NCERT SOLUTIONS FOR CLASS 9 MATHS CHAPTER 1 NUMBER SYSTEMS

The National Council of Educational Research and Training (NCERT) is an autonomous body of the Indian government that formulates the curricula for schools in India that are governed by the Central Board of Secondary Education (CBSE) and certain state boards. Therefore, students who will be taking the Class 10 tests administered by various boards should consult this NCERT Syllabus in order to prepare for those examinations, which in turn will assist those students get a passing score.

When working through the exercises in the NCERT textbook, if you run into any type of difficulty or uncertainty, you may use the swc NCERT Solutions for class 9 as a point of reference. While you are reading the theory form textbook, it is imperative that you always have notes prepared. You should make an effort to understand things from the very beginning so that you may create a solid foundation in the topic. Use the NCERT as your parent book to ensure that you have a strong foundation. After you have finished reading the theoretical section of the textbook, you should go to additional reference books.

NCERT Solutions for Class 9 Maths Exercise 1.1

Question 1. Is zero a rational number? Can you write it in the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, where p and q are integers andNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png?
Solution : Consider the definition of a rational number.

A rational number is the one that can be written in the form ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, where p and q are integers andNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png.

Zero can be written asNCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png.

So, we arrive at the conclusion that 0 can be written in the form ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, where q is any integer.

Therefore, zero is a rational number.

Question 2. Find six rational numbers between 3 and 4.
Solution : We know that there are infinite rational numbers between any two numbers.

A rational number is the one that can be written in the form ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, where p and q are

Integers and NCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png.

We know that the numbers NCERT solution for class 9 Maths Chapter-1 Number Systems/image004.pngall lie between 3 and 4.

We need to rewrite the numbersNCERT solution for class 9 Maths Chapter-1 Number Systems/image004.pngin NCERT solution for class 9 Maths Chapter-1 Number Systems/image001.pngform to get the rational numbers between 3 and 4.

So, after converting, we getNCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png.

We can further convert the rational numbers NCERT solution for class 9 Maths Chapter-1 Number Systems/image006.pnginto lowest fractions.

On converting the fractions into lowest fractions, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image007.png.

Therefore, six rational numbers between 3 and 4 are

NCERT solution for class 9 Maths Chapter-1 Number Systems/image008.png.

Question 3. Find five rational numbers betweenNCERT solution for class 9 Maths Chapter-1 Number Systems/image009.png.
Solution : We know that there are infinite rational numbers between any two numbers.

A rational number is the one that can be written in the form ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, where p and q are

Integers andNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png.

We know that the numbers NCERT solution for class 9 Maths Chapter-1 Number Systems/image011.pngcan also be written asNCERT solution for class 9 Maths Chapter-1 Number Systems/image010.png.

We can conclude that the numbersNCERT solution for class 9 Maths Chapter-1 Number Systems/image011.pngall lie betweenNCERT solution for class 9 Maths Chapter-1 Number Systems/image010.png

We need to rewrite the numbersNCERT solution for class 9 Maths Chapter-1 Number Systems/image011.pngin NCERT solution for class 9 Maths Chapter-1 Number Systems/image001.pngform to get the rational numbers between 3 and 4.

So, after converting, we getNCERT solution for class 9 Maths Chapter-1 Number Systems/image012.png.

We can further convert the rational numbers NCERT solution for class 9 Maths Chapter-1 Number Systems/image013.pnginto lowest fractions.

On converting the fractions, we getNCERT solution for class 9 Maths Chapter-1 Number Systems/image014.png.

Therefore, six rational numbers between 3 and 4 areNCERT solution for class 9 Maths Chapter-1 Number Systems/image015.png.

Question 4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
Solution :
(i) Consider the whole numbers and natural numbers separately.

We know that whole number series isNCERT solution for class 9 Maths Chapter-1 Number Systems/image016.png.

We know that natural number series isNCERT solution for class 9 Maths Chapter-1 Number Systems/image017.png.

So, we can conclude that every number of the natural number series lie in the whole number series.

Therefore, we conclude that, yes every natural number is a whole number.

(ii) Consider the integers and whole numbers separately.

We know that integers are those numbers that can be written in the form ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, whereNCERT solution for class 9 Maths Chapter-1 Number Systems/image018.png.

Now, considering the series of integers, we haveNCERT solution for class 9 Maths Chapter-1 Number Systems/image019.png.

We know that whole number series isNCERT solution for class 9 Maths Chapter-1 Number Systems/image016.png.

We can conclude that all the numbers of whole number series lie in the series of integers. But every number of series of integers does not appear in the whole number series.

Therefore, we conclude that every integer is not a whole number.

(iii) Consider the rational numbers and whole numbers separately.

We know that rational numbers are the numbers that can be written in the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, whereNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png.

We know that whole number series isNCERT solution for class 9 Maths Chapter-1 Number Systems/image016.png.

We know that every number of whole number series can be written in the form of NCERT solution for class 9 Maths Chapter-1 Number Systems/image001.pngasNCERT solution for class 9 Maths Chapter-1 Number Systems/image020.png.

We conclude that every number of the whole number series is a rational number. But, every rational number does not appear in the whole number series.

Therefore, we conclude that every rational number is not a whole number.

NCERT Solutions for Class 9 Maths Exercise 1.2

Question 1. State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is of the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, where m is a natural number.
(iii) Every real number is an irrational number.
Solution :
(i) Consider the irrational numbers and the real numbers separately.

We know that irrational numbers are the numbers that cannot be converted in the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png, where p and q are integers andNCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png.

We know that a real number is the collection of rational numbers and irrational numbers.

Therefore, we conclude that, yes every irrational number is a real number.

(ii) Consider a number line. We know that on a number line, we can represent negative as well as positive numbers.

We know that we cannot get a negative number after taking square root of any number.

Therefore, we conclude that not every number point on the number line is of the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png, where m is a natural number.

(iii) Consider the irrational numbers and the real numbers separately.

We know that irrational numbers are the numbers that cannot be converted in the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png, where p and q are integers andNCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png.

We know that a real number is the collection of rational numbers and irrational numbers.

So, we can conclude that every irrational number is a real number. But every real number is not an irrational number.

Therefore, we conclude that, every real number is not a rational number.

Question 2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Solution : We know that square root of every positive integer will not yield an integer.

We know thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image004.png is 2, which is an integer. But,NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.pngwill give an irrational number.

Therefore, we conclude that square root of every positive integer is not an irrational number.

Question 3. Show how √5 can be represented on the number line
Solution :
Draw a number line and take point O and A on it such that OA = 1 unit. Draw BA ⊥ OA as BA = 1 unit. Join OB = √2 units.
Now draw BB1 ⊥ OB such that BB1 =1 unit. Join OB1 = √3 units.
Next, draw B1B2⊥ OB1such that B1B2 = 1 unit.
Join OB2 = units.
Again draw B2B3 ⊥OB2 such that B2B3 = 1 unit.
Join OB3 = √5 units.
NCERT Solutions for Class 9 Maths Chapter 1 Number Systems
Take O as centre and OB3 as radius, draw an arc which cuts the number line at D.
Point D
represents √5 on the number line.

NCERT Solutions for Class 9 Maths Exercise 1.3

Question 1. Write the following in decimal form and say what kind of decimal expansion each has: NCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png

(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image004.png

(iv) NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

(v) NCERT solution for class 9 Maths Chapter-1 Number Systems/image006.png

(vi) NCERT solution for class 9 Maths Chapter-1 Number Systems/image007.png

Solution :
(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png

On dividing 36 by 100, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image008.png

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image009.png, which is a terminating decimal.

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png

On dividing 1 by 11, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image010.png

We can observe that while dividing 1 by 11, we got the remainder as 1, which will continue to be 1.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image014.png, which is a non-terminating decimal and recurring decimal.

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image004.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image012.png

On dividing 33 by 8, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image013.png

We can observe that while dividing 33 by 8, we got the remainder as 0.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image014.png, which is a terminating decimal.

(iv) NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

On dividing 3 by 13, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image015.png

We can observe that while dividing 3 by 13 we got the remainder as 3, which will continue to be 3 after carrying out 6 continuous divisions.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image016.png, which is a non-terminating decimal and recurring decimal.

(v) NCERT solution for class 9 Maths Chapter-1 Number Systems/image006.png

On dividing 2 by 11, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image017.png

We can observe that while dividing 2 by 11, first we got the remainder as 2 and then 9, which will continue to be 2 and 9 alternately.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image018.png, which is a non-terminating decimal and recurring decimal.

(vi) NCERT solution for class 9 Maths Chapter-1 Number Systems/image007.png

On dividing 329 by 400, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image019.png

We can observe that while dividing 329 by 400, we got the remainder as 0.

Therefore, we conclude that NCERT solution for class 9 Maths Chapter-1 Number Systems/image020.png, which is a terminating decimal.

Question 2.
You know that NCERT solution for class 9 Maths Chapter-1 Number Systems/image021.png. Can you predict what the decimal expansions ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image022.pngare, without actually doing the long division? If so, how?

[Hint: Study the remainders while finding the value of NCERT solution for class 9 Maths Chapter-1 Number Systems/image023.pngcarefully.]

Solution :
We are given thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image024.png.

We need to find the values ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image025.png, without performing long division.

We know that, NCERT solution for class 9 Maths Chapter-1 Number Systems/image025.pngcan be rewritten as

NCERT solution for class 9 Maths Chapter-1 Number Systems/image026.png.

On substituting value of NCERT solution for class 9 Maths Chapter-1 Number Systems/image023.pngas NCERT solution for class 9 Maths Chapter-1 Number Systems/image027.png, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image034.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image029.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image030.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image031.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image032.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image033.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image034.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image035.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image036.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image037.png

Therefore, we conclude that, we can predict the values ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image025.png, without performing long division, to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image038.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image039.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image040.png

Question 3. Express the following in the form , where p and q are integers and q0.

(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image043.png

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image044.png

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image045.png

Solution :
(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image046.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image047.png

We need to multiply both sides by 10 to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image048.png

We need to subtract (a)from (b), to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image049.png

We can also writeNCERT solution for class 9 Maths Chapter-1 Number Systems/image050.pngas NCERT solution for class 9 Maths Chapter-1 Number Systems/image051.pngorNCERT solution for class 9 Maths Chapter-1 Number Systems/image052.png.

Therefore, on converting NCERT solution for class 9 Maths Chapter-1 Number Systems/image043.png in theNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get the answer a NCERT solution for class 9 Maths Chapter-1 Number Systems/image053.png.

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image054.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image055.png

We need to multiply both sides by 10 to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image056.png

We need to subtract (a)from (b), to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image057.png

We can also write NCERT solution for class 9 Maths Chapter-1 Number Systems/image058.pngasNCERT solution for class 9 Maths Chapter-1 Number Systems/image059.png orNCERT solution for class 9 Maths Chapter-1 Number Systems/image060.png.

Therefore, on convertingNCERT solution for class 9 Maths Chapter-1 Number Systems/image044.png in theNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get the answer as NCERT solution for class 9 Maths Chapter-1 Number Systems/image061.png.

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image062.pngNCERT solution for class 9 Maths Chapter-1 Number Systems/image063.png

We need to multiply both sides by 1000 to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image064.png

We need to subtract (a)from (b), to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image065.png

We can also write NCERT solution for class 9 Maths Chapter-1 Number Systems/image066.png as NCERT solution for class 9 Maths Chapter-1 Number Systems/image067.png

Therefore, on convertingNCERT solution for class 9 Maths Chapter-1 Number Systems/image045.png in theNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get the answer asNCERT solution for class 9 Maths Chapter-1 Number Systems/image068.png4

Question 4. Express  in the form. Are you surprised by your answer? Discuss why the answer makes sense with your teacher and classmates.
Solution :
NCERT solution for class 9 Maths Chapter-1 Number Systems/image070.png

We need to multiply both sides by 10 to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image071.png

We need to subtract (a)from (b), to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image072.png

We can also writeNCERT solution for class 9 Maths Chapter-1 Number Systems/image073.pngasNCERT solution for class 9 Maths Chapter-1 Number Systems/image074.png.

Therefore, on convertingNCERT solution for class 9 Maths Chapter-1 Number Systems/image075.png in theNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get the answer asNCERT solution for class 9 Maths Chapter-1 Number Systems/image076.png.

Yes, at a glance we are surprised at our answer.

But the answer makes sense when we observe that 0.9999……… goes on forever. SO there is not gap between 1 and 0.9999……. and hence they are equal.

Question 5. What can the maximum number of digits be in the recurring block of digits in the decimal expansion of? Perform the division to check your answer.
Solution :
We need to find the number of digits in the recurring block ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image077.png.

Let us perform the long division to get the recurring block ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image077.png.

We need to divide 1 by 17, to get

We can observe that while dividing 1 by 17 we got the remainder as 1, which will continue to be 1 after carrying out 16 continuous divisions.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image078.png, which is a non-terminating decimal and recurring decimal.

Question 6. Look at several examples of rational numbers in the form (q ≠ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?
Solution :
Let us consider the examples of the formNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.png that are terminating decimals.

NCERT solution for class 9 Maths Chapter-1 Number Systems/image079.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image080.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image081.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image082.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image083.png

We can observe that the denominators of the above rational numbers have powers of 2, 5 or both.

Therefore, we can conclude that the property, which q must satisfy inNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.png, so that the rational numberNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngis a terminating decimal is that q must have powers of 2, 5 or both.

Question 7. Write three numbers whose decimal expansions are non-terminating non-recurring.
Solution : The three numbers that have their expansions as non-terminating on recurring decimal are given below.

0.04004000400004  ….

0.07007000700007  ….

0.013001300013000013 ….

Question 8. Find three different irrational numbers between the rational numbersand.
Solution :
Let us convertNCERT solution for class 9 Maths Chapter-1 Number Systems/image086.pnginto decimal form, to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image087.pngand NCERT solution for class 9 Maths Chapter-1 Number Systems/image088.png

Three irrational numbers that lie betweenNCERT solution for class 9 Maths Chapter-1 Number Systems/image089.pngare:

0.73073007300073 ….

0.74074007400074 ….

0.76076007600076 ….

Question 9. Classify the following numbers as rational or irrational :

(i) 23

(ii) 225

(iii) 0.3796

(iv)7.478478 …

(v)1.101001000100001 …

Solution :
(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image090.png

We know that on finding the square root of 23, we will not get an integer.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image090.pngis an irrational number.

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image091.png

We know that on finding the square root of 225, we get 15, which is an integer.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image091.pngis a rational number.

(iii) 0.3796

We know that 0.3796 can be converted intoNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.png.

While, converting 0.3796 intoNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image092.png.

The rational numberNCERT solution for class 9 Maths Chapter-1 Number Systems/image093.pngcan be converted into lowest fractions, to getNCERT solution for class 9 Maths Chapter-1 Number Systems/image094.png.

We can observe that 0.3796 can be converted into a rational number.

Therefore, we conclude that 0.3796 is a rational number.

(iv) 7.478478….

We know that 7.478478…. is a non-terminating recurring decimal, which can be converted intoNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform.

While, converting 7.478478…. intoNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform, we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image095.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image096.png

While, subtracting (a) from (b), we get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image097.png

We know thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image098.pngcan also be written asNCERT solution for class 9 Maths Chapter-1 Number Systems/image099.png.

Therefore, we conclude that 7.478478…. is a rational number.

(v)1.101001000100001  ….

We can observe that the number 1.101001000100001…. is a non-terminating on recurring decimal.

We know that non-terminating and non-recurring decimals cannot be converted intoNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.pngform.

Therefore, we conclude that 1.101001000100001…. is an irrational number.

NCERT Solutions for Class 9 Maths Exercise 1.4

Question 1. Visualise 3.765 on the number line, using successive magnification.
Solution :
3.765 lies between 3 and 4.
NCERT solution for class 9 Maths Chapter-1 Number Systems/ A1

Question 2. Visualise 4. on the number line, upto 4 decimal places.
Solution :
4.NCERT solution for class 9 Maths Chapter-1 Number Systems or 4.2626 lies between 4 and 5.
NCERT solution for class 9 Maths Chapter-1 Number Systems/ A2

NCERT Solutions for Class 9 Maths Exercise 1.5

Question 1. Classify the following numbers as rational or irrational:
(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png

(iv) NCERT solution for class 9 Maths Chapter-1 Number Systems/image004.png

(v) NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

Solution :

(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image001.png

We know thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image006.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image007.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image008.png

which is also an irrational number.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image001.pngis an irrational number.

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png

NCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image009.png

= 3

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.pngis a rational number.

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png

We can cancelNCERT solution for class 9 Maths Chapter-1 Number Systems/image010.pngin the numerator and denominator, asNCERT solution for class 9 Maths Chapter-1 Number Systems/image010.pngis the common number in numerator as well as denominator, to get

NCERT solution for class 9 Maths Chapter-1 Number Systems/image011.png

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image003.pngis a rational number.

(iv) NCERT solution for class 9 Maths Chapter-1 Number Systems/image004.png

We know thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image012.png.

We can conclude that, when 1 is divided byNCERT solution for class 9 Maths Chapter-1 Number Systems/image013.png, we will get an irrational number.

Therefore, we conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image004.pngis an irrational number.

(v) NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image014.png

We can conclude that NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.pngwill also be an irrational number.

Therefore, we conclude that NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.pngis an irrational number.

Question 2. Simplify each of the following expressions
NCERT solution for class 9 Maths Chapter-1 Number Systems/ Q2
Solution :
(i) (3 + √3)(2 + √2)
= 2(3 + √3) + √2(3 + √3)
= 6 + 2√3 + 3√2 + √6
Thus, (3 + √3)(2 + √2) = 6 + 2√3 + 3√2 + √6
(ii) (3 + √3)(3 – √3) = (3)2 – (√3)2
= 9 – 3 = 6
Thus, (3 + √3)(3 – √3) = 6
(iii) (√5 + √2)2 = (√5)2 + (√2)2 + 2(√5)(√2)
= 5 + 2 + 2√10 = 7 + 2√10
Thus, (√5 + √2 )2 = 7 + 2√10
(iv) (√5 – √2)(√5 + √2) = (√5)2 – (√2)2 = 5 – 2 = 3
Thus, (√5 – √2) (√5 + √2) = 3

Question 3. Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is π = . This seems to contradict the fact that n is irrational. How will you resolve this contradiction?
Solution : When we measure the length of a line with a scale or with any other device, we only get an approximate ational value, i.e. c and d both are irrational.
∴ NCERT solution for class 9 Maths Chapter-1 Number Systems is irrational and hence π is irrational.
Thus, there is no contradiction in saying that it is irrational.

Question 4. Represent on the number line.
Solution :
Draw a line segment AB = 9.3 units and extend it to C such that BC = 1 unit.
Find mid point of AC and mark it as O.
Draw a semicircle taking O as centre and AO as radius. Draw BD ⊥ AC.
Draw an arc taking B as centre and BD as radius meeting AC produced at E such that BE = BD = NCERT solution for class 9 Maths Chapter-1 Number Systemsunits.
NCERT Solutions for Class 9 Maths Chapter 1 Number Systems

Question 5. Rationalise the denominator of the following
NCERT Solutions for Class 9 Maths Chapter 1 Number Systems
Solution :
Chapter-1 Number Systems/ A5

NCERT Solutions for Class 9 Maths Exercise 1.6

Question 1. Find :
(i)NCERT solution for class 9 Maths Chapter-1 Number Systems/image004.png

(ii)NCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png

(iii)NCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png

Solution :
(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image004.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

We conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image004.pngcan also be written as

NCERT solution for class 9 Maths Chapter-1 Number Systems/image006.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image007.png

Therefore, the value ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image004.pngwill be 8.

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image002.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

We conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.pngcan also be written as

NCERT solution for class 9 Maths Chapter-1 Number Systems/image008.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image009.png

Therefore, the value ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image002.pngwill be 2.

(iii)NCERT solution for class 9 Maths Chapter-1 Number Systems/image003.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

We conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image003.pngcan also be written as

NCERT solution for class 9 Maths Chapter-1 Number Systems/image010.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image011.png

Therefore, the value ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image003.pngwill be 5.

Question 2. Find:
(i)NCERT solution for class 9 Maths Chapter-1 Number Systems/image012.png

(ii)NCERT solution for class 9 Maths Chapter-1 Number Systems/image013.png

(iii)NCERT solution for class 9 Maths Chapter-1 Number Systems/image014.png

(iv)NCERT solution for class 9 Maths Chapter-1 Number Systems/image015.png

Solution :
(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image012.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

We conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image012.pngcan also be written as

NCERT solution for class 9 Maths Chapter-1 Number Systems/image016.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image017.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image018.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image019.png

=27

Therefore, the value ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image012.pngwill be 27.

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image013.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

We conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image013.pngcan also be written as

NCERT solution for class 9 Maths Chapter-1 Number Systems/image020.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image021.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image022.png

= 4

Therefore, the value ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image013.pngwill be 4.

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image014.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

We conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image014.pngcan also be written as

NCERT solution for class 9 Maths Chapter-1 Number Systems/image023.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image024.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image025.png

= 8

Therefore, the value ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image014.pngwill be 8.

(iv) NCERT solution for class 9 Maths Chapter-1 Number Systems/image026.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image027.png

We conclude thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image026.pngcan also be written asNCERT solution for class 9 Maths Chapter-1 Number Systems/image028.png, orNCERT solution for class 9 Maths Chapter-1 Number Systems/image029.png.

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image005.png

We know thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image029.pngcan also be written as

NCERT solution for class 9 Maths Chapter-1 Number Systems/image030.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image031.png

Therefore, the value ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image026.pngwill beNCERT solution for class 9 Maths Chapter-1 Number Systems/image032.png.

Question 3. Simplify :
(i)NCERT solution for class 9 Maths Chapter-1 Number Systems/image033.png

(ii)NCERT solution for class 9 Maths Chapter-1 Number Systems/image034.png

(iii)NCERT solution for class 9 Maths Chapter-1 Number Systems/image035.png

(iv)NCERT solution for class 9 Maths Chapter-1 Number Systems/image036.png

Solution :
(i) NCERT solution for class 9 Maths Chapter-1 Number Systems/image033.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image037.png.

We can conclude that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image038.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image039.png

Therefore, the value ofNCERT solution for class 9 Maths Chapter-1 Number Systems/image033.pngwill beNCERT solution for class 9 Maths Chapter-1 Number Systems/image040.png.

(ii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image034.png

We know thatNCERT solution for class 9 Maths Chapter-1 Number Systems/image041.png

We conclude that NCERT solution for class 9 Maths Chapter-1 Number Systems/image034.pngcan also be written asNCERT solution for class 9 Maths Chapter-1 Number Systems/image042.png.

(iii) NCERT solution for class 9 Maths Chapter-1 Number Systems/image035.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image043.png

We conclude that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image044.png.

NCERT solution for class 9 Maths Chapter-1 Number Systems/image045.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image035.png

Therefore, the value of NCERT solution for class 9 Maths Chapter-1 Number Systems/image035.pngwill beNCERT solution for class 9 Maths Chapter-1 Number Systems/image047.png.

(iv) NCERT solution for class 9 Maths Chapter-1 Number Systems/image036.png

We know that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image048.png.

We can conclude that

NCERT solution for class 9 Maths Chapter-1 Number Systems/image049.png
NCERT solution for class 9 Maths Chapter-1 Number Systems/image050.png

Therefore, the value of NCERT solution for class 9 Maths Chapter-1 Number Systems/image036.pngwill beNCERT solution for class 9 Maths Chapter-1 Number Systems/image051.png.

Conclusions for NCERT SOLUTIONS FOR CLASS 9 MATHS CHAPTER 1 NUMBER SYSTEMS

SWC academic staff has developed NCERT answers for this chapter of the ninth grade mathematics curriculum. We have solutions prepared for all of the exercises of this chapter. The answers, broken down into steps, to all of the questions included in the NCERT textbook’s chapter are provided here. Read this chapter on theory. Be certain that you have read the theory section of this chapter of the NCERT textbook and that you have learnt the formulas for the chapter that you are studying. 

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