1.Class 9 Rational and Irrational Number:

Class 9 Rational and Irrational Number

In the article “Class 9 Rational and Irrational Number”,we shall study about rational and irrational numbers.We shall the representation of terminating and non-terminating decimal numbers on the number line.

Also Read This Article:- Class 9 Maths Polynomials:16 Important Descriptive Type Questions

2.Class 9 Rational and Irrational Number Illustrations:

Illustration:1.Classify the following numbers as rational or irrational numbers:
Illustration:1(a). \sqrt{7}
Solution: \sqrt{7}=2.64575\ldots
\sqrt{7} is irrational number because it is non-terminating non-recurring number
Illustration:1(b). \sqrt{196}
Solution: \sqrt{196}=14
\sqrt{196} is a perfect number,so it is rational number.
Illustration:1(c).0.3797
Solution: 0.3797=\frac{3797}{10000}
Which is in the form \frac{p}{q} and q \neq 0 .so it is rational.
Illustration:1(d).7.4784478…..
Solution:7.4784478…..
It is non-terminating recurring.So it is rational number.
Illustration:1(e).1.101001000100001……
Solution:1.101001000100001……
It is non-terminating non-recurring.So it is irrational number.
Illustration:2.write three numbers,whose decimal expressions are non-terminating non-recurring.
Solution:0.05005000500005….
0.06006000600006…….
0.07007000700007…..
Illustration:3.Write the decimal expressions of each of the following numbers and tell what kind of decimal expression each has:
Illustration:3(a). \frac{36}{100}
Solution: \frac{36}{100}=0.36
Teminating
Illustration:3(b). \frac{1}{11}
Solution: \frac{1}{11}

\frac{1}{11}=0.090909 \ldots is non-terminating repeating
Illustration:3(c). 4\frac{1}{8}
Solution: 4\frac{1}{8}=\frac{33}{8}

\frac{33}{8}=4.125
Terminating
Illustration:3(d). \frac{3}{13}
Solution: \frac{3}{13}

\frac{3}{13}=0.\overline{230769} \ldots Non-terminating repeating
Illustration:3(e). \frac{2}{11}
Solution: \frac{2}{11}

\frac{2}{11}=0.\overline{18} Non-terminating repeating
Illustration:3(f). \frac{329}{400}
Solution: \frac{329}{400}

\frac{329}{400}=0.8225 Terminating
Illustration:4.Find the three irrational numbers between \frac{5}{7} and \frac{9}{11}
Solution: \frac{5}{7}

\frac{5}{7}=0.\overline{714285}

\frac{9}{11}=0.\overline{81}
So three irrational numbers between \frac{5}{7} and \frac{9}{11}
0.73073007300073000073…….
0.74074007400074000074…..
0.75075007500075000075……

Class 9 Rational and Irrational Number

Illustration:5.Express the following numbers in the form of \frac{p}{q} ,where p and q are both integers and q \neq 0
Illustration:5(a). 0.\overline{3}
Solution: 0.\overline{3}
Let x=0.333 …       …. (1)
Multyplying both sides by 10,we get
10x=3.333…     …. (2)
Subtract Equation (1) from Equation (2),we get
9x=3\\ \Rightarrow x=\frac{3}{9}=\frac{1}{3}
So 0.\overline{3}=\frac{1}{3}
Illustration:5(b). 0.\overline{47}
Solution: 0.\overline{47}
Let x=0.474747…     …. (1)
There are two repeating digits after decimal,Multyplying on both sides of equation by 100,we get
100x=47.474747…   ….. (2)
Subtract Equation (1) from Equation (2),we get
\Rightarrow 99x=47 \\ x=\frac{47}{99}
Illustration:5(c). 1.\overline{27}
Solution: 1.\overline{27}
Let x=1.272727…     …. (1)
There are two repeating digits after decimal,Multyplying on both sides of equation (1) by 100,we get
100x=127.272727….     … (2)
Subtract Equation (1) from Equation (2),we get
99x=126\\ \Rightarrow x=\frac{126}{99}=\frac{14}{11}
Illustration:5(d). 1.2\overline{35}
Solution: 1.2\overline{35}
Let x=1.2353535…    …… (1)
There is one non-repeating digits after decimal so multyplying on both sides of equation by 10,we get
10x=12.353535…     ….. (2)
There are two repeating digits after decimal so multyplying on both sides of equation by 100,we get
1000x=1235.353535…    …… (3)
Subtract Equation (2) from Equation (3),we get
990x=1223\\ x=\frac{1223}{990}
Illustration:6.Express 0.2\overline{45} as a fraction in simplest form.
Solution: Let x=0.2\overline{45}=0.2454545\ldots (1)
There is one non-repeating digits after decimal so multyplying on both sides of equation by 10,we get
10x=2.454545\ldots \ldots (2)
As there are two repeating digits after decimal so multyplying on both sides of equation (2) by 100,we get
1000x=245.454545\ldots \ldots (3)
Subtract Equation (2) from Equation (3),we get
990x=243\\ x=\frac{243}{990}\\ \Rightarrow x=\frac{27}{110}
Illustration:7.Write the decimal expression of \frac{1}{7} ,Hence,write the decimal expressions of \frac{2}{7} , \frac{3}{7} and \frac{5}{7}
Solution:The decimal expression of the given number is
\frac{1}{7}=0.\overline{142857}\\ \frac{2}{7}=0.\overline{285714}\\ \frac{3}{7}=0.\overline{428571}\\ \frac{5}{7}=0.\overline{714285}
With the above illustrations,one can understand the Class 9 Rational and Irrational Number.

Class 9 Rational and Irrational Number

Also Read This Article:- Rational number

3.Class 9 Rational and Irrational Number Practice Problems for Students:

Class 9 Rational and Irrational Number

(1.)Show that 2.152786 is a rational number or express 2.1527886 in the form of \frac{p}{q}, where p and q both integers and q \neq 0
(2.)Show that 0.12 \overline{3} can be expressed in the form of \frac{p}{q} ,where p and q both integers and q \neq 0
Answers:(1.) 2.152786=\frac{2152786}{1000000} which is rational number
(2.) 0.12\overline{3}=\frac{37}{300}
By solving the above questions,you can understand the Class 9 Rational and Irrational Number well because the concept is well understood when you solve it practically.

### 📢 If you liked this math article:
* 👥 **Share with Friends:** Knowledge grows by sharing,so be sure to share it with your friends.
* 🔔 **Follow the Website:** If this is your first time here,follow our **email subscription** so that you get instant notifications for every new article.
* 💬 **Give Your Suggestions:** If you have any issues or would like to make any suggestions,do let us know by **commenting** below.
*Welcome to read the full article!*

Also Read This Article:- Class 9 Maths Number System:13 Important Questions (MCQ and Descriptive)

4.Frequently Asked Questions Related to Class 9 Rational and Irrational Number:

This article has been prepared by **Satyam Coaching Centre** on the **Satyam Mathematics** blog.”*
\begin{array}{|c|} \hline \text{**छात्र-छात्राओं से आज का सवाल**} \\ \text{एक पुस्तक का वजन आधा किलो और } \\ \text{ आधी पुस्तक के बराबर है,तो पुस्तक का वजन बताओ।} \\  \text{दिनांक 19.06.2021 के प्रश्न का उत्तर: } \\ \text{लम्बाई=4 इकाई,चौड़ाई=4 इकाई} \\ \text{Let Length=x , Breadth=y} \\ 2(x+y)=xy \\ \Rightarrow 2x+2y-xy=0 \\ \Rightarrow 2x-xy=-2y \\ \Rightarrow x(2-y)=-2y \\ \Rightarrow x(y-2)=2y \\ \Rightarrow x=\frac{2y}{y-2} \\ \text{अतः y का मान 2 से अधिक होगा} \\ \text{ क्योंकि भुजाएँ ऋणात्मक नहीं हो सकती।} \\ \text{Suppose y=4 then x=4} \\ \text{आप y का 2 से अधिक ओर भी मान ले सकते हैं।} \\ \text{**Today's Questions for Students**} \\ \text{If the weight of a book is equal to} \\ \text{ half a kilo and half a book,} \\ \text{ then tell the weight of the book. }\\ \text{Answer to the question dated 19.06.2021:} \\ \text{Length=4 units, Width=4 units} \\ \text{Hence, the value of y will be greater than} \\ \text{ 2 because the sides cannot be negative.} \\ \text{Suppose y=4 then x=4} \\  \text{You can also assume y to be greater than 2.} \\ \hline \end{array}