutes. You stop for 2 minutes and then start walking home again. So, we say you took a time interval of 2 minutes while walking back home. 

The time interval can be defined as the time between two events.

We can show time intervals on a timeline:

Here, the timeline shows intervals of 1 hour. We can find the time between events using intervals. 

For example, Sam started playing soccer at 4:30 p.m. He finished the game at 6:00 p.m. For how long did he play?

We can draw a timeline as follows:

The interval from 4:30 p.m. to 5:00 p.m. is half an hour. Then, from 5:00 p.m. to 6:00 p.m. is an hour. So, that makes it an hour and a half. Sam played soccer for one and a half hours.

Fun Facts

•  The word interval is derived from Latin “Inter” meaning “between” and “vallum” meaning “space between ramparts”, i.e., the space between protective walls of a castle.

Solved Examples

Example 1. What is the interval between 2 p.m. and 6:45 p.m.? 

Solution: Let’s subtract the time values:

6:45 – 2:00 = 4:45

Thus, there are 4 hours and 45 minutes between 2 p.m. and 6:45 p.m.

So the interval will be 4 hours and 45 minutes. 

Example 2. Students between the age of 12 and 17 can be on the school soccer team. Give an interval that will represent this information. 

Solution. The permitted ages are between age 12 to age 17.

So, the interval is [12, 17]. 

Example 3: To pass a math exam, students need to score above 65 marks out of a total of 100. How will this be represented as an interval?

Solution. The students need to score above 65 marks and up to 100 marks.

The interval for this will be written as (65, 100].

Practice Problems