NCERT Solutions for Exercise 1.3 Class 9 Maths Chapter 1 - Number Systems

NCERT Solutions for Exercise 1.3 Class 9 Maths Chapter 1 - Number Systems

Edited By Vishal kumar | Updated on Sep 29, 2023 04:03 PM IST

NCERT Solutions for Class 9 Maths Chapter 1: Number Systems Exercise 1.3- Download Free PDF

NCERT Solutions for Class 9 Maths Chapter 1: Number Systems Exercise 1.3- Welcome to the updated 9th class maths exercise 1.3 answers created by subject experts of Careers360. NCERT Solutions for Class 9 Maths Exercise 1.3 offers the idea of real numbers and their decimal expansions. Numbers are divided into several categories in Mathematics, including rational numbers, irrational numbers, natural numbers, whole numbers, integers, and real numbers. Exercise 1.3 Class 9 Maths is one of the important exercises of the chapter. 1 Number System.

NCERT book Class 9 Maths chapter 1 exercise 1.3 consist of 9 questions, 7 of which are brief and 4 of which are lengthy solution kinds primarily based totally on rational and irrational numbers. In this Class 9 Maths chapter 1 exercise 1.3 the ideas linked to the number system are thoroughly discussed. The following activities are included along with Class 9 Maths Chapter 1 Exercise 1.3.

Q1 (i) Write the following in decimal form and say what kind of decimal expansion each has : (i) $\frac{36}{100}$

We can write $\frac{36}{100}$ as
$\Rightarrow \frac{36}{100}= 0.36$
Since the decimal expansion ends after a finite number of steps. Hence, it is terminating

We can rewrite $\frac{1}{11}$ as

$\Rightarrow \frac{1}{11}= 0.09090909..... = 0.\overline{09}$
Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.

We can rewrite $4\frac{1}{8}$ as

$\Rightarrow 4\frac{1}{8} = \frac{33}{8}= 4.125$
Since the decimal expansion ends after a finite number. Therefore, it is terminating

We can rewrite $\frac{3}{13}$ as

$\Rightarrow \frac{3}{13} = 0.230769230769 = 0.\overline{230769}$
Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.

We can rewrite $\frac{2}{11}$ as

$\Rightarrow \frac{2}{11} = 0.181818......= 0.\overline{18}$
Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.

We can rewrite $\frac{329}{400}$ as

$\Rightarrow \frac{329}{400}= 0.8225$
Since decimal expansion ends after finite no. of figures. Hence, it is terminating.

It is given that $\frac{1}{7}=0.\overline{142857}$

Therefore,

$\Rightarrow \frac{2}{7} = 2\times \frac{1}{7} = 2 \times 0.\overline{142857}= 0.\overline{285714}$

Similarly,

$\Rightarrow \frac{3}{7} = 3\times \frac{1}{7} = 3 \times 0.\overline{142857}= 0.\overline{428571}$

$\Rightarrow \frac{4}{7} = 4\times \frac{1}{7} = 4 \times 0.\overline{142857}= 0.\overline{571428}$

$\Rightarrow \frac{5}{7} = 5\times \frac{1}{7} = 5 \times 0.\overline{142857}= 0.\overline{714285}$

$\Rightarrow \frac{6}{7} = 6\times \frac{1}{7} = 6 \times 0.\overline{142857}= 0.\overline{857142}$

Let $x = 0.\overline6= 0.6666....$ -(i)

Now, multiply by 10 on both sides

$10x= 6.6666...$

$\Rightarrow 10x = 6 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (i))$

$\Rightarrow 9x = 6$

$\Rightarrow x = \frac{6}{9} = \frac{2}{3}$

Therefore, $\frac{p}{q}$ form of $0.\bar{6}$ is $\frac{2}{3}$

We can write $0.4\overline7$ as

$\Rightarrow 0.4\overline7 = \frac{4}{10}+ \frac{0.777..}{10}$ -(i)

Now,

Let $x = 0.\overline7= 0.7777....$ -(ii)

Now, multiply by 10 on both sides

$10x= 7.7777...$

$\Rightarrow 10x = 7 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (ii))$

$\Rightarrow 9x = 7$

$\Rightarrow x = \frac{7}{9}$
Now, put the value of x in equation (i). we will get

$\Rightarrow 0.4\overline7 = \frac{4}{10}+ \frac{7}{10\times 9}= \frac{4}{10}+ \frac{7}{90} = \frac{36+7}{90} = \frac{43}{90}$

Therefore, $\frac{p}{q}$ form of $0.4\overline7$ is $\frac{43}{90}$

Let $x = 0.\overline{001}= 0.001001....$ -(i)

Now, multiply by 1000 on both sides

$1000x= 1.001001...$

$\Rightarrow 1000x = 1 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (i))$

$\Rightarrow 999x = 1$

$\Rightarrow x = \frac{1}{999}$

Therefore, $\frac{p}{q}$ form of $0.\overline{001}$ is $\frac{1}{999}$

Let $x = 0.\overline{9}= 0.9999....$ -(i)

Now, multiply by 10 on both sides

$10x= 9.999....$

$\Rightarrow 10x = 9 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (i))$

$\Rightarrow 9x = 9$

$\Rightarrow x = \frac{9}{9} = 1$

Therefore, $\frac{p}{q}$ form of $0.999....$ is 1

The difference between 1 and 0.999999 is o.000001 which is almost negligible.

Therefore, 0.999 is too much closer to 1. Hence, we can write 0.999999.... as 1

We can rewrite $\frac{1}{17}$ as

$\Rightarrow \frac{1}{17} = 0.05882352941176470588235294117647= 0.\overline{0588235294117647}$
Therefore, there are total 16 number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$

We can observe that when q is 2, 4, 5, 8, 10… then the decimal expansion is terminating. For example:

$\frac{3}{2}= 1.5$ , denominator $q = 2^1$

$\frac{8}{5}= 1.6$ , denominator $q = 5^1$

$\frac{15}{10} = 1.5$ , denominator $q =10=2\times 5= 2^1 , 5^1$

Therefore,

It can be observed that the terminating decimal can be obtained in a condition where prime factorization of the denominator of the given fractions has the power of 2 only or 5 only or both.

Three numbers whose decimal expansions are non-terminating non-recurring are
1) 0.02002000200002......
2) 0.15115111511115.......
3) 0.27227222722227.......

We can write $\frac{5}{7}$ as

$\Rightarrow \frac{5}{7} = 0.714285714285.... = 0.\overline{714285}$

And $\frac{9}{11}$ as

$\Rightarrow \frac{9}{11} = 0.818181.... = 0.\overline{81}$
Therefore, three different irrational numbers between the rational numbers $\dpi{80} \frac{5}{7}$ and $\dpi{80} \frac{9}{11}$ are

1) 0.72737475....
2) 0.750760770780...
3) 0.790780770760....

We can rewrite $\sqrt{23}$ in decimal form as

$\Rightarrow \sqrt{23} = 4.7958152....$

Now, as the decimal expansion of this number is non-terminating non-recurring.

Therefore, it is an irrational number.

We can rewrite $\sqrt{225}$ as

$\Rightarrow \sqrt{225} = 15$
We can clearly see that it is a rational number because we can represent it in $\frac{p}{q}$ form

We can rewrite 0.3796 as

$\Rightarrow 0.3796 = \frac{3796}{10000}$
Now, we can clearly see that it is a rational number as the decimal expansion of this number is terminating and we can also write it in $\frac{p}{q}$ form.

We can rewrite 7.478478.... as

$\Rightarrow 7.478478.... = 7.\overline{478}$
Now, as the decimal expansion of this number is non-terminating recurring. Therefore, it is a rational number.

In the case of number 1.101001000100001...
As the decimal expansion of this number is non-terminating non-repeating. Therefore, it is an irrational number.

More About NCERT Solutions for Class 9 Maths Exercise 1.3

The topic of real numbers and their decimal expansions was the emphasis of the NCERT solutions for Class 9 Maths exercise 1.3. There are other questions based on decimal forms in Exercise 1.3 Class 9 Maths. Decimal numbers are fractional numbers written in a certain format. The decimal point is the dot that appears between the entire number and fraction parts.

Terminating decimal numbers are decimal numbers that end after a certain number of decimal places. With repeating decimal patterns, rational numbers can be both terminating and non-terminating decimals. Irrational numbers should never have terminating decimals and should instead have non-terminating decimals with no repeating decimal patterns.

Also Read| Number Systems Class 9 Notes

Benefits of NCERT Solutions for Class 9 Maths Exercise 1.3:

• NCERT syllabus Class 9 Maths Exercise 1.3 helps us to understand the basics of the number system in-depth, which is beneficial for us in understanding higher math as well.

• By solving the NCERT solution for Class 9 Maths Chapter 1 exercise 1.3 exercises, we can also study the representation of real numbers on the number line with the help of the decimal expansions.

• Exercise 1.3 Class 9 Maths is well-versed with the basic facts of decimals and their representation on the number line which helps us to solve the exercises smoothly.

Key Features of Class 9 Maths Chapter 1 Exercise 1.3

1. Easy-to-Understand: The 9th class maths exercise 1.3 answers are presented in a straightforward and easily comprehensible manner, making it accessible to students of various levels.

2. Free Access: These class 9 maths chapter 1 exercise 1.3 solution are freely accessible to all students, ensuring that cost is not a barrier to learning.

3. Clear Explanations: Each class 9 maths ex 1.3 solution includes clear explanations to help students understand the concepts and problem-solving techniques.

4. Step-by-Step Format: The ex 1.3 class 9 solutions are structured in a step-by-step format, guiding students through the problem-solving process.

5. Syllabus Alignment: The class 9 ex 1.3 solutions align with the prescribed syllabus, covering all relevant topics and concepts.

Also see-

Subject Wise NCERT Exemplar Solutions

1. What are irrational numbers, according to NCERT solutions for Class 9 Maths chapter 1 exercise 1.3?

Irrational numbers are numbers that cannot be stated in the form of a fraction with an integer in both the numerator and the denominator.

2. Is root (49) a rational number?

root(49)=7 which can be written as 7/1

Thus 49 is a rational number.

3. Is 0 a rational number?

Yes 0 is a rational number (since 0 can be written as 0/1 , 0/2 etc… )

4. What are the different types of decimals, According to NCERT solutions for Class 9 Maths chapter 1 exercise 1.3 ?

Decimal numbers are classified into

• Recurring Decimal Numbers (repeating or Non-Terminating Decimals)

• Non-Recurring Decimal Numbers (non Repeating or Terminating Decimals).

5. Write the expanded form of 64.3?

The expanded form of 64.3 is 60+4+3/10.

6. Is 0.033 a non-recurring decimal number?

Yes, 0.033 is a non-recurring decimal number. Since 0.033 is a non Repeating and Terminating number.

7. The dot which is present in between the whole number and fractions part is known as ________.

The dot which is present in between the whole number and fractions part is known as the decimal point.

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