In this chapter, we will learn to find irrational numbers between two given fractions.
After completing this chapter, you will develop skill to produce irrational number of your choice at any given point.
Irrational number between two fractions
We know that irrational numbers are decimals that are non terminating and non repeating in nature.
We will use the above property to frame the required irrational numbers between the given fractions.
To locate the irrational number, follow the below steps;
(a) Find the value of each fraction in decimal.
(b) Now write the decimal numbers between the two given fraction.
Make sure that the decimal framed is non repeating and non terminating in nature.
In the above two steps you can find any given irrational numbers between the two points.
Given below are some examples for your further understanding.
Finding irrational numbers between fractions – solved examples
Example 01
Locate two irrational number between \mathtt{\frac{3}{4} \ \&\ \frac{7}{3}}
Solution
Follow the below steps;
(a) find the decimal value of each fraction.
\mathtt{\frac{3}{4} =\ 0.75}\\\ \\ \mathtt{\frac{7}{3} =\ 2.33}
So, the irrational number lies between 0.75 and 2.33.
(b) Frame decimal number between 0.75 and 2.33 which are non repeating and non terminating in nature.
Two required numbers are;
⟹ 0.863783121177 . . . . .
⟹ 1.973947612 . . . . .
Note:
You can frame infinite number of irrational number between any two fractions.
Example 02
Locate two irrational number between fraction 3/7 and 14/4
Solution
To locate the irrational number, follow the below steps;
(a) find the decimal value of each fraction
\mathtt{\frac{3}{7} =\ 0.42}\\\ \\ \mathtt{\frac{14}{4} =\ 3.5}
(b) write decimal number between 0.42 and 3.5 which is non repeating and non terminating.
Two irrational numbers are;
⟹ 0.9867239384 . . . . .
⟹ 2.77609142 . . . .
Example 03
Locate three irrational numbers between 5/7 and 9/11
Solution
Follow the below steps;
(a) Find the decimal value of number.
\mathtt{\frac{5}{7} =\ 0.714}\\\ \\ \mathtt{\frac{9}{11} =\ 0.82}
(b) Write non terminating and non repeating decimal between 0.714 and 0.82
Two required irrational numbers are;
\mathtt{\Longrightarrow \ 0.762519032\ .\ \ .\ \ .}\\\ \\ \mathtt{\Longrightarrow \ 0.801922239\ .\ \ .\ \ .}
Next chapter : Steps to rationalize irrational number