In this chapter, we will learn to locate irrational numbers between two given rational numbers with solved examples.
At the end of the chapter, some problems are also given for your practice.
After completing this chapter, you can locate the irrational number between two integers in square root form.
Identify irrational numbers between two rational numbers
Locating irrational number between two given numbers is difficult as irrational numbers are present in square root or cube root form.
Let ” a ” and ” b” are two given integer numbers.
To identify the irrational numbers in between follow the below steps;
( i ) Square both the given integers.
After squaring we get \mathtt{a^{2} \ \&\ b^{2}}
(ii) Now locate the number between \mathtt{a^{2} \ \&\ b^{2}} which is not a perfect square.
Let ” c ” be the non perfect square number between \mathtt{a^{2} \ \&\ b^{2}} .
(iii) Take the square root of ” c “ and you will get required irrational number.
Note:
The square root of all non – perfect square number between \mathtt{a^{2} \ \&\ b^{2}} can be the solution of this problem.
I hope you understood the above process. Let us solve some examples for your better understanding.
Locating irrational numbers – Solved examples
Example 01
Find irrational numbers between number 4 and 5
Solution
To locate irrational numbers, follow below steps;
(a) Take square of both the numbers.
\mathtt{4^{2} \Longrightarrow 16\ }\\\ \\ \mathtt{5^{2} \Longrightarrow \ 25}
(b) Find all the non perfect square numbers between 16 and 25.
Required numbers are 17, 18, 19, 20, 21, 22, 23 and 24.
(c) Taking square root of all the numbers will get you the irrational numbers between 4 and 5.
Hence, \mathtt{\sqrt{17} ,\ \sqrt{18} ,\ \sqrt{19} ,\ \sqrt{20} ,\ \sqrt{21} \ ,\ \sqrt{22} ,\ \sqrt{23} \ \&\sqrt{24}} are the required irrational numbers.
Validation;
Since we are looking irrational number between 4 and 5, the decimal value of all the above irrational numbers will be between 4 and 5.
Observing the decimal values of irrational numbers.
\mathtt{\sqrt{17} \ =4.123}\\\ \\ \mathtt{\sqrt{18} \ =\ 4.243}\\\ \\ \mathtt{\sqrt{19} \ =\ 4.359}\\\ \\ \mathtt{\sqrt{20} \ =\ 4.472}\\\ \\ \mathtt{\ \sqrt{21} \ \ =\ 4.583}\\\ \\ \mathtt{\ \sqrt{22} \ =4.69}\\\ \\ \mathtt{\sqrt{23} \ =\ 4.796}\\\ \\ \mathtt{\sqrt{24} \ =4.899}
Note that all the value of irrational number lies between 4 and 5. Hence, the above process is correct.
Example 02
Locate the irrational numbers in square root form between 2 and 3.
Solution
To find the irrational numbers follow the below steps;
(a) Square the given numbers
\mathtt{2^{2} \Longrightarrow 4\ }\\\ \\ \mathtt{3^{2} \Longrightarrow \ 9}
(b) Write all the numbers between 4 and 9 except the perfect square.
Number 5, 6, 7 and 8 are the required numbers.
(c) Now take square root of all the numbers and you will get the irrational numbers.
Hence, \mathtt{\sqrt{5} ,\ \sqrt{6} ,\ \sqrt{7} \ \&\ \sqrt{8}} are the required irrational number that lies between 2 and 3.
Example 03
Find irrational number between 1 and 2.
Solution
Follow the below steps;
(a) Square the given numbers
\mathtt{1^{2} \Longrightarrow 1}\\\ \\ \mathtt{2^{2} \Longrightarrow \ 4}
(b) Find numbers between 1 & 4 that are not perfect squares.
Here 2 and 3 are the required numbers.
(c) Now take the square root of these numbers.
Hence, \mathtt{\sqrt{2} \ \ \&\ \sqrt{3}} are the irrational numbers that lie between 1 and 2.
Next chapter : Finding irrational number between two fractions