Types of Intervals and Interval Notation with Examples
An interval in math is a range of numbers between two assigned numbers and includes all of the real numbers between those two numbers. If you already know, real numbers are pretty much any number you can imagine about: 4.67, 151, √6, -0.143, π, etc. Intervals can be written with the help of inequalities, a number line, or in interval notation! There are also unique techniques to indicate whether the two assigned numbers, called endpoints, are included in the interval.
Example of Intervals
You must have heard that. A weather forecaster just made a prediction that there is going to be a rainfall with at least 4 but less than 9mm of rain! Thus, what are the different amounts of rain that we could anticipate based on those numbers? Well, go get your wipers ready and then let's find out.
When the forecaster said that there would be at least 4 but less than 9 millimeters of rain, he described the amount of rain in an interval!
Using Inequalities in Identifying Intervals
Inequalities in math are the symbols that represent mathematical signs such as greater than, and greater than or equal to, less than, and less than or equal to. Now considering these symbols, let’s decompose our weather forecast down and write the interval with the help of inequalities:
The 1st part is that it will rain at least 4mm. That implies that the amount of rain, which we will depict with the variable x since its unknown, will be at least equal to 4 but could be greater than 4. The 2nd part of our interval is that the amount of rain, x, will be less than 9mm.
Thus, we will read this inequality in intervals like this: 4 is less than or equal to x, which is less than 9. Thus, this interval is including endpoint 4 (since it's equal to), but not including endpoint 9. Therefore, x is any real number from 4 all the way to the last real number prior to 9.
Identifying Intervals Using the Number Line
To describe an interval on a number line, you have to first construct two circles at the two endpoints of the interval. So, we will construct two circles at 4 and 9. Now, construct a line to join the two circles! The last step is to color inside the circles only if the endpoint is included in the interval.
(Image will be uploaded soon)
Types of Intervals
1. Finite Interval
A finite interval (bounded interval) is an interval, whose both endpoints are numbers (also variables, which as you know describe unknown numbers).
Example: (-5,2]
Inequality:
-5 < x ≤ 2
These kinds of endpoints (numbers) are what we call finite endpoints.
2. Infinite Intervals
An infinite interval is that whose minimum one endpoint is infinity. The infinity interval is represented by the symbol ∞. You would be surprised to know that there is also minus infinity denoted as -∞ as well as the plus infinity denoted as +∞.
3. Open and Closed Intervals
Usually, it’s all about the types of brackets: a square bracket or a parenthesis.
You might already be familiar when there is a square bracket – the endpoint is related to the interval. Whereas, when there is a parenthesis - the endpoint does not pertain to the interval. That being said,
In the 1st possibility (a square bracket) the endpoint is closed,
In the 2nd possibility (a parenthesis) – the endpoint is open.
The interval can be referred to as:
Closed interval – Both brackets include the square brackets, for example: [2, 9].
Left-Closed Interval – Only the left side bracket is a square bracket, for example: [1, 5).
Right-Closed Interval – Only the right side bracket is a square bracket, for example: (-4, 7].
Open Interval – Both brackets include parentheses, for example: (-6, -2).
Left-Open Interval – Only the left side bracket is a parenthesis, for example: (-2, 8].
Right-Open Interval – only the right side bracket is a parenthesis, for example: [7, 11).
For most intervals, two descriptions are proper at the same time.
FAQs on Understanding Intervals in Mathematics
1. What is an interval in Maths?
An interval in Maths is a set of numbers between two given values on the number line. It represents all real numbers that lie within specified boundaries.
- An interval can include or exclude its endpoints.
- It is commonly used in inequalities, set notation, and calculus.
- Example: The interval from 2 to 5 including both endpoints is written as [2, 5].
2. What are the different types of intervals?
The main types of intervals are open, closed, half-open (or half-closed), and infinite intervals. Each type depends on whether the endpoints are included.
- Open interval: (a, b) – excludes both endpoints.
- Closed interval: [a, b] – includes both endpoints.
- Half-open interval: [a, b) or (a, b] – includes one endpoint.
- Infinite interval: (a, ∞) or (−∞, b] – extends indefinitely.
3. What is the difference between an open and a closed interval?
The difference is that a closed interval includes its endpoints, while an open interval excludes them.
- Closed interval: [a, b] means a ≤ x ≤ b.
- Open interval: (a, b) means a < x < b.
- On a number line, closed endpoints are shown with filled circles, and open endpoints with hollow circles.
4. How do you write intervals in interval notation?
Intervals are written using brackets and parentheses to show whether endpoints are included.
- Use [ ] if the endpoint is included (≤ or ≥).
- Use ( ) if the endpoint is excluded (< or >).
- Example: x ≥ 3 is written as [3, ∞).
- Example: 1 < x < 4 is written as (1, 4).
5. How do you convert inequalities into interval notation?
To convert an inequality into interval notation, identify the boundary numbers and determine whether they are included or excluded.
- Step 1: Find the critical values (boundary points).
- Step 2: Use [ ] for ≤ or ≥, and ( ) for < or >.
- Example: x > 2 becomes (2, ∞).
- Example: −1 ≤ x ≤ 3 becomes [−1, 3].
6. What is an infinite interval?
An infinite interval is an interval that extends indefinitely in one or both directions on the number line.
- Infinity is always written with a parenthesis: (a, ∞) or (−∞, b].
- Infinity symbols (∞, −∞) are never included as endpoints.
- Example: x < 5 is written as (−∞, 5).
7. What is the union and intersection of intervals?
The union of intervals combines all values from both sets, while the intersection includes only the common values.
- Union symbol: ∪.
- Intersection symbol: ∩.
- Example: [1, 4] ∪ [3, 6] = [1, 6].
- Example: [1, 4] ∩ [3, 6] = [3, 4].
8. How do you represent intervals on a number line?
Intervals are represented on a number line using shaded regions and endpoint markers.
- Draw a filled circle for included endpoints (closed interval).
- Draw an open circle for excluded endpoints (open interval).
- Shade the region between the endpoints.
- For infinite intervals, extend the shading with an arrow.
9. Can you give an example of solving a problem using intervals?
Yes, solving inequalities often results in an interval as the solution set.
- Example: Solve 2x − 3 < 5.
- Step 1: Add 3 → 2x < 8.
- Step 2: Divide by 2 → x < 4.
- Solution in interval notation: (−∞, 4).
10. Why are intervals important in calculus and functions?
Intervals are important in calculus and functions because they describe where a function is defined, increasing, decreasing, or continuous.
- The domain and range of a function are written using intervals.
- Intervals show where a function is positive or negative.
- In calculus, intervals define limits of integration such as [a, b].
