Number System Class 9 Worksheet have been designed as per the latest pattern for CBSE, NCERT and KVS for Grade 9. Students are always suggested to solve printable worksheets for Mathematics Number System Grade 9 as they can be really helpful to clear their concepts and improve problem solving skills. We at worksheetsbag.com have provided here free PDF worksheets for students in standard 9 so that you can easily take print of these test sheets and use them daily for practice. All worksheets are easy to download and have been designed by teachers of Class 9 for benefit of students and is available for free download.

## Mathematics Number System Worksheets for Class 9

We have provided **chapter-wise worksheets for class 9 Mathematics Number System** which the students can download in Pdf format for free. This is the best collection of Mathematics Number System standard 9th worksheets with important questions and answers for each grade 9th Mathematics Number System chapter so that the students are able to properly practice and gain more marks in Class 9 Mathematics Number System class tests and exams.

### Chapter-wise Class 9 Mathematics Number System Worksheets Pdf Download

1. Rational Numbers

2. Irrational Numbers

3. Real Numbers and their Decimal Expansions

4. Operations on Real Numbers

5. Laws of Exponents for Real Numbers

**•** Natural numbers are – 1, 2, 3, ……………. denoted by N.**•** Whole numbers are – 0, 1, 2, 3, ……………… denoted by W.**•** Integers – ……. -3, -2, -1, 0, 1, 2, 3, ……………… denoted by Z.**•** Rational numbers – All the numbers which can be written in the form r / s p / q, are called rational numbers where p and q are integers.**•** Irrational numbers – A number s is called irrational, if it cannot be written in the form p / q where p and q are integers and**•** The decimal expansion of a rational number is either terminating or non-terminating recurring. Thus we say that a number whose decimal expansion is either terminating or nonterminating recurring is a rational number.**• **The decimal expansion of a irrational number is non terminating non-recurring.**•** All the rational numbers and irrational numbers taken together.**•** Make a collection of real number.**•** A real no is either rational or irrational.**•** If r is rational and s is irrational then r+s, r–s, r.s are always irrational numbers but r/s may be rational or irrational.**• **Every irrational number can be represented on a number line using Pythagoras theorem.**•** Rationalization means to remove square root from the denominator.

rationalization factor a ∓ √b

**Question.****Every rational number is**

(A) a natural number

(B) an integer

(C) a real number

(D) a whole number

**Answer**

C

**Question.****The value of 1.999…. in the form, p/q where p and q are integers and q ≠ 0, is**

(a) 19/10

(b) 1999/1000

(c) 2

(d) 1/9

**Answer**

C

** Question. 2√3 + √3 is equal to**(a) 2√6

(b) 6

(c) 3√3

(d) 4√6

**Answer**

C

**Question.****√10 × √15 is equal to**

(a) 6√5

(b) 5√6

(c) √25

(d) 10√5

**Answer**

B

**Question.** The number obtained on rationalizing the denominator of 1/√7 − 2 is

(a) √7 + 2 / 3

(b) √7 – 2 / 3

(c) √7 + 2 / 5

(d) √7 + 2 / 45

**Answer**

D

**Question.** 1 / √9 − √8 is equal to

(a) 1 / 2 (3 – 2√2)

(b) 1 / 3 + 2√2

(c) 3− 2√2

(d) 3+ 2√2

**Answer**

D

**Question.** After rationalizing the denominator of 7 / 3√3 − 2√2 we get the denominator as

(a) 13

(b) 19

(c) 5

(d) 35

**Answer**

B

**Question.** Between two rational numbers

(A) there is no rational number

(B) there is exactly one rational number

(C) there are infinitely many rational numbers

(D) there are only rational numbers and no irrational numbers

**Answer**

C

**Question.** Decimal representation of a rational number cannot be

(A) terminating

(B) non-terminating

(C) non-terminating repeating

(D) non-terminating non-repeating

**Answer**

D

**Question.****The product of any two irrational numbers is**

(A) always an irrational number

(B) always a rational number

(C) always an integer

(D) sometimes rational, sometimes irrational

**Answer**

D

**Question.** The decimal expansion of the number 2 is

(A) a finite decimal

(B) 1.41421

(C) non-terminating recurring

(D) non-terminating non-recurring

**Answer**

B

**Question**. Which of the following is irrational?

**Answer**

A

**Question.****Which of the following is irrational?**

(A) 0.14

(B) 0.1416

(C) 0.1416

(D) 0.4014001400014…

**Answer**

D

**Question.****A rational number between √2 and √3 is**

(a) √2 + √3 / 2

(b) √2 √3 / 2

(c) 1.5

(d) 1.8

**Answer**

D

**Question.****The value of √32 + √48 / √8 + √12 is equal to**

(a) √2

(b) 2

(c) 4

(d) 8

**Answer**

B

**Question.** If √2 = 1.4142, then

(a) 2.4142

(b) 5.8282

(c) 0.4142

(d) 0.1718

**Answer**

C

**Question.** 4√3√2√2 equal

(a) 2^{1/6}

(b) 2^{-6}

(c) 2^{1/6}

(d) 2^{6}

**Answer**

C

**Question.** The product 3√2 4√2 .12√32 equals

(a) √2

(b) 2

(c) 12 √2

(d) 12 √32

**Answer**

B

**Question.** Value of 4 √(81)^{-2} is

(a) 1/9

(b) 1/3

(c) 9

(d) 1/81

**Answer**

A

**Question.** Value of (256)** ^{0.16}** × (256)

**is**

^{0.09}(a) 4

(b) 16

(c) 64

(d) 256.25

**Answer**

A

**Question.** Which of the following is equal to x?

(a) x^{12/7} – x^{5/7}

(b) 12√(x^{4})^{1/3}

(c) (√x^{3})^{2/3}

(d) x^{12/7} X x^{7/12}

**Answer**

A

**Question. Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.****Answer.** Yes, x+ y is necessary an irrational number.

Let x = 5 and y = √2.

Then, x + y = 5 + √2 = 5 + 1.4142…..= 6.4142….. which is non – terminating and non-repeating.

Hence, x + y is an irrational number.

**Question. Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example.****Answer.** Let x = 0 (a rational number) and y = √3 be an irrational number. Then,

xy = 0(√3) = 0, which is not an irrational number.

Hence, xy is not necessarily an irrational number.

**Question. State whether the following statements are true or false? Justify your answer.****(i) √2/3 is a rational number.****(ii) There are infinitely many integers between any two integers.****(iii) Number of rational numbers between 15 and 18 is finite.****(iv) There are numbers which cannot be written in the form, (p/q) q ≠ 0, p, q both are integers.****(v) The square of an irrational number is always rational.****(vi) √12/√3 is not a rational number as √12 and √3 are not integers.****(vii) √15/√3 is written in the form, p/q where q ≠ 0 so it is a rational number.****Answer.** (i) The given statement is false. √2/3 is of the form p/q but p = √2 is not an integer.

(ii) The given statement is false. Consider two integers 3 and 4. There is no integers between 3 and 4.

(iii) The given statement is false. There lies infinitely many rational numbers between any two rational number. Hence, number of rational numbers between 15 and 18 are infinite.

(iv) The given statement is true. For example, √3/√5 is of the form p/q but p = √3 and q = √5 are not integers.

(v) The given statement is false. Consider an irrational number 4√2. Then, its square (4√2)2 = √2, which is not a rational number.

(vi) The given statement is false.

(vii) The given statement is false.

where p = √5 is irrational number.

**Question.****Classify the following numbers as rational or irrational with justification:(i) √196(ii) 3√18(iii) √9/27(iv) √28/√343(v) −√0.4(vi) √12/√75(vii) 0.5918(viii) (1+ √5) − (4 + √5)(ix) 10.124124…(x) 1.010010001….**

**Answer.**(i) √196 = 14, which is a rational number.

(ii) 3√18 = 3√9 × 2 = 3 × 3√2,= 9√2, which is the product of a rational and an irrational number.

Hence, 3√18 is an irrational number.

(iii) √9/27 = 1/√3 which of the quotient of a rational and an irrational number and therefore an irrational number.

(iv) √28/√343 = √4/49 = 2/7 which is a rational number.

(v) -√0.4 = – 2/√10 which is a quotient of a rational and an irrational number and so it is an irrational number.

(vi) √12/√75 = √4/25 = 2/5 which is a rational number.

(vii) 0.5918 is a terminating decimal expansion. Hence, it is rational number.

(viii) (1 + √5) − (4 + √5) = −3, which is a rational number.

(ix) 10.124124… is a decimal expansion which non-terminating recurring. Hence, it is a rational number.

(x) 1.010010001… is a decimal expansion which is non-terminating non-recurring. Hence, it is an irrational number.

**Question.****Find which of the variables x, y, z and u represent rational numbers and which irrational numbers:(i) x ^{2 }= 5(ii) 2 y = 9(iii) 2 z = 0.4(iv) u^{2} = 17/4**

**Answer.**(i) x

^{2}= 5⇒ x = √5, which is an irrational number.

(ii) y

^{2}= 9 ⇒ y = √9 = 3, which is a rational number.

(iii) z

^{2}= .04⇒ z = √04 = 0.2, which is a terminating decimal. Hence, it is rational number.

(iv) u

^{2 }= 17/4 ⇒ u = √17/4 = √17/2 which is of the form, p/q where p = √17 is not an integer. Hence, u is an irrational number.

**Question.****Find three rational numbers between****(i) –1 and –2****(ii) 0.1 and 0.11****(iii) 5/6 and 6/7****(iv) 1/4 and 1/5****Answer. **(i) -1.1, -1.2, -1.3 (terminating decimals) are three rational numbers lying between – 1 and – 2.

(ii) 0.101, 0.102, 0.103 (terminating decimals) are three rational numbers which lie between 0.1 and 0.11.

(iii) 5/7 = 5/7 x 10/10 = 50/70 and 6/7 = 6/7 x 10/10 = 60/70

⇒ 51/70, 52/70, 53/70

⇒ are three rational numbers lying and between 50/70 and 60/70 and therefore lie between 5/7 and 6/7

(iv) 1/4 = 1/4 x 20/20 = 20/80 and 1/5 = 1/5 x 16/16 = 16/80

Now, √2 x √3(18/80) (= 9/40) 19/80 are three rational numbers lying between 1/4 and 1/5

**Question.** Insert a rational number and an irrational number between the following:**(i) 2 and 3****(ii) 0 and 0.1(iii) 1/3 and 1/2**

**(iv) -2/5 and 1/2**

(v) 0.15 and 0.16

(vi) √2 and √3

(vii) 2.357 and 3.121

(viii) 0.0001 and 0.001

(ix) 3.623623 and 0.484848

(x) 6.375289 and 6.375738

(v) 0.15 and 0.16

(vi) √2 and √3

(vii) 2.357 and 3.121

(viii) 0.0001 and 0.001

(ix) 3.623623 and 0.484848

(x) 6.375289 and 6.375738

**Answer.**(i) A rational number between 2 and 3 is 2 + 3/2 = 5/2 = 2.5. Also, 2.1 (terminating decimal) is a rational between 2 and 3. Again, 2.010010001… (a non-terminating and non-recurring decimal) is an irrational number between 2 and 3.

(ii) 0.04 is a terminating decimal and also it is lies between 0 and 0.1. Hence, 0.04 is a rational number which lies between 0 and 0.1. Again 0.003000300003… is a non-terminating and non-recurring decimal which lies between 0 and 0.1. Hence, 0.003000300003… is an irrational number between 0 and 0.1.

Now, 5/12 is a rational number between 4/12 and 6/12 So, 5/12 is a rational number lying

between 1/3 and 1/2

Again, 1/3 = 0.33333…and 1/2 = 0.5.

Now, 0.414114111… is a non-terminating and non-recurring decimal.

Hence, 0.414114111… is an irrational number lying between 1/3 and 1/2

(iv) -2/5 = -0.4 and 1/2 and 1/2 = 0.5

Now, 0 is a rational number between -0.4 and 0.5 i.e., 0 is a rational number lying between -2/5 and 1/2

Again, 0.131131113…. is a non – terminating and non – recurring decimal which lies between – 0.4 and 0.5.

Hence, 0.131131113… is an irrational number lying between -2/5 and 1/2

(v) 0.151 is a rational number between 0.15 and 0.16. Similarly, 0.153, 0.157, etc. are rational number lying between 0.15 and 0.16.

Again, 0.151151115… (a non-terminating and non-recurring decimal) is an irrational number between 0.15 and 0.16.

(vi) √2 =1.4142135…. and √3 =1.732050807…

Now, 1.5 (a terminating decimal) which lies between 1.4142135… and 1.732050807….

Hence, 1.5 is a rational number between √2 and √3.

Again, 1.575575557… (a non – terminating and non – recurring decimal) is an irrational number lying between √2 and √3.

(vii) 3 is a rational number between 2.357 and 3.121.

Again, 3.101101110… (a non-terminating and non-recurring decimal) is an irrational number lying between 2.357 and 3.121.

(viii) 0.00011 is a rational number 0.0001 and 0.001.

Again, 0.0001131331333…. (a non-terminating and non-recurring decimal) is an irrational number between 0.0001 and 0.001.

(ix) 1 is a rational number between 0.484848 and 3.623623.

Again, 1.909009000… (a non-terminating and non-recurring decimal) is an irrational number lying between 0.484848 and 3.623623.

(x) 6.3753 (a terminating decimal) is a rational number between 6.375289 and 6.375738.

Again, 6.375414114111… (a non-terminating and non-recurring decimal) is an irrational lying between 6.375289 and 6.375738.

**Question.** Represent the following numbers on the number line:**7, 7, 2, -3/2, -12/5****Answer.**

**Question.****Locate √5, √10 and √17 on the number line:****Answer.** Presentation of 5 on number line:

We write √5 as the sum of the square of two natural numbers:5 =1+ 4 =12 + 22

On the number line, take OA = 2 units.

Draw BA = 1 unit, perpendicular to OA. Join OB.

By Pythagoras theorem, OB= √5

Using a compass with centre O and radius OB, draw an arc which intersects the number line at the point C. Then, C corresponds to √5

Presentation of √10 on the number line:

We write 10 as the sun of the square of two natural numbers: 10 =1+ 9 =12 + 32

On the number line, taken OA = 3 units.

Draw BA = 1 unit, perpendicular to OA, Join OB.

By Pythagoras theorem, OB = √10

Using a compass with centre O and radius OB, draw an arc which intersects the number line at the point C. Then, C corresponds to √10.

Presentation of √17 on the number line:

We write 17 as the sum of the square of two natural numbers: 17 =1+16 =12 + 42

On the number line, take OA = 4 units.

Draw BA = 1 units, perpendicular to OA. Join OB.

By Pythagoras theorem, OB = √17

Using a compass with centre O and radius OB, draw an arc which intersects the number line at the point C. Then, C corresponds to √17.

**Question.** Represent geometrically the following numbers on the number line:**(A) √4.5****(B) √5.6****(C) √8.1****(D) √2.3****Answer. **(i) √4.5

Presentation of √4.5 on number line:

Mark the distance 4.5 units from a fixed point A on a given line to obtain a point B such that AB = 4.5 units. From B, mark a distance of 1 units and mark the new points as C. Find the mid-point of AC and mark that points as O. Draw a semicircle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then, BD = √4.5.

Now, draw an arc with centre B and radius BD, which intersects the number line in E.

Thus, E represent √4.5.

(ii) √5.6

Presentation of √5.6 on number line:

Mark the distance 5.6 units from a fixed points A on a given line to obtain a point B such that AB = 5.6 units. From B, mark a distance of 1 unit and mark the new points as C. Find the mid-point of AC and mark the points as O. Draw a semicircle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then BD = √5.6.

Now, draw an arc with centre B and radius BD, which intersects the number line in E.

Thus, E represent √5.6.

(iii) √8.1

Presentation of √8.1 on number line:

Mark the distance 8.1 units from a fixed point A on a given line to obtain a point B such that AB = 8.1 units. From B, mark a distance of 1 unit and mark the new points as C. Find the mid-point of AC and mark that point as O. Draw a semicircle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then, BD = √8.1.

Now, draw an arc with centre B and radius BD, which intersects the number line in E.

Thus, E represents √8.1

(iv) √2.3

Presentation of √2.3 on number line:

Mark the distance 2.3 units from a fixed points A on a given line to obtain a point B such that AB = 2.3 units. From B mark, a distance of 1 unit and mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semicircle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then, BD = √2.3.

Now, draw an arc with centre B and radius BD, which intersects the number line in E.

Thus, E represents √2.3.

**Question**. Express the following in the form, p/q where p and q are integers and q ≠ 0.

(i) 0.2

(ii) 0.888…

(iii) 5.2

(iv) 0.2555…

(viii) 0.404040…**Answer.** (i) 0.2 .= 2/10 = 10 5

(ii) Let x = 0.888… = 0.8¯ …(1)

10x = 8.8¯ …(2)

Subtracting (1) and (2), we get

9x = 8

Hence, x = 8/9

(iii) let x = 5.2 = 5.2222… …(1)

Multiplying both sides by 10, we get

10 = 52.222… = 52.2¯ …(2)

If we subtract 5.2 from 52.2 , the repeating portion the decimal cancels out.

Subtracting (1) and (2), we get

10 x = 47 ⇒ 9x = 47 ⇒ 47/9

Hence, 5.2¯ = 47/9

(iv) Let x = 0.2555… = 0.25. …(1)

10x = 2.5… …(2)

And 100 x = 25.5 …(3)

Subtracting (2) from (3), we get

90x = 23

x = 23/90

(viii) Let x = 0.404040… = 0.40¯ . …(1)

100x = 40.40¯ …(2)

Subtracting (1) from (2), we get

99x = 40

x = 40/99

**Question.****Show that 0.142857142857… = 1/7****Answer.** Let x = 0.142857142857… …(1)

**Question.****Simplify:(i) √45 – 3 √20 + 4 √5**

**(ii) √24/8 + √54/9**

(iii) 4√12 X 6√7

(iv) 4√28 ÷ 3√7

(v) 3√3 + 2√27 + 7/√3

(vi) (√3 – √2)2

(viii) 3/√8 + 1/√2

(ix) 2√3/3 – √3/6

(iii) 4√12 X 6√7

(iv) 4√28 ÷ 3√7

(v) 3√3 + 2√27 + 7/√3

(vi) (√3 – √2)2

(viii) 3/√8 + 1/√2

(ix) 2√3/3 – √3/6

**Answer.**(i) √45 − 3√20 + 4√5 = √9×5 −3√4×5 + 4 √5

= 3√5 − 3× 2√5 + 4√5 = (3 − 6 + 4) √5 = √5

**Question.** Rationalize the denominator of the following:

**Answer.**

**Question.****Find the value of a and b in each of the following:**

**Answer.**

**Question.** If a = 2 + √3, then find the value of a -1/a.**Answer.** We have a = 2 +√3

**Question.****Simplify: (256)-(4 ^{3/2})**

**Answer.**

**Question.****Rationalize the denominator in each of the following and hence evaluate by taking √2 = 1.414, √3 = 1.732 and √5 = 2.236, upto three places of decimal.**

**Answer.**

**Question.** If √2 = 1.414, √3 = 1.732, then find the value of**4/3√3 – 2√2 + 3/3√3 + 2√2**

**Answer.**

**Question.****Express 0.6 + 0.7¯ + 0.47¯ in the form, p/q where p and q are integers and q ≠ 0.****Answer.** We have = 6/10 …(1)

Let x = 0.7¯ = 0.777… …(2)

Subtracting (1) from (2), we get

9x = 7 ⇒ x = 7/9 or 0.7¯ = 7/9

**Question. Simplify:**

**Answer.**

**Question.****If a = 3 + √5/2 then find the value of a ^{2} + 1/a^{2}** We have,

Answer.

**Question.****If x = √3 + √2 / √3 – √2 and y = 3 – √2 / √3 – √2 then find the value of x ^{2} + y^{2}.**

**Answer.**

**Question.****Find the value of**

**Answer.** We have,

## Mathematics Number System Worksheets for Class 9 as per CBSE NCERT pattern

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- CBSE Class 9 Mathematics Number System Workbook will surely help to improve knowledge of this subject

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