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Intervals

What is an Interval?

In English, we define an interval as a period between events. But do we have the same meaning of intervals in maths as well? Let us find out.

We know what we mean by intervals in the English language. But, in mathematics, it is not the same meaning. In maths, an interval means a group of numbers that falls in a certain range. In other words, an interval is a range of numbers between two given numbers and includes all of the real numbers between those two numbers.  For example, let us consider all the whole numbers less than 10. We know that the whole numbers start from 0 with the next number being 1, 2, 3 and so on. So, the intervals where a variable x is less than 10 will be the numbers from 1 to 9. Note here that we have not considered the number 10. This is because according to the condition of the interval, the whole numbers which are less than 10 are to be considered. The “less than “condition here means that the number 10 is not included in the interval. This also makes us point out that there may be more than one type of interval depending upon the conditions of the group of numbers that are to be included in the interval. 

How to Represent Intervals

There are many ways to represent an interval, such as 

  • Using notations
  • Number Line
  • Inequalities

Let us learn about them and their use in representing intervals.

Representation of Intervals on a Number Line

If we consider the above condition where we have whole numbers less than 10, how do we represent them on a number line? Let us check.

Let us consider the above number line. Now we need to plot the whole numbers less than 10 on this number line. This set of numbers, say X will be equal to – 

X = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Let us represent this set on the number line.

The range from 0 to 9 has been represented in the above number line. But, is this the only way of representing an interval or we would have done it in any other way? Let us find out.

Before learning about different representations on a number line, we must first learn how to write them in mathematical form. 

Therefore, let us learn how to write an interval in notation form. 

Notation of an Interval

Now, that we have learnt, what we mean by an interval, we also understand that it is not always possible to give a detailed description of the values that are contained in an interval. So, there has to be a way to write the interval in mathematical form. This mathematical form of describing an interval is called its notation.  We will discuss more about it when we learn about different types of intervals. 

Types of Intervals

How many types of intervals are there? Intervals are of three types, namely, closed intervals, open intervals and semi open or semi closed intervals. Let us discuss about them. But, before discussing the types of intervals, it is important to learn about an important term – the endpoint.

In intervals, an endpoint is a value that is either the start value or the end value of the interval.  For example, in the example, we have been discussing above the values 0 and 9 are the end points of the given interval as 0 is the starting point while 9 is the end point of the interval X. But, could there be any other end point of this interval?

What if we say that the end points of this interval X were -1 and 10. Now, -1 is not a whole number and 10 was not considered in the condition of the interval. So, how can these two values be the end points of this interval? This leads us to defining the different types of intervals that are used to denote the intervals. 

Closed Interval

As the name suggests, a closed interval is a closed group where the end points are included in the interval. Therefore, if we consider the above example, it would be a closed interval if we consider its end points as 0 and 9. So, we can say that, the interval X = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } or X = { 0, 1 …………, 9 }. How do you represent it mathematically? Let us find out

Notation of a Closed Interval

The values in a closed interval are presented between square brackets, [ ]. This means that a square bracket is used to represent an interval when both the endpoints are included in the set. For instance, in the above example where we have an interval X that contains, whole numbers less than 10, we can represent it as –

X = [0, 9]

The expansion of this interval will be X = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Now, let us see how to represent a closed interval on a number line. 

Representation of a Closed Interval on a Number Line

We know how to represent numbers on a number line. How, do we represent the interval of a specific group of numbers?

Let us again consider the same example, where we have X = {  0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } and plot this interval on the number line. We will get,

Note the closed circles at the end points of the interval. This means that the said values are included in the interval and hence is a closed interval.

Open Interval

As the name suggests, a closed interval is an open group where the end points are not included in the interval. Therefore, if we consider the above example, it would be an open interval if we consider its end points as -1 and 10. How do you represent it mathematically? Let us find out

Notation of an Open Interval

The values in a closed interval are presented between square brackets, ( ). This means that a round bracket is used to represent an interval when both the endpoints are not included in the set. For instance, in the above example where we have an interval X that contains, whole numbers less than 10, we can represent it as –

X = ( -1, 10 ), where x $\in$ W

The above representation means that X includes all numbers between -1 and 10 but not including these two end points.

The expansion of this interval will still be X = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Now, let us see how to represent an open interval on a number line. 

Representation of an Open Interval on a Number Line

Let us again consider the same example, where we have X = {  0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } and plot this as an open interval on the number line. We will get,

Note, the open circles at the end points -1 and 10. This represents that -1 and 10 are excluded from the interval but all numbers between -1 and 10 are included. Now, since the there is no whole number between -1 and 0 and between 9 and 10, the first and the last numbers to be included will be 0 and 10. So the end points -1 and 10 have been shown on the number line but in an open circle form indicating their exclusion from the interval. Therefore, this is how we represent an open interval on the number line.

Semi Open or Semi Closed Interval

 As the name suggests, a semi open or a semi closed interval is a semi open or a semi closed group where one of the end points has been included in the interval while the other end point has been excluded from the intervals. Therefore, if we consider the above example, how can it be presented an a semi open or a semi closed interval? It would be a semi open or a semi closed interval if we consider its end points as -1 and 9, or 0 and 10. How do you represent it mathematically? Let us find out

Notation of a Semi-Open or a Semi Closed Interval

The values in a semi open or a semi closed interval are presented between a combination of a square and closed [ ). This means that a round bracket that is used to represent the end point that has been excluded while a square bracket is sued to represent the interval that has been included.  For instance, in the above example where we have an interval X that contains, whole numbers less than 10, we can represent it as –

X = [0, 10), where x $\in$ W or

X = (-1, 9]

The expansion of this interval will still be X = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }

Now, let us see how to represent an open interval on a number line. 

Representation of a Semi-Open or a Semi Closed Interval on a Number Line

Let us again consider the same example, where we have X = {  0, 1, 2, 3, 4, 5, 6, 7, 8, 9 } and plot this as a semi-open or a semi closed Interval on the number line. We will get,

It can also be represented as 

Note the open circles at one of the end points in the numbers lines above that denote the exclusion of that number from the interval. 

Let us understand the above representation through an example.

Example Use interval notation to represent the interval notation shown on the number line below.

Solution We can see that the interval includes values between -6 and 3, but does not include 3. Therefore, the correct interval notation is [-6,3)

Intervals and Inequalities

Intervals can be written using inequalities as well. Let us recall that math inequalities are the symbols that stand for less than, less than or equal to, greater than, and greater than or equal to. So we have

< symbol for less than

> symbol for greater than

symbol for greater than or equal to

symbol for less than or equal to

How can the above symbols be sued to represent inequalities as intervals? Let us find out.

Let us now consider an algebraic inequality, say, x   2. This inequality is read as “ x is greater than or equal to 2”. This inequality will have many solutions for x. some of the solutions of this inequality will be x = 3, x = 3.5, x = 3.8 and so on. Since it is impossible to list all of the solutions, a system is needed that allows a clear communication of this infinite set. This is can be done using graphical representation or using intervals.  Let us plot this inequality on a graph, we will get,

Now, if we wish to represent this inequality as an interval it will be represented as [ 2, ]. 

Following steps should be considered while plotting an inequality on a number line or determining its interval – 

  1. Determine the interval notation after graphing the solution set on a number line. 
  2. The numbers in interval notation should be written in the same order as they appear on the number line, with smaller numbers in the set appearing first. 
  3. If there is any inclusive inequality, which means that one of the end points is included in the interval, you denote with a square bracket. 
  4. If there is any non-inclusive inequality, which means that one of the end points is not included in the interval, you denote with a round bracket. 
  5. The symbol ( ) is read as infinity and indicates that the set is unbounded to the right on a number line. Interval notation requires a parenthesis to enclose infinity. 
  6.  Infinity is an upper bound to the real numbers but is not itself a real number: it cannot be included in the solution set.

Let us learn about bounded and unbounded interval in inequalities.

Bounded and Unbounded Intervals

An interval is said to be bounded if both of its endpoints are real numbers. Bounded intervals are also commonly known as finite intervals. Conversely, if neither endpoint is a real number, the interval is said to be unbounded. For example, the interval, (2 , 11 ) is a bounded interval while ( – , ) is an unbounded interval.

Solved Examples

 Example 1 Write x 5 as an interval.

Solution We have been given the inequality x 5. Since no specific set for x has been defined we shall consider it as a set of real numbers.  Now we know that there are infinite real numbers greater than or equal to 5. This means there would be no end to this set and hence it would be an open interval towards the end. Also, since we have that x 5as well, this means that we have a start of the set in the form of x = 5. Therefore at the start, this interval will be a closed one.

We will, therefore, write this interval as x = [5, )

Example 2 Write x <  5 and x > 2 as an interval

Solution We have bene given the inequality, x <  5 and x > 2. . Since no specific set for x has been defined we shall consider it as set of real numbers.  Now we know that there are infinite real numbers less than 5. This means there would be no number that can be considered as the end of this set and hence it would be an open interval towards the end. Also, we have been given that x > 2. Again, there are infinitely many numbers that will be greater than 2. But the number has to be less than 5. Therefore, the given set lies between 2 and 5 while excluding these two numbers. Hence, we will have an open set as x = 2 < x < 5 which in notation form will be written as x = (2, 5 ).

Key Facts and Summary

  1. An interval is a range of numbers between two given numbers and includes all of the real numbers between those two numbers. 
  2. In intervals, an end point is the value that is either the start value or the end value of the interval.  
  3. The mathematical form of describing an interval is called its notation.  
  4. A square bracket [ ] is used to represent an interval when both the endpoints are included in the set.
  5. A round bracket ( ) that is used to represent an interval when both the endpoints are not included in the set.
  6. The values in a semi open or a semi closed interval are presented between a combination of a square and closed [ ).
  7. An interval is said to be bounded if both of its endpoints are real numbers. Bounded intervals are also commonly known as finite intervals.
  8. If neither endpoint is a real number, the interval is said to be unbounded.
  9. Inequalities usually have infinitely many solutions, so rather than presenting an impossibly large list, we present such solutions sets either graphically on a number line or textually using interval notation.

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