Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental topic that learners are required learn owing to the fact that it becomes more critical as you grow to more difficult arithmetic.
If you see advances mathematics, something like differential calculus and integral, in front of you, then being knowledgeable of interval notation can save you time in understanding these theories.
This article will talk about what interval notation is, what it’s used for, and how you can decipher it.
What Is Interval Notation?
The interval notation is simply a method to express a subset of all real numbers along the number line.
An interval means the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)
Basic problems you encounter mainly composed of one positive or negative numbers, so it can be challenging to see the utility of the interval notation from such simple utilization.
Though, intervals are usually used to denote domains and ranges of functions in advanced mathematics. Expressing these intervals can progressively become difficult as the functions become progressively more tricky.
Let’s take a simple compound inequality notation as an example.
x is greater than negative four but less than 2
Up till now we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. However, it can also be denoted with interval notation (-4, 2), denoted by values a and b separated by a comma.
So far we understand, interval notation is a method of writing intervals concisely and elegantly, using set principles that make writing and understanding intervals on the number line less difficult.
In the following section we will discuss about the rules of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Several types of intervals place the base for denoting the interval notation. These interval types are important to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are used when the expression does not contain the endpoints of the interval. The prior notation is a fine example of this.
The inequality notation {x | -4 < x < 2} describes x as being more than -4 but less than 2, meaning that it does not include either of the two numbers mentioned. As such, this is an open interval denoted with parentheses or a round bracket, such as the following.
(-4, 2)
This implies that in a given set of real numbers, such as the interval between -4 and 2, those two values are not included.
On the number line, an unshaded circle denotes an open value.
Closed
A closed interval is the opposite of the previous type of interval. Where the open interval does exclude the values mentioned, a closed interval does. In text form, a closed interval is expressed as any value “higher than or equal to” or “less than or equal to.”
For example, if the previous example was a closed interval, it would read, “x is greater than or equal to negative four and less than or equal to 2.”
In an inequality notation, this would be expressed as {x | -4 < x < 2}.
In an interval notation, this is written with brackets, or [-4, 2]. This states that the interval contains those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized to denote an included open value.
Half-Open
A half-open interval is a combination of previous types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the last example as a guide, if the interval were half-open, it would read as “x is greater than or equal to negative four and less than 2.” This implies that x could be the value -4 but couldn’t possibly be equal to the value 2.
In an inequality notation, this would be expressed as {x | -4 < x < 2}.
A half-open interval notation is written with both a bracket and a parenthesis, or [-4, 2).
On the number line, the shaded circle denotes the number included in the interval, and the unshaded circle indicates the value which are not included from the subset.
Symbols for Interval Notation and Types of Intervals
To recap, there are different types of interval notations; open, closed, and half-open. An open interval doesn’t contain the endpoints on the real number line, while a closed interval does. A half-open interval includes one value on the line but does not include the other value.
As seen in the last example, there are numerous symbols for these types under the interval notation.
These symbols build the actual interval notation you develop when stating points on a number line.
( ): The parentheses are employed when the interval is open, or when the two endpoints on the number line are excluded from the subset.
[ ]: The square brackets are employed when the interval is closed, or when the two points on the number line are included in the subset of real numbers.
( ]: Both the parenthesis and the square bracket are employed when the interval is half-open, or when only the left endpoint is excluded in the set, and the right endpoint is included. Also called a left open interval.
[ ): This is also a half-open notation when there are both included and excluded values between the two. In this case, the left endpoint is included in the set, while the right endpoint is not included. This is also known as a right-open interval.
Number Line Representations for the Various Interval Types
Apart from being denoted with symbols, the various interval types can also be described in the number line using both shaded and open circles, relying on the interval type.
The table below will show all the different types of intervals as they are described in the number line.
Practice Examples for Interval Notation
Now that you’ve understood everything you need to know about writing things in interval notations, you’re prepared for a few practice problems and their accompanying solution set.
Example 1
Transform the following inequality into an interval notation: {x | -6 < x < 9}
This sample problem is a straightforward conversion; simply use the equivalent symbols when stating the inequality into an interval notation.
In this inequality, the a-value (-6) is an open interval, while the b value (9) is a closed one. Thus, it’s going to be expressed as (-6, 9].
Example 2
For a school to take part in a debate competition, they require at least three teams. Represent this equation in interval notation.
In this word question, let x stand for the minimum number of teams.
Because the number of teams required is “three and above,” the number 3 is consisted in the set, which means that three is a closed value.
Additionally, since no maximum number was stated with concern to the number of maximum teams a school can send to the debate competition, this value should be positive to infinity.
Thus, the interval notation should be denoted as [3, ∞).
These types of intervals, where there is one side of the interval that stretches to either positive or negative infinity, are called unbounded intervals.
Example 3
A friend wants to undertake a diet program constraining their daily calorie intake. For the diet to be successful, they must have minimum of 1800 calories regularly, but maximum intake restricted to 2000. How do you describe this range in interval notation?
In this word problem, the value 1800 is the minimum while the number 2000 is the maximum value.
The problem suggest that both 1800 and 2000 are included in the range, so the equation is a close interval, expressed with the inequality 1800 ≤ x ≤ 2000.
Therefore, the interval notation is written as [1800, 2000].
When the subset of real numbers is confined to a range between two values, and doesn’t stretch to either positive or negative infinity, it is called a bounded interval.
Interval Notation FAQs
How To Graph an Interval Notation?
An interval notation is fundamentally a way of representing inequalities on the number line.
There are laws of expressing an interval notation to the number line: a closed interval is expressed with a shaded circle, and an open integral is expressed with an unfilled circle. This way, you can quickly check the number line if the point is excluded or included from the interval.
How To Convert Inequality to Interval Notation?
An interval notation is basically a different technique of expressing an inequality or a combination of real numbers.
If x is higher than or lower than a value (not equal to), then the value should be written with parentheses () in the notation.
If x is higher than or equal to, or less than or equal to, then the interval is denoted with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are used.
How To Exclude Numbers in Interval Notation?
Values excluded from the interval can be written with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which means that the number is ruled out from the set.
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