How to Identify and Use Intervals in Problem Solving
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.
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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 Intervals in Maths: Definition, Types & Examples
1. What is an interval in mathematics, with an example?
In mathematics, an interval is a set that contains all the real numbers lying between two specific numbers, which are known as the endpoints. It represents a continuous range of values on the number line. For example, the set of all numbers x such that 2 ≤ x ≤ 5 is an interval. It includes 2, 5, and all the real numbers in between, like 2.1, 3, and 4.99.
2. What are the main types of intervals in maths?
There are four primary types of intervals based on whether the endpoints are included or excluded:
Closed Interval: [a, b]. Includes both endpoints 'a' and 'b'.
Open Interval: (a, b). Excludes both endpoints 'a' and 'b'.
Semi-Open/Semi-Closed Interval: This can be [a, b), which includes 'a' but excludes 'b', or (a, b], which excludes 'a' but includes 'b'.
Unbounded Interval: An interval that extends to infinity in one or both directions, such as (a, ∞) or (-∞, b].
3. How is interval notation written and what do the different brackets mean?
Interval notation uses specific brackets to define the set. Square brackets [ ] are used to indicate that an endpoint is included in the interval (closed). Parentheses ( ) are used to indicate that an endpoint is excluded from the interval (open). For example, the interval [3, 7] means 'all numbers between 3 and 7, including 3 and 7', while (3, 7) means 'all numbers strictly between 3 and 7, excluding 3 and 7'.
4. How do you represent an interval on a number line?
To represent an interval on a number line, you mark the endpoints. A solid or filled circle (●) is used for an endpoint that is included (part of a closed interval, using [ or ]). An open or unfilled circle (○) is used for an endpoint that is excluded (part of an open interval, using ( or )). A line is then drawn between the two circles to show all the numbers contained within the interval.
5. What is the key difference between an open interval and a closed interval?
The key difference lies in the inclusion of the endpoints. A closed interval, written as [a, b], contains its endpoints. If a number x is in this interval, it satisfies the inequality a ≤ x ≤ b. In contrast, an open interval, written as (a, b), does not contain its endpoints. For a number x in this interval, it must satisfy the inequality a < x < b.
6. Why are intervals considered subsets of real numbers (R)?
Intervals are considered subsets of real numbers (R) because they describe a continuous section of the number line. The set of real numbers (R) represents the entire, unbroken number line, including all integers, fractions, and irrational numbers. An interval is simply a connected 'piece' or a segment of this complete line. Therefore, any interval, by its very definition, is a smaller collection of numbers taken from the larger set R, making it a subset of R.
7. In which mathematical topics are intervals most commonly used?
Intervals are a fundamental concept used across various mathematical topics, especially in higher secondary classes. Key applications include:
Linear Inequalities: Expressing the solution sets for inequalities. For instance, the solution to 2x - 4 > 6 is x > 5, which is written in interval notation as (5, ∞).
Functions: Defining the domain and range of functions, which specifies the set of all possible input and output values.
Calculus: Describing intervals where a function is increasing, decreasing, or constant, and for defining the convergence of series.
8. Is it possible for an interval to contain only a single number?
Yes, but only in a very specific case. A closed interval where the starting and ending endpoints are the same, such as [c, c], represents the set containing only the single number 'c'. This is because the condition is c ≤ x ≤ c, which is only true for x = c. However, an open interval with the same endpoints, like (c, c), represents an empty set because there are no numbers strictly between 'c' and 'c'.
