Mathematics is a challenging subject that comprises Arithmetic, Algebra, Trigonometry, Geometry, and more. It is an area for integrating quantities, spaces, and structures that help in calculating the basic aspects of an arrangement. Today, let’s look at the basic explanation of the term interval and its notation. Learning is such a great part of life!
Defining an interval
An interval is a group of real numbers denoted by two numbers, which are the endpoints of an interval. It is generally used as two numbers that lie at the endpoints of the set constructed. Furthermore, it is dependent on the interval notation as to whether the set ought to contain the numbers or not.
Understanding interval notation is often easier with the help of an example. Suppose there is a set of numbers x which satisfies -2≤a≤2. This is an interval that consists of the numbers between -2 and 2. Sometimes, within the interval, the numbers are not used among the endpoints and can be figured out as -2<a<2.
It generally signifies that the interval contains the real numbers between -2 and 2, but not the specific numbers -2 and 2. Intervals are quite often used in mathematics to generally explain the measure of the number lying between the sets.
What is interval notation and its attributes?
An interval can be denoted in many ways, depending on the properties of the two endpoints (x,y). So, they are known as interval notation.
Thus the interval between the numbers x and y can be denoted as [x,y], which includes both x and y. Similarly, the interval between the numbers x and y can also be denoted as [x,y] which excludes both x and y. Eventually, semicolons are also used instead of commas in many different mathematical aspects.
Intervals are generally put into three main categories:
- Open Interval
- Half-Open Interval
- Closed Interval
Let’s look at each of them, next.
1. What is an open interval?
An open interval can be defined as a set of numbers within the endpoint x and y, but it does not comprise the endpoints within the interval. This depicts that the interval [x,y] is created using all the numbers within the interval between x and y.
Thus, you can use an equation like x<a<y. Often, writing an interval for interval notation involves using brackets like (x,y). This topic is one of those that you might learn if you go into a career with numbers.
2. What is a half-open interval?
A half-open interval can also be called a semi-open interval or a semi-closed interval. Comprising endpoints x and y, a half-open interval can include only one of them within the interval. That means there may be a semi-open interval to the left or a semi-open interval to the right.
With endpoints x and y, a half-open interval to the left can be denoted as [x,y] where the numbers are greater than x and less or equal to y. So, you can form that “a” belongs to an interval only if x<a≤y.
Similarly, with endpoints x and y, a half-open interval to the right can be denoted as [x,y] where the numbers are greater than or equal to x but less than y. Thus, you can write that “a” belongs to the interval only if x≤a,y.
The use of round brackets and square brackets are common when writing the interval in interval notation. A square bracket signifies the inclusion of the endpoint (x,y], and a round bracket signifies exclusion of the endpoint [x,y).
3. What is a closed interval?
A closed interval is an interval that includes both the endpoints x and y. Thus, the interval [x,y] consists of all the numbers between x and y. Hence, you can use the equation that “a” belongs to the interval only if x≤a≤y. Generally, you will see the use of square brackets to write the interval within interval notation.
For further assistance over interval notations, visit Cuemath for the best online math practices. Cuemath is a leading online platform for math classes across many countries.